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Theorem oa3-6lem 980
Description: Lemma for 3-OA(6). Equivalence with substitution into 4-OA.
Assertion
Ref Expression
oa3-6lem ((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))))))
Proof of Theorem oa3-6lem
StepHypRef Expression
1 an1 106 . . . . . . . . 9 (a ^ 1) = a
2 1i1 274 . . . . . . . . . . 11 (1 ->1 c) = c
32lan 77 . . . . . . . . . 10 ((a ->1 c) ^ (1 ->1 c)) = ((a ->1 c) ^ c)
4 u1lemab 610 . . . . . . . . . 10 ((a ->1 c) ^ c) = ((a ^ c) v (a' ^ c))
53, 4ax-r2 36 . . . . . . . . 9 ((a ->1 c) ^ (1 ->1 c)) = ((a ^ c) v (a' ^ c))
61, 52or 72 . . . . . . . 8 ((a ^ 1) v ((a ->1 c) ^ (1 ->1 c))) = (a v ((a ^ c) v (a' ^ c)))
7 ax-a3 32 . . . . . . . . 9 ((a v (a ^ c)) v (a' ^ c)) = (a v ((a ^ c) v (a' ^ c)))
87ax-r1 35 . . . . . . . 8 (a v ((a ^ c) v (a' ^ c))) = ((a v (a ^ c)) v (a' ^ c))
9 orabs 120 . . . . . . . . 9 (a v (a ^ c)) = a
109ax-r5 38 . . . . . . . 8 ((a v (a ^ c)) v (a' ^ c)) = (a v (a' ^ c))
116, 8, 103tr 65 . . . . . . 7 ((a ^ 1) v ((a ->1 c) ^ (1 ->1 c))) = (a v (a' ^ c))
12 an1 106 . . . . . . . . 9 (b ^ 1) = b
132lan 77 . . . . . . . . . 10 ((b ->1 c) ^ (1 ->1 c)) = ((b ->1 c) ^ c)
14 u1lemab 610 . . . . . . . . . 10 ((b ->1 c) ^ c) = ((b ^ c) v (b' ^ c))
1513, 14ax-r2 36 . . . . . . . . 9 ((b ->1 c) ^ (1 ->1 c)) = ((b ^ c) v (b' ^ c))
1612, 152or 72 . . . . . . . 8 ((b ^ 1) v ((b ->1 c) ^ (1 ->1 c))) = (b v ((b ^ c) v (b' ^ c)))
17 ax-a3 32 . . . . . . . . 9 ((b v (b ^ c)) v (b' ^ c)) = (b v ((b ^ c) v (b' ^ c)))
1817ax-r1 35 . . . . . . . 8 (b v ((b ^ c) v (b' ^ c))) = ((b v (b ^ c)) v (b' ^ c))
19 orabs 120 . . . . . . . . 9 (b v (b ^ c)) = b
2019ax-r5 38 . . . . . . . 8 ((b v (b ^ c)) v (b' ^ c)) = (b v (b' ^ c))
2116, 18, 203tr 65 . . . . . . 7 ((b ^ 1) v ((b ->1 c) ^ (1 ->1 c))) = (b v (b' ^ c))
2211, 212an 79 . . . . . 6 (((a ^ 1) v ((a ->1 c) ^ (1 ->1 c))) ^ ((b ^ 1) v ((b ->1 c) ^ (1 ->1 c)))) = ((a v (a' ^ c)) ^ (b v (b' ^ c)))
2322lor 70 . . . . 5 (((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 ^ b) v ((a ->1 c) ^ (b ->1 c))) v ((a v (a' ^ c)) ^ (b v (b' ^ c))))
24 or32 82 . . . . 5 (((a ^ b) v ((a ->1 c) ^ (b ->1 c))) v ((a v (a' ^ c)) ^ (b v (b' ^ c)))) = (((a ^ b) v ((a v (a' ^ c)) ^ (b v (b' ^ c)))) v ((a ->1 c) ^ (b ->1 c)))
25 leo 158 . . . . . . . . 9 a =< (a v (a' ^ c))
26 leo 158 . . . . . . . . 9 b =< (b v (b' ^ c))
2725, 26le2an 169 . . . . . . . 8 (a ^ b) =< ((a v (a' ^ c)) ^ (b v (b' ^ c)))
2827df-le2 131 . . . . . . 7 ((a ^ b) v ((a v (a' ^ c)) ^ (b v (b' ^ c)))) = ((a v (a' ^ c)) ^ (b v (b' ^ c)))
29 ax-a1 30 . . . . . . . . . 10 a = a''
3029ax-r5 38 . . . . . . . . 9 (a v (a' ^ c)) = (a'' v (a' ^ c))
31 df-i1 44 . . . . . . . . . 10 (a' ->1 c) = (a'' v (a' ^ c))
3231ax-r1 35 . . . . . . . . 9 (a'' v (a' ^ c)) = (a' ->1 c)
3330, 32ax-r2 36 . . . . . . . 8 (a v (a' ^ c)) = (a' ->1 c)
34 ax-a1 30 . . . . . . . . . 10 b = b''
3534ax-r5 38 . . . . . . . . 9 (b v (b' ^ c)) = (b'' v (b' ^ c))
36 df-i1 44 . . . . . . . . . 10 (b' ->1 c) = (b'' v (b' ^ c))
3736ax-r1 35 . . . . . . . . 9 (b'' v (b' ^ c)) = (b' ->1 c)
3835, 37ax-r2 36 . . . . . . . 8 (b v (b' ^ c)) = (b' ->1 c)
3933, 382an 79 . . . . . . 7 ((a v (a' ^ c)) ^ (b v (b' ^ c))) = ((a' ->1 c) ^ (b' ->1 c))
4028, 39ax-r2 36 . . . . . 6 ((a ^ b) v ((a v (a' ^ c)) ^ (b v (b' ^ c)))) = ((a' ->1 c) ^ (b' ->1 c))
4140ax-r5 38 . . . . 5 (((a ^ b) v ((a v (a' ^ c)) ^ (b v (b' ^ c)))) v ((a ->1 c) ^ (b ->1 c))) = (((a' ->1 c) ^ (b' ->1 c)) v ((a ->1 c) ^ (b ->1 c)))
4223, 24, 413tr 65 . . . 4 (((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) ^ (b' ->1 c)) v ((a ->1 c) ^ (b ->1 c)))
4342lan 77 . . 3 (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)))))) = (b ^ (((a' ->1 c) ^ (b' ->1 c)) v ((a ->1 c) ^ (b ->1 c))))
4443lor 70 . 2 (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 v (b ^ (((a' ->1 c) ^ (b' ->1 c)) v ((a ->1 c) ^ (b ->1 c)))))
4544lan 77 1 ((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))))))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->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:  oa3-6to3  987
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