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Theorem woml6 436
Description: Variant of weakly orthomodular law.
Assertion
Ref Expression
woml6 ((a ->1 b)' v (a ->2 b)) = 1
Proof of Theorem woml6
StepHypRef Expression
1 df-i1 44 . . . . . 6 (a ->1 b) = (a' v (a ^ b))
2 df-a 40 . . . . . . 7 (a ^ b) = (a' v b')'
32lor 70 . . . . . 6 (a' v (a ^ b)) = (a' v (a' v b')')
41, 3ax-r2 36 . . . . 5 (a ->1 b) = (a' v (a' v b')')
54ax-r4 37 . . . 4 (a ->1 b)' = (a' v (a' v b')')'
6 df-a 40 . . . . 5 (a ^ (a' v b')) = (a' v (a' v b')')'
76ax-r1 35 . . . 4 (a' v (a' v b')')' = (a ^ (a' v b'))
85, 7ax-r2 36 . . 3 (a ->1 b)' = (a ^ (a' v b'))
9 df-i2 45 . . 3 (a ->2 b) = (b v (a' ^ b'))
108, 92or 72 . 2 ((a ->1 b)' v (a ->2 b)) = ((a ^ (a' v b')) v (b v (a' ^ b')))
11 ax-a2 31 . . . . 5 ((a ^ (a' v b')) v b) = (b v (a ^ (a' v b')))
12 ancom 74 . . . . . 6 (a ^ (a' v b')) = ((a' v b') ^ a)
1312lor 70 . . . . 5 (b v (a ^ (a' v b'))) = (b v ((a' v b') ^ a))
1411, 13ax-r2 36 . . . 4 ((a ^ (a' v b')) v b) = (b v ((a' v b') ^ a))
1514ax-r5 38 . . 3 (((a ^ (a' v b')) v b) v (a' ^ b')) = ((b v ((a' v b') ^ a)) v (a' ^ b'))
16 ax-a3 32 . . 3 (((a ^ (a' v b')) v b) v (a' ^ b')) = ((a ^ (a' v b')) v (b v (a' ^ b')))
17 1b 117 . . . . 5 (1 == ((b v ((a' v b') ^ a)) v (a' ^ b'))) = ((b v ((a' v b') ^ a)) v (a' ^ b'))
1817ax-r1 35 . . . 4 ((b v ((a' v b') ^ a)) v (a' ^ b')) = (1 == ((b v ((a' v b') ^ a)) v (a' ^ b')))
19 wcomorr 412 . . . . . . . . . . . 12 C (b', (b' v a')) = 1
20 ax-a2 31 . . . . . . . . . . . . 13 (b' v a') = (a' v b')
2120bi1 118 . . . . . . . . . . . 12 ((b' v a') == (a' v b')) = 1
2219, 21wcbtr 411 . . . . . . . . . . 11 C (b', (a' v b')) = 1
2322wcomcom 414 . . . . . . . . . 10 C ((a' v b'), b') = 1
2423wcomcom3 416 . . . . . . . . 9 C ((a' v b')', b') = 1
2524wcomcom5 420 . . . . . . . 8 C ((a' v b'), b) = 1
26 wcomorr 412 . . . . . . . . . . 11 C (a', (a' v b')) = 1
2726wcomcom 414 . . . . . . . . . 10 C ((a' v b'), a') = 1
2827wcomcom3 416 . . . . . . . . 9 C ((a' v b')', a') = 1
2928wcomcom5 420 . . . . . . . 8 C ((a' v b'), a) = 1
3025, 29wfh4 426 . . . . . . 7 ((b v ((a' v b') ^ a)) == ((b v (a' v b')) ^ (b v a))) = 1
3130wr5-2v 366 . . . . . 6 (((b v ((a' v b') ^ a)) v (a' ^ b')) == (((b v (a' v b')) ^ (b v a)) v (a' ^ b'))) = 1
32 or12 80 . . . . . . . . . . . . 13 (b v (a' v b')) = (a' v (b v b'))
33 df-t 41 . . . . . . . . . . . . . . 15 1 = (b v b')
3433lor 70 . . . . . . . . . . . . . 14 (a' v 1) = (a' v (b v b'))
3534ax-r1 35 . . . . . . . . . . . . 13 (a' v (b v b')) = (a' v 1)
36 or1 104 . . . . . . . . . . . . 13 (a' v 1) = 1
3732, 35, 363tr 65 . . . . . . . . . . . 12 (b v (a' v b')) = 1
3837ran 78 . . . . . . . . . . 11 ((b v (a' v b')) ^ (b v a)) = (1 ^ (b v a))
39 ancom 74 . . . . . . . . . . 11 (1 ^ (b v a)) = ((b v a) ^ 1)
4038, 39ax-r2 36 . . . . . . . . . 10 ((b v (a' v b')) ^ (b v a)) = ((b v a) ^ 1)
41 an1 106 . . . . . . . . . 10 ((b v a) ^ 1) = (b v a)
42 ax-a2 31 . . . . . . . . . 10 (b v a) = (a v b)
4340, 41, 423tr 65 . . . . . . . . 9 ((b v (a' v b')) ^ (b v a)) = (a v b)
44 anor3 90 . . . . . . . . 9 (a' ^ b') = (a v b)'
4543, 442or 72 . . . . . . . 8 (((b v (a' v b')) ^ (b v a)) v (a' ^ b')) = ((a v b) v (a v b)')
46 df-t 41 . . . . . . . . 9 1 = ((a v b) v (a v b)')
4746ax-r1 35 . . . . . . . 8 ((a v b) v (a v b)') = 1
4845, 47ax-r2 36 . . . . . . 7 (((b v (a' v b')) ^ (b v a)) v (a' ^ b')) = 1
4948bi1 118 . . . . . 6 ((((b v (a' v b')) ^ (b v a)) v (a' ^ b')) == 1) = 1
5031, 49wr2 371 . . . . 5 (((b v ((a' v b') ^ a)) v (a' ^ b')) == 1) = 1
5150wr1 197 . . . 4 (1 == ((b v ((a' v b') ^ a)) v (a' ^ b'))) = 1
5218, 51ax-r2 36 . . 3 ((b v ((a' v b') ^ a)) v (a' ^ b')) = 1
5315, 16, 523tr2 64 . 2 ((a ^ (a' v b')) v (b v (a' ^ b'))) = 1
5410, 53ax-r2 36 1 ((a ->1 b)' v (a ->2 b)) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8   ->1 wi1 12   ->2 wi2 13
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-wom 361
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134
This theorem is referenced by:  lem3.4.1  1075
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