Theorem wle2an 386

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

Step	Hyp	Ref	Expression
1		wle2.1	. . 3 (a ≤2 b) = 1
2	1	wleran 376	. 2 ((a ∩ c) ≤2 (b ∩ c)) = 1
3		wle2.2	. . . 4 (c ≤2 d) = 1
4	3	wleran 376	. . 3 ((c ∩ b) ≤2 (d ∩ b)) = 1
5		ancom 68	. . . 4 (b ∩ c) = (c ∩ b)
6	5	bi1 110	. . 3 ((b ∩ c) ≡ (c ∩ b)) = 1
7		ancom 68	. . . 4 (b ∩ d) = (d ∩ b)
8	7	bi1 110	. . 3 ((b ∩ d) ≡ (d ∩ b)) = 1
9	4, 6, 8	wle3tr1 381	. 2 ((b ∩ c) ≤2 (b ∩ d)) = 1
10	2, 9	wletr 378	1 ((a ∩ c) ≤2 (b ∩ d)) = 1

Syntax hints:   = wb 1   ∩ wa 7  1wt 9   ≤₂ wle2 11

This theorem is referenced by:  wledi 387  wledio 388  wlem14 412

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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