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Theorem wledi 405
Description: Half of distributive law.
Assertion
Ref Expression
wledi (((a ^ b) v (a ^ c)) =<2 (a ^ (b v c))) = 1
Proof of Theorem wledi
StepHypRef Expression
1 anidm 111 . . . 4 (((a ^ b) v (a ^ c)) ^ ((a ^ b) v (a ^ c))) = ((a ^ b) v (a ^ c))
21bi1 118 . . 3 ((((a ^ b) v (a ^ c)) ^ ((a ^ b) v (a ^ c))) == ((a ^ b) v (a ^ c))) = 1
32wr1 197 . 2 (((a ^ b) v (a ^ c)) == (((a ^ b) v (a ^ c)) ^ ((a ^ b) v (a ^ c)))) = 1
4 wlea 388 . . . . 5 ((a ^ b) =<2 a) = 1
5 wlea 388 . . . . 5 ((a ^ c) =<2 a) = 1
64, 5wle2or 403 . . . 4 (((a ^ b) v (a ^ c)) =<2 (a v a)) = 1
7 oridm 110 . . . . 5 (a v a) = a
87bi1 118 . . . 4 ((a v a) == a) = 1
96, 8wlbtr 398 . . 3 (((a ^ b) v (a ^ c)) =<2 a) = 1
10 ancom 74 . . . . . 6 (a ^ b) = (b ^ a)
1110bi1 118 . . . . 5 ((a ^ b) == (b ^ a)) = 1
12 wlea 388 . . . . 5 ((b ^ a) =<2 b) = 1
1311, 12wbltr 397 . . . 4 ((a ^ b) =<2 b) = 1
14 ancom 74 . . . . . 6 (a ^ c) = (c ^ a)
1514bi1 118 . . . . 5 ((a ^ c) == (c ^ a)) = 1
16 wlea 388 . . . . 5 ((c ^ a) =<2 c) = 1
1715, 16wbltr 397 . . . 4 ((a ^ c) =<2 c) = 1
1813, 17wle2or 403 . . 3 (((a ^ b) v (a ^ c)) =<2 (b v c)) = 1
199, 18wle2an 404 . 2 ((((a ^ b) v (a ^ c)) ^ ((a ^ b) v (a ^ c))) =<2 (a ^ (b v c))) = 1
203, 19wbltr 397 1 (((a ^ b) v (a ^ c)) =<2 (a ^ (b v c))) = 1
Colors of variables: term
Syntax hints:   = wb 1   v wo 6   ^ wa 7  1wt 8   =<2 wle2 10
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-wom 361
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131
This theorem is referenced by:  wfh1  423  wfh2  424
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