Theorem lel 151

Description: Add conjunct to left of l.e. (Contributed by NM, 27-Aug-1997.)

Hypothesis

Ref	Expression
le.1	a ≤ b

Assertion

Ref	Expression
lel	(a ∩ c) ≤ b

Proof of Theorem lel

Step	Hyp	Ref	Expression
1		an32 83	. . 3 ((a ∩ c) ∩ b) = ((a ∩ b) ∩ c)
2		le.1	. . . . 5 a ≤ b
3	2	df2le2 136	. . . 4 (a ∩ b) = a
4	3	ran 78	. . 3 ((a ∩ b) ∩ c) = (a ∩ c)
5	1, 4	ax-r2 36	. 2 ((a ∩ c) ∩ b) = (a ∩ c)
6	5	df2le1 135	1 (a ∩ c) ≤ b

This proof depends on syntax axioms:   ≤ wle 2   ∩ wa 7

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This proof depends on definitions:  df-a 40  df-le1 130  df-le2 131

This theorem is used by:  negantlem9  859  neg3antlem2  865  marsdenlem3  882  cancellem  891  oa3moa3  1029  lem3.4.3  1076