Theorem wle2an 404

Description: Conjunction of 2 l.e.'s. (Contributed by NM, 13-Oct-1997.)

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

Step	Hyp	Ref	Expression
1		wle2.1	. . 3 (a ≤2 b) = 1
2	1	wleran 394	. 2 ((a ∩ c) ≤2 (b ∩ c)) = 1
3		wle2.2	. . . 4 (c ≤2 d) = 1
4	3	wleran 394	. . 3 ((c ∩ b) ≤2 (d ∩ b)) = 1
5		ancom 74	. . . 4 (b ∩ c) = (c ∩ b)
6	5	bi1 118	. . 3 ((b ∩ c) ≡ (c ∩ b)) = 1
7		ancom 74	. . . 4 (b ∩ d) = (d ∩ b)
8	7	bi1 118	. . 3 ((b ∩ d) ≡ (d ∩ b)) = 1
9	4, 6, 8	wle3tr1 399	. 2 ((b ∩ c) ≤2 (b ∩ d)) = 1
10	2, 9	wletr 396	1 ((a ∩ c) ≤2 (b ∩ d)) = 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:  wledi  405  wledio  406  wlem14  430