Theorem wleran 394

Description: Add conjunct to right of both sides. (Contributed by NM, 13-Oct-1997.)

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 115	. . . . 5 ((a ∩ b) ∩ c) = ((a ∩ c) ∩ (b ∩ c))
2	1	bi1 118	. . . 4 (((a ∩ b) ∩ c) ≡ ((a ∩ c) ∩ (b ∩ c))) = 1
3	2	wr1 197	. . 3 (((a ∩ c) ∩ (b ∩ c)) ≡ ((a ∩ b) ∩ c)) = 1
4		wle.1	. . . . 5 (a ≤2 b) = 1
5	4	wdf2le2 386	. . . 4 ((a ∩ b) ≡ a) = 1
6	5	wran 369	. . 3 (((a ∩ b) ∩ c) ≡ (a ∩ c)) = 1
7	3, 6	wr2 371	. 2 (((a ∩ c) ∩ (b ∩ c)) ≡ (a ∩ c)) = 1
8	7	wdf2le1 385	1 ((a ∩ c) ≤2 (b ∩ c)) = 1

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