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Theorem negantlem2 849
 Description: Lemma for negated antecedent identity.
Hypothesis
Ref Expression
negant.1 (a1 c) = (b1 c)
Assertion
Ref Expression
negantlem2 a ≤ (b1 c)

Proof of Theorem negantlem2
StepHypRef Expression
1 leo 158 . 2 a ≤ (a ∪ (b1 c))
2 i1orni1 847 . . . . . 6 ((b1 c) ∪ (b1 c)) = 1
32lan 77 . . . . 5 ((a ∪ (b1 c)) ∩ ((b1 c) ∪ (b1 c))) = ((a ∪ (b1 c)) ∩ 1)
43ax-r1 35 . . . 4 ((a ∪ (b1 c)) ∩ 1) = ((a ∪ (b1 c)) ∩ ((b1 c) ∪ (b1 c)))
5 an1 106 . . . . 5 ((a ∪ (b1 c)) ∩ 1) = (a ∪ (b1 c))
65ax-r1 35 . . . 4 (a ∪ (b1 c)) = ((a ∪ (b1 c)) ∩ 1)
7 u1lemc6 706 . . . . 5 (b1 c) C (b1 c)
8 negant.1 . . . . . . 7 (a1 c) = (b1 c)
98negantlem1 848 . . . . . 6 a C (b1 c)
109comcom 453 . . . . 5 (b1 c) C a
117, 10fh4rc 482 . . . 4 ((a ∩ (b1 c)) ∪ (b1 c)) = ((a ∪ (b1 c)) ∩ ((b1 c) ∪ (b1 c)))
124, 6, 113tr1 63 . . 3 (a ∪ (b1 c)) = ((a ∩ (b1 c)) ∪ (b1 c))
13 ancom 74 . . . . . . . 8 (a ∩ (a1 c)) = ((a1 c) ∩ a)
148lan 77 . . . . . . . 8 (a ∩ (a1 c)) = (a ∩ (b1 c))
15 u1lemaa 600 . . . . . . . 8 ((a1 c) ∩ a) = (ac)
1613, 14, 153tr2 64 . . . . . . 7 (a ∩ (b1 c)) = (ac)
17 lear 161 . . . . . . 7 (ac) ≤ c
1816, 17bltr 138 . . . . . 6 (a ∩ (b1 c)) ≤ c
19 lear 161 . . . . . 6 (a ∩ (b1 c)) ≤ (b1 c)
2018, 19ler2an 173 . . . . 5 (a ∩ (b1 c)) ≤ (c ∩ (b1 c))
21 lea 160 . . . . . . . 8 (bc) ≤ b
22 ax-a1 30 . . . . . . . 8 b = b
2321, 22lbtr 139 . . . . . . 7 (bc) ≤ b
2423leror 152 . . . . . 6 ((bc) ∪ (bc)) ≤ (b ∪ (bc))
25 ancom 74 . . . . . . 7 (c ∩ (b1 c)) = ((b1 c) ∩ c)
26 u1lemab 610 . . . . . . 7 ((b1 c) ∩ c) = ((bc) ∪ (bc))
2725, 26ax-r2 36 . . . . . 6 (c ∩ (b1 c)) = ((bc) ∪ (bc))
28 df-i1 44 . . . . . 6 (b1 c) = (b ∪ (bc))
2924, 27, 28le3tr1 140 . . . . 5 (c ∩ (b1 c)) ≤ (b1 c)
3020, 29letr 137 . . . 4 (a ∩ (b1 c)) ≤ (b1 c)
31 leid 148 . . . 4 (b1 c) ≤ (b1 c)
3230, 31lel2or 170 . . 3 ((a ∩ (b1 c)) ∪ (b1 c)) ≤ (b1 c)
3312, 32bltr 138 . 2 (a ∪ (b1 c)) ≤ (b1 c)
341, 33letr 137 1 a ≤ (b1 c)
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →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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