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Theorem mlaconjolem 885
Description: Lemma for OML proof of Mladen's conjecture,
Assertion
Ref Expression
mlaconjolem ((a == c) v (b == c)) =< ((c ^ (a v b)) v (c' ^ (a' v b')))
Proof of Theorem mlaconjolem
StepHypRef Expression
1 orbile 843 . 2 ((a == c) v (b == c)) =< (((a ^ b) ->2 c) ^ (c ->1 (a v b)))
2 df-i2 45 . . . . 5 ((a ^ b) ->2 c) = (c v ((a ^ b)' ^ c'))
3 oran3 93 . . . . . . . 8 (a' v b') = (a ^ b)'
43ran 78 . . . . . . 7 ((a' v b') ^ c') = ((a ^ b)' ^ c')
54lor 70 . . . . . 6 (c v ((a' v b') ^ c')) = (c v ((a ^ b)' ^ c'))
65ax-r1 35 . . . . 5 (c v ((a ^ b)' ^ c')) = (c v ((a' v b') ^ c'))
72, 6ax-r2 36 . . . 4 ((a ^ b) ->2 c) = (c v ((a' v b') ^ c'))
8 df-i1 44 . . . 4 (c ->1 (a v b)) = (c' v (c ^ (a v b)))
97, 82an 79 . . 3 (((a ^ b) ->2 c) ^ (c ->1 (a v b))) = ((c v ((a' v b') ^ c')) ^ (c' v (c ^ (a v b))))
10 comor1 461 . . . . 5 (c v ((a' v b') ^ c')) C c
1110comcom2 183 . . . 4 (c v ((a' v b') ^ c')) C c'
12 leao1 162 . . . . . 6 (c ^ (a v b)) =< (c v ((a' v b') ^ c'))
1312lecom 180 . . . . 5 (c ^ (a v b)) C (c v ((a' v b') ^ c'))
1413comcom 453 . . . 4 (c v ((a' v b') ^ c')) C (c ^ (a v b))
1511, 14fh1 469 . . 3 ((c v ((a' v b') ^ c')) ^ (c' v (c ^ (a v b)))) = (((c v ((a' v b') ^ c')) ^ c') v ((c v ((a' v b') ^ c')) ^ (c ^ (a v b))))
16 ancom 74 . . . . . . . 8 ((a' v b') ^ c') = (c' ^ (a' v b'))
1716lor 70 . . . . . . 7 (c v ((a' v b') ^ c')) = (c v (c' ^ (a' v b')))
1817ran 78 . . . . . 6 ((c v ((a' v b') ^ c')) ^ c') = ((c v (c' ^ (a' v b'))) ^ c')
19 ancom 74 . . . . . 6 ((c v (c' ^ (a' v b'))) ^ c') = (c' ^ (c v (c' ^ (a' v b'))))
20 omlan 448 . . . . . 6 (c' ^ (c v (c' ^ (a' v b')))) = (c' ^ (a' v b'))
2118, 19, 203tr 65 . . . . 5 ((c v ((a' v b') ^ c')) ^ c') = (c' ^ (a' v b'))
22 ancom 74 . . . . . 6 ((c v ((a' v b') ^ c')) ^ (c ^ (a v b))) = ((c ^ (a v b)) ^ (c v ((a' v b') ^ c')))
2312df2le2 136 . . . . . 6 ((c ^ (a v b)) ^ (c v ((a' v b') ^ c'))) = (c ^ (a v b))
2422, 23ax-r2 36 . . . . 5 ((c v ((a' v b') ^ c')) ^ (c ^ (a v b))) = (c ^ (a v b))
2521, 242or 72 . . . 4 (((c v ((a' v b') ^ c')) ^ c') v ((c v ((a' v b') ^ c')) ^ (c ^ (a v b)))) = ((c' ^ (a' v b')) v (c ^ (a v b)))
26 ax-a2 31 . . . 4 ((c' ^ (a' v b')) v (c ^ (a v b))) = ((c ^ (a v b)) v (c' ^ (a' v b')))
2725, 26ax-r2 36 . . 3 (((c v ((a' v b') ^ c')) ^ c') v ((c v ((a' v b') ^ c')) ^ (c ^ (a v b)))) = ((c ^ (a v b)) v (c' ^ (a' v b')))
289, 15, 273tr 65 . 2 (((a ^ b) ->2 c) ^ (c ->1 (a v b))) = ((c ^ (a v b)) v (c' ^ (a' v b')))
291, 28lbtr 139 1 ((a == c) v (b == c)) =< ((c ^ (a v b)) v (c' ^ (a' v b')))
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   == tb 5   v wo 6   ^ wa 7   ->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-r3 439
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  df-c1 132  df-c2 133
This theorem is referenced by:  mlaconjo  886
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