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Theorem wledio 406
 Description: Half of distributive law.
Assertion
Ref Expression
wledio ((a ∪ (bc)) ≤2 ((ab) ∩ (ac))) = 1

Proof of Theorem wledio
StepHypRef Expression
1 anidm 111 . . . . . 6 (aa) = a
21bi1 118 . . . . 5 ((aa) ≡ a) = 1
32wr1 197 . . . 4 (a ≡ (aa)) = 1
4 wleo 387 . . . . 5 (a2 (ab)) = 1
5 wleo 387 . . . . 5 (a2 (ac)) = 1
64, 5wle2an 404 . . . 4 ((aa) ≤2 ((ab) ∩ (ac))) = 1
73, 6wbltr 397 . . 3 (a2 ((ab) ∩ (ac))) = 1
8 wleo 387 . . . . 5 (b2 (ba)) = 1
9 ax-a2 31 . . . . . 6 (ba) = (ab)
109bi1 118 . . . . 5 ((ba) ≡ (ab)) = 1
118, 10wlbtr 398 . . . 4 (b2 (ab)) = 1
12 wleo 387 . . . . 5 (c2 (ca)) = 1
13 ax-a2 31 . . . . . 6 (ca) = (ac)
1413bi1 118 . . . . 5 ((ca) ≡ (ac)) = 1
1512, 14wlbtr 398 . . . 4 (c2 (ac)) = 1
1611, 15wle2an 404 . . 3 ((bc) ≤2 ((ab) ∩ (ac))) = 1
177, 16wle2or 403 . 2 ((a ∪ (bc)) ≤2 (((ab) ∩ (ac)) ∪ ((ab) ∩ (ac)))) = 1
18 oridm 110 . . 3 (((ab) ∩ (ac)) ∪ ((ab) ∩ (ac))) = ((ab) ∩ (ac))
1918bi1 118 . 2 ((((ab) ∩ (ac)) ∪ ((ab) ∩ (ac))) ≡ ((ab) ∩ (ac))) = 1
2017, 19wlbtr 398 1 ((a ∪ (bc)) ≤2 ((ab) ∩ (ac))) = 1
 Colors of variables: term Syntax hints:   = wb 1   ∪ 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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