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Theorem wledi 405
 Description: Half of distributive law. (Contributed by NM, 13-Oct-1997.)
Assertion
Ref Expression
wledi (((ab) ∪ (ac)) ≤2 (a ∩ (bc))) = 1

Proof of Theorem wledi
StepHypRef Expression
1 anidm 111 . . . 4 (((ab) ∪ (ac)) ∩ ((ab) ∪ (ac))) = ((ab) ∪ (ac))
21bi1 118 . . 3 ((((ab) ∪ (ac)) ∩ ((ab) ∪ (ac))) ≡ ((ab) ∪ (ac))) = 1
32wr1 197 . 2 (((ab) ∪ (ac)) ≡ (((ab) ∪ (ac)) ∩ ((ab) ∪ (ac)))) = 1
4 wlea 388 . . . . 5 ((ab) ≤2 a) = 1
5 wlea 388 . . . . 5 ((ac) ≤2 a) = 1
64, 5wle2or 403 . . . 4 (((ab) ∪ (ac)) ≤2 (aa)) = 1
7 oridm 110 . . . . 5 (aa) = a
87bi1 118 . . . 4 ((aa) ≡ a) = 1
96, 8wlbtr 398 . . 3 (((ab) ∪ (ac)) ≤2 a) = 1
10 ancom 74 . . . . . 6 (ab) = (ba)
1110bi1 118 . . . . 5 ((ab) ≡ (ba)) = 1
12 wlea 388 . . . . 5 ((ba) ≤2 b) = 1
1311, 12wbltr 397 . . . 4 ((ab) ≤2 b) = 1
14 ancom 74 . . . . . 6 (ac) = (ca)
1514bi1 118 . . . . 5 ((ac) ≡ (ca)) = 1
16 wlea 388 . . . . 5 ((ca) ≤2 c) = 1
1715, 16wbltr 397 . . . 4 ((ac) ≤2 c) = 1
1813, 17wle2or 403 . . 3 (((ab) ∪ (ac)) ≤2 (bc)) = 1
199, 18wle2an 404 . 2 ((((ab) ∪ (ac)) ∩ ((ab) ∪ (ac))) ≤2 (a ∩ (bc))) = 1
203, 19wbltr 397 1 (((ab) ∪ (ac)) ≤2 (a ∩ (bc))) = 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:  wfh1  423  wfh2  424
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