Article HILBERT1, MML version 4.99.1005
:: HILBERT1:attrnot 1 => HILBERT1:attr 1
definition
let a1 be set;
attr a1 is with_VERUM means
<*0*> in a1;
end;
:: HILBERT1:dfs 1
definiens
let a1 be set;
To prove
a1 is with_VERUM
it is sufficient to prove
thus <*0*> in a1;
:: HILBERT1:def 1
theorem
for b1 being set holds
b1 is with_VERUM
iff
<*0*> in b1;
:: HILBERT1:attrnot 2 => HILBERT1:attr 2
definition
let a1 be set;
attr a1 is with_implication means
for b1, b2 being Relation-like Function-like FinSequence-like set
st b1 in a1 & b2 in a1
holds (<*1*> ^ b1) ^ b2 in a1;
end;
:: HILBERT1:dfs 2
definiens
let a1 be set;
To prove
a1 is with_implication
it is sufficient to prove
thus for b1, b2 being Relation-like Function-like FinSequence-like set
st b1 in a1 & b2 in a1
holds (<*1*> ^ b1) ^ b2 in a1;
:: HILBERT1:def 2
theorem
for b1 being set holds
b1 is with_implication
iff
for b2, b3 being Relation-like Function-like FinSequence-like set
st b2 in b1 & b3 in b1
holds (<*1*> ^ b2) ^ b3 in b1;
:: HILBERT1:attrnot 3 => HILBERT1:attr 3
definition
let a1 be set;
attr a1 is with_conjunction means
for b1, b2 being Relation-like Function-like FinSequence-like set
st b1 in a1 & b2 in a1
holds (<*2*> ^ b1) ^ b2 in a1;
end;
:: HILBERT1:dfs 3
definiens
let a1 be set;
To prove
a1 is with_conjunction
it is sufficient to prove
thus for b1, b2 being Relation-like Function-like FinSequence-like set
st b1 in a1 & b2 in a1
holds (<*2*> ^ b1) ^ b2 in a1;
:: HILBERT1:def 3
theorem
for b1 being set holds
b1 is with_conjunction
iff
for b2, b3 being Relation-like Function-like FinSequence-like set
st b2 in b1 & b3 in b1
holds (<*2*> ^ b2) ^ b3 in b1;
:: HILBERT1:attrnot 4 => HILBERT1:attr 4
definition
let a1 be set;
attr a1 is with_propositional_variables means
for b1 being Element of NAT holds
<*3 + b1*> in a1;
end;
:: HILBERT1:dfs 4
definiens
let a1 be set;
To prove
a1 is with_propositional_variables
it is sufficient to prove
thus for b1 being Element of NAT holds
<*3 + b1*> in a1;
:: HILBERT1:def 4
theorem
for b1 being set holds
b1 is with_propositional_variables
iff
for b2 being Element of NAT holds
<*3 + b2*> in b1;
:: HILBERT1:attrnot 5 => HILBERT1:attr 5
definition
let a1 be set;
attr a1 is HP-closed means
a1 c= NAT * & a1 is with_VERUM & a1 is with_implication & a1 is with_conjunction & a1 is with_propositional_variables;
end;
:: HILBERT1:dfs 5
definiens
let a1 be set;
To prove
a1 is HP-closed
it is sufficient to prove
thus a1 c= NAT * & a1 is with_VERUM & a1 is with_implication & a1 is with_conjunction & a1 is with_propositional_variables;
:: HILBERT1:def 5
theorem
for b1 being set holds
b1 is HP-closed
iff
b1 c= NAT * & b1 is with_VERUM & b1 is with_implication & b1 is with_conjunction & b1 is with_propositional_variables;
:: HILBERT1:condreg 1
registration
cluster HP-closed -> non empty with_VERUM with_implication with_conjunction with_propositional_variables (set);
end;
:: HILBERT1:condreg 2
registration
cluster with_VERUM with_implication with_conjunction with_propositional_variables -> HP-closed (Element of bool (NAT *));
end;
:: HILBERT1:funcnot 1 => HILBERT1:func 1
