Theorem wleran 376

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

Step	Hyp	Ref	Expression
1		anandir 107	. . . . 5 ((a ^ b) ^ c) = ((a ^ c) ^ (b ^ c))
2	1	bi1 110	. . . 4 (((a ^ b) ^ c) == ((a ^ c) ^ (b ^ c))) = 1
3	2	wr1 189	. . 3 (((a ^ c) ^ (b ^ c)) == ((a ^ b) ^ c)) = 1
4		wle.1	. . . . 5 (a =<2 b) = 1
5	4	wdf2le2 368	. . . 4 ((a ^ b) == a) = 1
6	5	wran 351	. . 3 (((a ^ b) ^ c) == (a ^ c)) = 1
7	3, 6	wr2 353	. 2 (((a ^ c) ^ (b ^ c)) == (a ^ c)) = 1
8	7	wdf2le1 367	1 ((a ^ c) =<2 (b ^ c)) = 1

Syntax hints:   = wb 1   ^ wa 7  1wt 9   =<2 wle2 11

This theorem is referenced by:  wle2an 386

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-wom 343

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le 121  df-le1 122  df-le2 123

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