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Theorem oa3-6to3 987
Description: Derivation of 3-OA variant (3) from (6).
Hypothesis
Ref Expression
oa3-6to3.1 ((a ->1 c) ^ (a v (b ^ (((a' ->1 c) ^ (b' ->1 c)) v ((a ->1 c) ^ (b ->1 c)))))) =< c
Assertion
Ref Expression
oa3-6to3 (a' ^ (a v (b ^ ((a == b) v ((a' ->1 c) ^ (b' ->1 c)))))) =< c
Proof of Theorem oa3-6to3
StepHypRef Expression
1 oa3-3lem 981 . . 3 (a' ^ (a v (b ^ (((a ^ b) v (a' ^ b')) v (((a ^ 1) v (a' ^ c)) ^ ((b ^ 1) v (b' ^ c))))))) = (a' ^ (a v (b ^ ((a == b) v ((a' ->1 c) ^ (b' ->1 c))))))
21ax-r1 35 . 2 (a' ^ (a v (b ^ ((a == b) v ((a' ->1 c) ^ (b' ->1 c)))))) = (a' ^ (a v (b ^ (((a ^ b) v (a' ^ b')) v (((a ^ 1) v (a' ^ c)) ^ ((b ^ 1) v (b' ^ c)))))))
3 leid 148 . . 3 a' =< a'
4 leid 148 . . 3 b' =< b'
5 df-f 42 . . . . 5 0 = 1'
65ax-r1 35 . . . 4 1' = 0
7 le0 147 . . . 4 0 =< c
86, 7bltr 138 . . 3 1' =< c
9 ancom 74 . . . . . . . 8 (1 ^ c) = (c ^ 1)
10 an1 106 . . . . . . . 8 (c ^ 1) = c
119, 10ax-r2 36 . . . . . . 7 (1 ^ c) = c
12 dff 101 . . . . . . . . . 10 0 = (a ^ a')
13 dff 101 . . . . . . . . . 10 0 = (b ^ b')
1412, 132or 72 . . . . . . . . 9 (0 v 0) = ((a ^ a') v (b ^ b'))
1514ax-r1 35 . . . . . . . 8 ((a ^ a') v (b ^ b')) = (0 v 0)
16 or0 102 . . . . . . . 8 (0 v 0) = 0
1715, 16ax-r2 36 . . . . . . 7 ((a ^ a') v (b ^ b')) = 0
1811, 172or 72 . . . . . 6 ((1 ^ c) v ((a ^ a') v (b ^ b'))) = (c v 0)
19 or0 102 . . . . . 6 (c v 0) = c
2018, 19ax-r2 36 . . . . 5 ((1 ^ c) v ((a ^ a') v (b ^ b'))) = c
2120ax-r1 35 . . . 4 c = ((1 ^ c) v ((a ^ a') v (b ^ b')))
22 ax-a2 31 . . . 4 ((1 ^ c) v ((a ^ a') v (b ^ b'))) = (((a ^ a') v (b ^ b')) v (1 ^ c))
2321, 22ax-r2 36 . . 3 c = (((a ^ a') v (b ^ b')) v (1 ^ c))
24 oa3-6lem 980 . . . 4 ((a ->1 c) ^ (a v (b ^ (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ 1) v ((a ->1 c) ^ (1 ->1 c))) ^ ((b ^ 1) v ((b ->1 c) ^ (1 ->1 c)))))))) = ((a ->1 c) ^ (a v (b ^ (((a' ->1 c) ^ (b' ->1 c)) v ((a ->1 c) ^ (b ->1 c))))))
25 oa3-6to3.1 . . . 4 ((a ->1 c) ^ (a v (b ^ (((a' ->1 c) ^ (b' ->1 c)) v ((a ->1 c) ^ (b ->1 c)))))) =< c
2624, 25bltr 138 . . 3 ((a ->1 c) ^ (a v (b ^ (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v (((a ^ 1) v ((a ->1 c) ^ (1 ->1 c))) ^ ((b ^ 1) v ((b ->1 c) ^ (1 ->1 c)))))))) =< c
273, 4, 8, 23, 26oa4to6dual 964 . 2 (a' ^ (a v (b ^ (((a ^ b) v (a' ^ b')) v (((a ^ 1) v (a' ^ c)) ^ ((b ^ 1) v (b' ^ c))))))) =< c
282, 27bltr 138 1 (a' ^ (a v (b ^ ((a == b) v ((a' ->1 c) ^ (b' ->1 c)))))) =< c
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8  0wf 9   ->1 wi1 12
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-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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