definition
func HP-WFF -> set means
it is HP-closed &
(for b1 being set
st b1 is HP-closed
holds it c= b1);
end;
:: HILBERT1:def 6
theorem
for b1 being set holds
b1 = HP-WFF
iff
b1 is HP-closed &
(for b2 being set
st b2 is HP-closed
holds b1 c= b2);
:: HILBERT1:funcreg 1
registration
cluster HP-WFF -> HP-closed;
end;
:: HILBERT1:exreg 1
registration
cluster non empty HP-closed set;
end;
:: HILBERT1:funcreg 2
registration
cluster HP-WFF -> functional;
end;
:: HILBERT1:condreg 3
registration
cluster -> FinSequence-like (Element of HP-WFF);
end;
:: HILBERT1:modenot 1
definition
mode HP-formula is Element of HP-WFF;
end;
:: HILBERT1:funcnot 2 => HILBERT1:func 2
definition
func VERUM -> Element of HP-WFF equals
<*0*>;
end;
:: HILBERT1:def 7
theorem
VERUM = <*0*>;
:: HILBERT1:funcnot 3 => HILBERT1:func 3
definition
let a1, a2 be Element of HP-WFF;
func A1 => A2 -> Element of HP-WFF equals
(<*1*> ^ a1) ^ a2;
end;
:: HILBERT1:def 8
theorem
for b1, b2 being Element of HP-WFF holds
b1 => b2 = (<*1*> ^ b1) ^ b2;
:: HILBERT1:funcnot 4 => HILBERT1:func 4
definition
let a1, a2 be Element of HP-WFF;
func A1 '&' A2 -> Element of HP-WFF equals
(<*2*> ^ a1) ^ a2;
end;
:: HILBERT1:def 9
theorem
for b1, b2 being Element of HP-WFF holds
b1 '&' b2 = (<*2*> ^ b1) ^ b2;
:: HILBERT1:attrnot 6 => HILBERT1:attr 6
definition
let a1 be Element of bool HP-WFF;
attr a1 is Hilbert_theory means
VERUM in a1 &
(for b1, b2, b3 being Element of HP-WFF holds
b1 => (b2 => b1) in a1 &
(b1 => (b2 => b3)) => ((b1 => b2) => (b1 => b3)) in a1 &
(b1 '&' b2) => b1 in a1 &
(b1 '&' b2) => b2 in a1 &
b1 => (b2 => (b1 '&' b2)) in a1 &
(b1 in a1 & b1 => b2 in a1 implies b2 in a1));
end;
:: HILBERT1:dfs 10
definiens
let a1 be Element of bool HP-WFF;
To prove
a1 is Hilbert_theory
it is sufficient to prove
thus VERUM in a1 &
(for b1, b2, b3 being Element of HP-WFF holds
b1 => (b2 => b1) in a1 &
(b1 => (b2 => b3)) => ((b1 => b2) => (b1 => b3)) in a1 &
(b1 '&' b2) => b1 in a1 &
(b1 '&' b2) => b2 in a1 &
b1 => (b2 => (b1 '&' b2)) in a1 &
(b1 in a1 & b1 => b2 in a1 implies b2 in a1));
:: HILBERT1:def 10
theorem
for b1 being Element of bool HP-WFF holds
b1 is Hilbert_theory
iff
VERUM in b1 &
(for b2, b3, b4 being Element of HP-WFF holds
b2 => (b3 => b2) in b1 &
(b2 => (b3 => b4)) => ((b2 => b3) => (b2 => b4)) in b1 &
(b2 '&' b3) => b2 in b1 &
(b2 '&' b3) => b3 in b1 &
b2 => (b3 => (b2 '&' b3)) in b1 &
(b2 in b1 & b2 => b3 in b1 implies b3 in b1));
:: HILBERT1:funcnot 5 => HILBERT1:func 5
definition
let a1 be Element of bool HP-WFF;
func CnPos A1 -> Element of bool HP-WFF means
for b1 being Element of HP-WFF holds
b1 in it
iff
for b2 being Element of bool HP-WFF
st b2 is Hilbert_theory & a1 c= b2
holds b1 in b2;
end;
:: HILBERT1:def 11
theorem
for b1, b2 being Element of bool HP-WFF holds
b2 = CnPos b1
iff
for b3 being Element of HP-WFF holds
b3 in b2
iff
for b4 being Element of bool HP-WFF
st b4 is Hilbert_theory & b1 c= b4
holds b3 in b4;
:: HILBERT1:funcnot 6 => HILBERT1:func 6
definition
func HP_TAUT -> Element of bool HP-WFF equals
CnPos {} HP-WFF;
end;
:: HILBERT1:def 12
theorem
HP_TAUT = CnPos {} HP-WFF;
