Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  negantlem9 GIF version

Theorem negantlem9 859
 Description: Negated antecedent identity.
Hypothesis
Ref Expression
negant.1 (a1 c) = (b1 c)
Assertion
Ref Expression
negantlem9 (a3 c) ≤ (b3 c)

Proof of Theorem negantlem9
StepHypRef Expression
1 leao4 165 . . . . 5 (ac) ≤ (bc)
2 leor 159 . . . . . 6 (ac) ≤ (a ∪ (ac))
3 negant.1 . . . . . . . . 9 (a1 c) = (b1 c)
43sac 835 . . . . . . . 8 (a1 c) = (b1 c)
5 df-i1 44 . . . . . . . . 9 (a1 c) = (a ∪ (ac))
6 ax-a1 30 . . . . . . . . . . 11 a = a
76ax-r5 38 . . . . . . . . . 10 (a ∪ (ac)) = (a ∪ (ac))
87ax-r1 35 . . . . . . . . 9 (a ∪ (ac)) = (a ∪ (ac))
95, 8ax-r2 36 . . . . . . . 8 (a1 c) = (a ∪ (ac))
10 df-i1 44 . . . . . . . . 9 (b1 c) = (b ∪ (bc))
11 ax-a1 30 . . . . . . . . . . 11 b = b
1211ax-r5 38 . . . . . . . . . 10 (b ∪ (bc)) = (b ∪ (bc))
1312ax-r1 35 . . . . . . . . 9 (b ∪ (bc)) = (b ∪ (bc))
1410, 13ax-r2 36 . . . . . . . 8 (b1 c) = (b ∪ (bc))
154, 9, 143tr2 64 . . . . . . 7 (a ∪ (ac)) = (b ∪ (bc))
16 leo 158 . . . . . . . 8 b ≤ (b ∪ (bc ))
1716leror 152 . . . . . . 7 (b ∪ (bc)) ≤ ((b ∪ (bc )) ∪ (bc))
1815, 17bltr 138 . . . . . 6 (a ∪ (ac)) ≤ ((b ∪ (bc )) ∪ (bc))
192, 18letr 137 . . . . 5 (ac) ≤ ((b ∪ (bc )) ∪ (bc))
201, 19ler2an 173 . . . 4 (ac) ≤ ((bc) ∩ ((b ∪ (bc )) ∪ (bc)))
21 leao1 162 . . . . . 6 (ac ) ≤ (ac)
223negantlem8 856 . . . . . 6 (ac) = (bc)
2321, 22lbtr 139 . . . . 5 (ac ) ≤ (bc)
243negantlem5 853 . . . . . 6 (ac ) = (bc )
25 leor 159 . . . . . . 7 (bc ) ≤ (b ∪ (bc ))
2625ler 149 . . . . . 6 (bc ) ≤ ((b ∪ (bc )) ∪ (bc))
2724, 26bltr 138 . . . . 5 (ac ) ≤ ((b ∪ (bc )) ∪ (bc))
2823, 27ler2an 173 . . . 4 (ac ) ≤ ((bc) ∩ ((b ∪ (bc )) ∪ (bc)))
2920, 28lel2or 170 . . 3 ((ac) ∪ (ac )) ≤ ((bc) ∩ ((b ∪ (bc )) ∪ (bc)))
30 lear 161 . . . . 5 (a ∩ (ac)) ≤ (ac)
3130, 22lbtr 139 . . . 4 (a ∩ (ac)) ≤ (bc)
32 leo 158 . . . . . . 7 a ≤ (a ∪ (ac))
3332, 15lbtr 139 . . . . . 6 a ≤ (b ∪ (bc))
3433, 17letr 137 . . . . 5 a ≤ ((b ∪ (bc )) ∪ (bc))
3534lel 151 . . . 4 (a ∩ (ac)) ≤ ((b ∪ (bc )) ∪ (bc))
3631, 35ler2an 173 . . 3 (a ∩ (ac)) ≤ ((bc) ∩ ((b ∪ (bc )) ∪ (bc)))
3729, 36lel2or 170 . 2 (((ac) ∪ (ac )) ∪ (a ∩ (ac))) ≤ ((bc) ∩ ((b ∪ (bc )) ∪ (bc)))
38 df-i3 46 . 2 (a3 c) = (((ac) ∪ (ac )) ∪ (a ∩ (ac)))
39 dfi3b 499 . 2 (b3 c) = ((bc) ∩ ((b ∪ (bc )) ∪ (bc)))
4037, 38, 39le3tr1 140 1 (a3 c) ≤ (b3 c)
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12   →3 wi3 14 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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  negant3  860
 Copyright terms: Public domain W3C validator