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Theorem oalem1 1005
Description: Lemma.
Assertion
Ref Expression
oalem1 ((b v c) v ((b v c)' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))))) =< (a ->2 (b v c))
Proof of Theorem oalem1
StepHypRef Expression
1 anidm 111 . . . . . . . . 9 (b' ^ b') = b'
21ran 78 . . . . . . . 8 ((b' ^ b') ^ c') = (b' ^ c')
32ax-r1 35 . . . . . . 7 (b' ^ c') = ((b' ^ b') ^ c')
4 anor3 90 . . . . . . 7 (b' ^ c') = (b v c)'
5 an32 83 . . . . . . . 8 ((b' ^ b') ^ c') = ((b' ^ c') ^ b')
64ran 78 . . . . . . . 8 ((b' ^ c') ^ b') = ((b v c)' ^ b')
75, 6ax-r2 36 . . . . . . 7 ((b' ^ b') ^ c') = ((b v c)' ^ b')
83, 4, 73tr2 64 . . . . . 6 (b v c)' = ((b v c)' ^ b')
98ran 78 . . . . 5 ((b v c)' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) = (((b v c)' ^ b') ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))))
10 anass 76 . . . . . 6 (((b v c)' ^ b') ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) = ((b v c)' ^ (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))))
11 oalii 1002 . . . . . . 7 (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< a'
1211lelan 167 . . . . . 6 ((b v c)' ^ (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))))) =< ((b v c)' ^ a')
1310, 12bltr 138 . . . . 5 (((b v c)' ^ b') ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< ((b v c)' ^ a')
149, 13bltr 138 . . . 4 ((b v c)' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< ((b v c)' ^ a')
15 ancom 74 . . . 4 ((b v c)' ^ a') = (a' ^ (b v c)')
1614, 15lbtr 139 . . 3 ((b v c)' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< (a' ^ (b v c)')
1716lelor 166 . 2 ((b v c) v ((b v c)' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))))) =< ((b v c) v (a' ^ (b v c)'))
18 df-i2 45 . . 3 (a ->2 (b v c)) = ((b v c) v (a' ^ (b v c)'))
1918ax-r1 35 . 2 ((b v c) v (a' ^ (b v c)')) = (a ->2 (b v c))
2017, 19lbtr 139 1 ((b v c) v ((b v c)' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))))) =< (a ->2 (b v c))
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->2 wi2 13
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  ax-r3 439  ax-3oa 998
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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