:: HILBERT1:th 1
theorem
for b1 being Element of bool HP-WFF holds
VERUM in CnPos b1;
:: HILBERT1:th 2
theorem
for b1 being Element of bool HP-WFF
for b2, b3 being Element of HP-WFF holds
b2 => (b3 => (b2 '&' b3)) in CnPos b1;
:: HILBERT1:th 3
theorem
for b1 being Element of bool HP-WFF
for b2, b3, b4 being Element of HP-WFF holds
(b2 => (b3 => b4)) => ((b2 => b3) => (b2 => b4)) in CnPos b1;
:: HILBERT1:th 4
theorem
for b1 being Element of bool HP-WFF
for b2, b3 being Element of HP-WFF holds
b2 => (b3 => b2) in CnPos b1;
:: HILBERT1:th 5
theorem
for b1 being Element of bool HP-WFF
for b2, b3 being Element of HP-WFF holds
(b2 '&' b3) => b2 in CnPos b1;
:: HILBERT1:th 6
theorem
for b1 being Element of bool HP-WFF
for b2, b3 being Element of HP-WFF holds
(b2 '&' b3) => b3 in CnPos b1;
:: HILBERT1:th 7
theorem
for b1 being Element of bool HP-WFF
for b2, b3 being Element of HP-WFF
st b2 in CnPos b1 & b2 => b3 in CnPos b1
holds b3 in CnPos b1;
:: HILBERT1:th 8
theorem
for b1, b2 being Element of bool HP-WFF
st b1 is Hilbert_theory & b2 c= b1
holds CnPos b2 c= b1;
:: HILBERT1:th 9
theorem
for b1 being Element of bool HP-WFF holds
b1 c= CnPos b1;
:: HILBERT1:th 10
theorem
for b1, b2 being Element of bool HP-WFF
st b1 c= b2
holds CnPos b1 c= CnPos b2;
:: HILBERT1:th 11
theorem
for b1 being Element of bool HP-WFF holds
CnPos CnPos b1 = CnPos b1;
:: HILBERT1:funcreg 3
registration
let a1 be Element of bool HP-WFF;
cluster CnPos a1 -> Hilbert_theory;
end;
:: HILBERT1:th 12
theorem
for b1 being Element of bool HP-WFF holds
b1 is Hilbert_theory
iff
CnPos b1 = b1;
:: HILBERT1:th 13
theorem
for b1 being Element of bool HP-WFF
st b1 is Hilbert_theory
holds HP_TAUT c= b1;
:: HILBERT1:funcreg 4
registration
cluster HP_TAUT -> Hilbert_theory;
end;
:: HILBERT1:th 14
theorem
for b1 being Element of HP-WFF holds
b1 => b1 in HP_TAUT;
:: HILBERT1:th 15
theorem
for b1, b2 being Element of HP-WFF
st b1 in HP_TAUT
holds b2 => b1 in HP_TAUT;
:: HILBERT1:th 16
theorem
for b1 being Element of HP-WFF holds
b1 => VERUM in HP_TAUT;
:: HILBERT1:th 17
theorem
for b1, b2 being Element of HP-WFF holds
(b1 => b2) => (b1 => b1) in HP_TAUT;
:: HILBERT1:th 18
theorem
for b1, b2 being Element of HP-WFF holds
(b1 => b2) => (b2 => b2) in HP_TAUT;
:: HILBERT1:th 19
theorem
for b1, b2, b3 being Element of HP-WFF holds
(b1 => b2) => ((b3 => b1) => (b3 => b2)) in HP_TAUT;
:: HILBERT1:th 20
theorem
for b1, b2, b3 being Element of HP-WFF
st b1 => (b2 => b3) in HP_TAUT
holds b2 => (b1 => b3) in HP_TAUT;
:: HILBERT1:th 21
theorem
for b1, b2, b3 being Element of HP-WFF holds
(b1 => b2) => ((b2 => b3) => (b1 => b3)) in HP_TAUT;
:: HILBERT1:th 22
theorem
for b1, b2, b3 being Element of HP-WFF
st b1 => b2 in HP_TAUT
holds (b2 => b3) => (b1 => b3) in HP_TAUT;
:: HILBERT1:th 23
theorem
for b1, b2, b3 being Element of HP-WFF
st b1 => b2 in HP_TAUT & b2 => b3 in HP_TAUT
holds b1 => b3 in HP_TAUT;
:: HILBERT1:th 24
theorem
for b1, b2, b3, b4 being Element of HP-WFF holds
(b1 => (b2 => b3)) => ((b4 => b2) => (b1 => (b4 => b3))) in HP_TAUT;
:: HILBERT1:th 25
theorem
for b1, b2, b3 being Element of HP-WFF holds
