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Theorem wleran 394
Description: Add conjunct to right of both sides.
Hypothesis
Ref Expression
wle.1 (a =<2 b) = 1
Assertion
Ref Expression
wleran ((a ^ c) =<2 (b ^ c)) = 1
Proof of Theorem wleran
StepHypRef Expression
1 anandir 115 . . . . 5 ((a ^ b) ^ c) = ((a ^ c) ^ (b ^ c))
21bi1 118 . . . 4 (((a ^ b) ^ c) == ((a ^ c) ^ (b ^ c))) = 1
32wr1 197 . . 3 (((a ^ c) ^ (b ^ c)) == ((a ^ b) ^ c)) = 1
4 wle.1 . . . . 5 (a =<2 b) = 1
54wdf2le2 386 . . . 4 ((a ^ b) == a) = 1
65wran 369 . . 3 (((a ^ b) ^ c) == (a ^ c)) = 1
73, 6wr2 371 . 2 (((a ^ c) ^ (b ^ c)) == (a ^ c)) = 1
87wdf2le1 385 1 ((a ^ c) =<2 (b ^ c)) = 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:  wle2an  404
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