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Theorem u1lemab 610
Description: Lemma for Sasaki implication study. Equation 4.10 of [MegPav2000] p. 23. This is the second part of the equation.
Assertion
Ref Expression
u1lemab ((a ->1 b) ^ b) = ((a ^ b) v (a' ^ b))
Proof of Theorem u1lemab
StepHypRef Expression
1 df-i1 44 . . 3 (a ->1 b) = (a' v (a ^ b))
21ran 78 . 2 ((a ->1 b) ^ b) = ((a' v (a ^ b)) ^ b)
3 ax-a2 31 . . . . 5 (a' v (a ^ b)) = ((a ^ b) v a')
43ran 78 . . . 4 ((a' v (a ^ b)) ^ b) = (((a ^ b) v a') ^ b)
5 coman2 186 . . . . 5 (a ^ b) C b
6 coman1 185 . . . . . 6 (a ^ b) C a
76comcom2 183 . . . . 5 (a ^ b) C a'
85, 7fh2r 474 . . . 4 (((a ^ b) v a') ^ b) = (((a ^ b) ^ b) v (a' ^ b))
94, 8ax-r2 36 . . 3 ((a' v (a ^ b)) ^ b) = (((a ^ b) ^ b) v (a' ^ b))
10 anass 76 . . . . 5 ((a ^ b) ^ b) = (a ^ (b ^ b))
11 anidm 111 . . . . . 6 (b ^ b) = b
1211lan 77 . . . . 5 (a ^ (b ^ b)) = (a ^ b)
1310, 12ax-r2 36 . . . 4 ((a ^ b) ^ b) = (a ^ b)
1413ax-r5 38 . . 3 (((a ^ b) ^ b) v (a' ^ b)) = ((a ^ b) v (a' ^ b))
159, 14ax-r2 36 . 2 ((a' v (a ^ b)) ^ b) = ((a ^ b) v (a' ^ b))
162, 15ax-r2 36 1 ((a ->1 b) ^ b) = ((a ^ b) v (a' ^ b))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->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:  u1lemnonb  675  i1com  708  u1lem3  749  u1lem11  780  sadm3  838  negantlem2  849  negantlem3  850  negantlem10  861  neg3antlem1  864  oa4to4u  973  oa3-6lem  980  oa3-u1lem  985  oa3-u2lem  986  oa3-1to5  993  lem4.6.2e2  1081
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