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Theorem negantlem3 850
 Description: Lemma for negated antecedent identity.
Hypothesis
Ref Expression
negant.1 (a1 c) = (b1 c)
Assertion
Ref Expression
negantlem3 (ac) ≤ (b1 c)

Proof of Theorem negantlem3
StepHypRef Expression
1 leo 158 . . . 4 a ≤ (a ∪ (ac))
2 df-i1 44 . . . . . 6 (a1 c) = (a ∪ (ac))
32ax-r1 35 . . . . 5 (a ∪ (ac)) = (a1 c)
4 negant.1 . . . . 5 (a1 c) = (b1 c)
53, 4ax-r2 36 . . . 4 (a ∪ (ac)) = (b1 c)
61, 5lbtr 139 . . 3 a ≤ (b1 c)
76leran 153 . 2 (ac) ≤ ((b1 c) ∩ c)
8 lea 160 . . . 4 (bc) ≤ b
98leror 152 . . 3 ((bc) ∪ (bc)) ≤ (b ∪ (bc))
10 u1lemab 610 . . 3 ((b1 c) ∩ c) = ((bc) ∪ (bc))
11 df-i1 44 . . . 4 (b1 c) = (b ∪ (bc))
12 ax-a1 30 . . . . . 6 b = b
1312ax-r5 38 . . . . 5 (b ∪ (bc)) = (b ∪ (bc))
1413ax-r1 35 . . . 4 (b ∪ (bc)) = (b ∪ (bc))
1511, 14ax-r2 36 . . 3 (b1 c) = (b ∪ (bc))
169, 10, 15le3tr1 140 . 2 ((b1 c) ∩ c) ≤ (b1 c)
177, 16letr 137 1 (ac) ≤ (b1 c)
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →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:  negantlem4  851
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