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Theorem nom21 314
Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom21 (a ==1 (a ^ b)) = (a ->1 b)
Proof of Theorem nom21
StepHypRef Expression
1 ancom 74 . . 3 ((a' v (a v b')) ^ (a' v (a ^ b))) = ((a' v (a ^ b)) ^ (a' v (a v b')))
2 or12 80 . . . . 5 (a' v (a v b')) = (a v (a' v b'))
3 oran3 93 . . . . . 6 (a' v b') = (a ^ b)'
43lor 70 . . . . 5 (a v (a' v b')) = (a v (a ^ b)')
52, 4ax-r2 36 . . . 4 (a' v (a v b')) = (a v (a ^ b)')
6 anidm 111 . . . . . . . 8 (a ^ a) = a
76ran 78 . . . . . . 7 ((a ^ a) ^ b) = (a ^ b)
87ax-r1 35 . . . . . 6 (a ^ b) = ((a ^ a) ^ b)
9 anass 76 . . . . . 6 ((a ^ a) ^ b) = (a ^ (a ^ b))
108, 9ax-r2 36 . . . . 5 (a ^ b) = (a ^ (a ^ b))
1110lor 70 . . . 4 (a' v (a ^ b)) = (a' v (a ^ (a ^ b)))
125, 112an 79 . . 3 ((a' v (a v b')) ^ (a' v (a ^ b))) = ((a v (a ^ b)') ^ (a' v (a ^ (a ^ b))))
13 lea 160 . . . . . 6 (a ^ b) =< a
14 leo 158 . . . . . 6 a =< (a v b')
1513, 14letr 137 . . . . 5 (a ^ b) =< (a v b')
1615lelor 166 . . . 4 (a' v (a ^ b)) =< (a' v (a v b'))
1716df2le2 136 . . 3 ((a' v (a ^ b)) ^ (a' v (a v b'))) = (a' v (a ^ b))
181, 12, 173tr2 64 . 2 ((a v (a ^ b)') ^ (a' v (a ^ (a ^ b)))) = (a' v (a ^ b))
19 df-id1 50 . 2 (a ==1 (a ^ b)) = ((a v (a ^ b)') ^ (a' v (a ^ (a ^ b))))
20 df-i1 44 . 2 (a ->1 b) = (a' v (a ^ b))
2118, 19, 203tr1 63 1 (a ==1 (a ^ b)) = (a ->1 b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   ==1 wid1 18
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-id1 50  df-le1 130  df-le2 131
This theorem is referenced by:  nom34  323  nom52  333
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