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Theorem u1lemaa 600
Description: Lemma for Sasaki implication study. Equation 4.10 of [MegPav2000] p. 23. This is the second part of the equation.
Assertion
Ref Expression
u1lemaa ((a ->1 b) ^ a) = (a ^ b)
Proof of Theorem u1lemaa
StepHypRef Expression
1 df-i1 44 . . 3 (a ->1 b) = (a' v (a ^ b))
21ran 78 . 2 ((a ->1 b) ^ a) = ((a' v (a ^ b)) ^ a)
3 comid 187 . . . . 5 a C a
43comcom2 183 . . . 4 a C a'
5 comanr1 464 . . . 4 a C (a ^ b)
64, 5fh1r 473 . . 3 ((a' v (a ^ b)) ^ a) = ((a' ^ a) v ((a ^ b) ^ a))
7 ax-a2 31 . . . . 5 ((a' ^ a) v ((a ^ b) ^ a)) = (((a ^ b) ^ a) v (a' ^ a))
8 an32 83 . . . . . . 7 ((a ^ b) ^ a) = ((a ^ a) ^ b)
9 anidm 111 . . . . . . . 8 (a ^ a) = a
109ran 78 . . . . . . 7 ((a ^ a) ^ b) = (a ^ b)
118, 10ax-r2 36 . . . . . 6 ((a ^ b) ^ a) = (a ^ b)
12 ancom 74 . . . . . . 7 (a' ^ a) = (a ^ a')
13 dff 101 . . . . . . . 8 0 = (a ^ a')
1413ax-r1 35 . . . . . . 7 (a ^ a') = 0
1512, 14ax-r2 36 . . . . . 6 (a' ^ a) = 0
1611, 152or 72 . . . . 5 (((a ^ b) ^ a) v (a' ^ a)) = ((a ^ b) v 0)
177, 16ax-r2 36 . . . 4 ((a' ^ a) v ((a ^ b) ^ a)) = ((a ^ b) v 0)
18 or0 102 . . . 4 ((a ^ b) v 0) = (a ^ b)
1917, 18ax-r2 36 . . 3 ((a' ^ a) v ((a ^ b) ^ a)) = (a ^ b)
206, 19ax-r2 36 . 2 ((a' v (a ^ b)) ^ a) = (a ^ b)
212, 20ax-r2 36 1 ((a ->1 b) ^ a) = (a ^ b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  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:  u1lemnona  665  u12lembi  726  u1lem5  761  negantlem2  849  kb10iii  893  oas  925  oau  929  oaur  930  oa6to4  958  oa8to5  972
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