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Theorem wledio 406
Description: Half of distributive law.
Assertion
Ref Expression
wledio ((a v (b ^ c)) =<2 ((a v b) ^ (a v c))) = 1
Proof of Theorem wledio
StepHypRef Expression
1 anidm 111 . . . . . 6 (a ^ a) = a
21bi1 118 . . . . 5 ((a ^ a) == a) = 1
32wr1 197 . . . 4 (a == (a ^ a)) = 1
4 wleo 387 . . . . 5 (a =<2 (a v b)) = 1
5 wleo 387 . . . . 5 (a =<2 (a v c)) = 1
64, 5wle2an 404 . . . 4 ((a ^ a) =<2 ((a v b) ^ (a v c))) = 1
73, 6wbltr 397 . . 3 (a =<2 ((a v b) ^ (a v c))) = 1
8 wleo 387 . . . . 5 (b =<2 (b v a)) = 1
9 ax-a2 31 . . . . . 6 (b v a) = (a v b)
109bi1 118 . . . . 5 ((b v a) == (a v b)) = 1
118, 10wlbtr 398 . . . 4 (b =<2 (a v b)) = 1
12 wleo 387 . . . . 5 (c =<2 (c v a)) = 1
13 ax-a2 31 . . . . . 6 (c v a) = (a v c)
1413bi1 118 . . . . 5 ((c v a) == (a v c)) = 1
1512, 14wlbtr 398 . . . 4 (c =<2 (a v c)) = 1
1611, 15wle2an 404 . . 3 ((b ^ c) =<2 ((a v b) ^ (a v c))) = 1
177, 16wle2or 403 . 2 ((a v (b ^ c)) =<2 (((a v b) ^ (a v c)) v ((a v b) ^ (a v c)))) = 1
18 oridm 110 . . 3 (((a v b) ^ (a v c)) v ((a v b) ^ (a v c))) = ((a v b) ^ (a v c))
1918bi1 118 . 2 ((((a v b) ^ (a v c)) v ((a v b) ^ (a v c))) == ((a v b) ^ (a v c))) = 1
2017, 19wlbtr 398 1 ((a v (b ^ c)) =<2 ((a v b) ^ (a 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: (None)
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