((b1 => b2) => b3) => (b2 => b3) in HP_TAUT;
:: HILBERT1:th 26
theorem
for b1, b2, b3 being Element of HP-WFF holds
(b1 => (b2 => b3)) => (b2 => (b1 => b3)) in HP_TAUT;
:: HILBERT1:th 27
theorem
for b1, b2 being Element of HP-WFF holds
(b1 => (b1 => b2)) => (b1 => b2) in HP_TAUT;
:: HILBERT1:th 28
theorem
for b1, b2 being Element of HP-WFF holds
b1 => ((b1 => b2) => b2) in HP_TAUT;
:: HILBERT1:th 29
theorem
for b1, b2, b3 being Element of HP-WFF
st b1 => (b2 => b3) in HP_TAUT & b2 in HP_TAUT
holds b1 => b3 in HP_TAUT;
:: HILBERT1:th 30
theorem
for b1 being Element of HP-WFF holds
b1 => (b1 '&' b1) in HP_TAUT;
:: HILBERT1:th 31
theorem
for b1, b2 being Element of HP-WFF holds
b1 '&' b2 in HP_TAUT
iff
b1 in HP_TAUT & b2 in HP_TAUT;
:: HILBERT1:th 32
theorem
for b1, b2 being Element of HP-WFF holds
b1 '&' b2 in HP_TAUT
iff
b2 '&' b1 in HP_TAUT;
:: HILBERT1:th 33
theorem
for b1, b2, b3 being Element of HP-WFF holds
((b1 '&' b2) => b3) => (b1 => (b2 => b3)) in HP_TAUT;
:: HILBERT1:th 34
theorem
for b1, b2, b3 being Element of HP-WFF holds
(b1 => (b2 => b3)) => ((b1 '&' b2) => b3) in HP_TAUT;
:: HILBERT1:th 35
theorem
for b1, b2, b3 being Element of HP-WFF holds
(b1 => b2) => ((b1 => b3) => (b1 => (b2 '&' b3))) in HP_TAUT;
:: HILBERT1:th 36
theorem
for b1, b2 being Element of HP-WFF holds
((b1 => b2) '&' b1) => b2 in HP_TAUT;
:: HILBERT1:th 37
theorem
for b1, b2, b3 being Element of HP-WFF holds
(((b1 => b2) '&' b1) '&' b3) => b2 in HP_TAUT;
:: HILBERT1:th 38
theorem
for b1, b2, b3 being Element of HP-WFF holds
(b1 => b2) => ((b3 '&' b1) => b2) in HP_TAUT;
:: HILBERT1:th 39
theorem
for b1, b2, b3 being Element of HP-WFF holds
(b1 => b2) => ((b1 '&' b3) => b2) in HP_TAUT;
:: HILBERT1:th 40
theorem
for b1, b2, b3 being Element of HP-WFF holds
((b1 '&' b2) => b3) => ((b1 '&' b2) => (b3 '&' b2)) in HP_TAUT;
:: HILBERT1:th 41
theorem
for b1, b2, b3 being Element of HP-WFF holds
(b1 => b2) => ((b1 '&' b3) => (b2 '&' b3)) in HP_TAUT;
:: HILBERT1:th 42
theorem
for b1, b2, b3 being Element of HP-WFF holds
((b1 => b2) '&' (b1 '&' b3)) => (b2 '&' b3) in HP_TAUT;
:: HILBERT1:th 43
theorem
for b1, b2 being Element of HP-WFF holds
(b1 '&' b2) => (b2 '&' b1) in HP_TAUT;
:: HILBERT1:th 44
theorem
for b1, b2, b3 being Element of HP-WFF holds
((b1 => b2) '&' (b1 '&' b3)) => (b3 '&' b2) in HP_TAUT;
:: HILBERT1:th 45
theorem
for b1, b2, b3 being Element of HP-WFF holds
(b1 => b2) => ((b1 '&' b3) => (b3 '&' b2)) in HP_TAUT;
:: HILBERT1:th 46
theorem
for b1, b2, b3 being Element of HP-WFF holds
(b1 => b2) => ((b3 '&' b1) => (b3 '&' b2)) in HP_TAUT;
:: HILBERT1:th 47
theorem
for b1, b2, b3 being Element of HP-WFF holds
(b1 '&' (b2 '&' b3)) => (b1 '&' (b3 '&' b2)) in HP_TAUT;
:: HILBERT1:th 48
theorem
for b1, b2, b3 being Element of HP-WFF holds
((b1 => b2) '&' (b1 => b3)) => (b1 => (b2 '&' b3)) in HP_TAUT;
:: HILBERT1:th 49
theorem
for b1, b2, b3 being Element of HP-WFF holds
((b1 '&' b2) '&' b3) => (b1 '&' (b2 '&' b3)) in HP_TAUT;
:: HILBERT1:th 50
theorem
for b1, b2, b3 being Element of HP-WFF holds
(b1 '&' (b2 '&' b3)) => ((b1 '&' b2) '&' b3) in HP_TAUT;