Theorem comi1 709

Description: Commutation expressed with →₁ . (Contributed by NM, 1-Dec-1999.)

Hypothesis

Ref	Expression
comi1.1	a C b

Assertion

Ref	Expression
comi1	b ≤ (a →1 b)

Proof of Theorem comi1

Step	Hyp	Ref	Expression
1		ancom 74	. . . . 5 (b ∩ a) = (a ∩ b)
2	1	ax-r5 38	. . . 4 ((b ∩ a) ∪ (b ∩ a⊥ )) = ((a ∩ b) ∪ (b ∩ a⊥ ))
3		ax-a2 31	. . . 4 ((a ∩ b) ∪ (b ∩ a⊥ )) = ((b ∩ a⊥ ) ∪ (a ∩ b))
4	2, 3	ax-r2 36	. . 3 ((b ∩ a) ∪ (b ∩ a⊥ )) = ((b ∩ a⊥ ) ∪ (a ∩ b))
5		lear 161	. . . 4 (b ∩ a⊥ ) ≤ a⊥
6	5	leror 152	. . 3 ((b ∩ a⊥ ) ∪ (a ∩ b)) ≤ (a⊥ ∪ (a ∩ b))
7	4, 6	bltr 138	. 2 ((b ∩ a) ∪ (b ∩ a⊥ )) ≤ (a⊥ ∪ (a ∩ b))
8		comi1.1	. . . 4 a C b
9	8	comcom 453	. . 3 b C a
10	9	df-c2 133	. 2 b = ((b ∩ a) ∪ (b ∩ a⊥ ))
11		df-i1 44	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
12	7, 10, 11	le3tr1 140	1 b ≤ (a →1 b)

Syntax hints:   ≤ wle 2   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12

This theorem was proved from 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  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)