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Theorem wle2an 404
Description: Conjunction of 2 l.e.'s.
Hypotheses
Ref Expression
wle2.1 (a =<2 b) = 1
wle2.2 (c =<2 d) = 1
Assertion
Ref Expression
wle2an ((a ^ c) =<2 (b ^ d)) = 1
Proof of Theorem wle2an
StepHypRef Expression
1 wle2.1 . . 3 (a =<2 b) = 1
21wleran 394 . 2 ((a ^ c) =<2 (b ^ c)) = 1
3 wle2.2 . . . 4 (c =<2 d) = 1
43wleran 394 . . 3 ((c ^ b) =<2 (d ^ b)) = 1
5 ancom 74 . . . 4 (b ^ c) = (c ^ b)
65bi1 118 . . 3 ((b ^ c) == (c ^ b)) = 1
7 ancom 74 . . . 4 (b ^ d) = (d ^ b)
87bi1 118 . . 3 ((b ^ d) == (d ^ b)) = 1
94, 6, 8wle3tr1 399 . 2 ((b ^ c) =<2 (b ^ d)) = 1
102, 9wletr 396 1 ((a ^ c) =<2 (b ^ d)) = 1
Colors of variables: term
Syntax hints:   = wb 1   ^ 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:  wledi  405  wledio  406  wlem14  430
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