Theorem comi1 691

Description: Commutation expressed with →₁ .

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 68	. . . . 5 (b ∩ a) = (a ∩ b)
2	1	ax-r5 37	. . . 4 ((b ∩ a) ∪ (b ∩ a⊥ )) = ((a ∩ b) ∪ (b ∩ a⊥ ))
3		ax-a2 30	. . . 4 ((a ∩ b) ∪ (b ∩ a⊥ )) = ((b ∩ a⊥ ) ∪ (a ∩ b))
4	2, 3	ax-r2 35	. . 3 ((b ∩ a) ∪ (b ∩ a⊥ )) = ((b ∩ a⊥ ) ∪ (a ∩ b))
5		lear 153	. . . 4 (b ∩ a⊥ ) ≤ a⊥
6	5	leror 144	. . 3 ((b ∩ a⊥ ) ∪ (a ∩ b)) ≤ (a⊥ ∪ (a ∩ b))
7	4, 6	bltr 130	. 2 ((b ∩ a) ∪ (b ∩ a⊥ )) ≤ (a⊥ ∪ (a ∩ b))
8		comi1.1	. . . 4 a C b
9	8	comcom 435	. . 3 b C a
10	9	df-c2 125	. 2 b = ((b ∩ a) ∪ (b ∩ a⊥ ))
11		df-i1 43	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
12	7, 10, 11	le3tr1 132	1 b ≤ (a →1 b)

Syntax hints:   ≤ wle 2   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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