Theorem comi12 689

Description: Commutation theorem for →₁ and →₂ .

Assertion

Ref	Expression
comi12	(a →1 b) C (c →2 a)

Proof of Theorem comi12

Step	Hyp	Ref	Expression
1		df-i1 43	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
2		lea 152	. . . . . . . 8 (a⊥ ∩ (c⊥ ∩ a⊥ )⊥ ) ≤ a⊥
3		leo 150	. . . . . . . 8 a⊥ ≤ (a⊥ ∪ (a ∩ b))
4	2, 3	letr 129	. . . . . . 7 (a⊥ ∩ (c⊥ ∩ a⊥ )⊥ ) ≤ (a⊥ ∪ (a ∩ b))
5	4	lecom 172	. . . . . 6 (a⊥ ∩ (c⊥ ∩ a⊥ )⊥ ) C (a⊥ ∪ (a ∩ b))
6	5	comcom 435	. . . . 5 (a⊥ ∪ (a ∩ b)) C (a⊥ ∩ (c⊥ ∩ a⊥ )⊥ )
7		anor3 82	. . . . 5 (a⊥ ∩ (c⊥ ∩ a⊥ )⊥ ) = (a ∪ (c⊥ ∩ a⊥ ))⊥
8	6, 7	cbtr 174	. . . 4 (a⊥ ∪ (a ∩ b)) C (a ∪ (c⊥ ∩ a⊥ ))⊥
9	8	comcom7 442	. . 3 (a⊥ ∪ (a ∩ b)) C (a ∪ (c⊥ ∩ a⊥ ))
10		df-i2 44	. . . 4 (c →2 a) = (a ∪ (c⊥ ∩ a⊥ ))
11	10	ax-r1 34	. . 3 (a ∪ (c⊥ ∩ a⊥ )) = (c →2 a)
12	9, 11	cbtr 174	. 2 (a⊥ ∪ (a ∩ b)) C (c →2 a)
13	1, 12	bctr 173	1 (a →1 b) C (c →2 a)

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13   →₂ wi2 14

This theorem is referenced by:  orbi 824

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-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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