Theorem comi12 707

Description: Commutation theorem for →₁ and →₂ . (Contributed by NM, 5-Jul-2000.)

Assertion

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

Proof of Theorem comi12

Step	Hyp	Ref	Expression
1		df-i1 44	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
2		lea 160	. . . . . . . 8 (a⊥ ∩ (c⊥ ∩ a⊥ )⊥ ) ≤ a⊥
3		leo 158	. . . . . . . 8 a⊥ ≤ (a⊥ ∪ (a ∩ b))
4	2, 3	letr 137	. . . . . . 7 (a⊥ ∩ (c⊥ ∩ a⊥ )⊥ ) ≤ (a⊥ ∪ (a ∩ b))
5	4	lecom 180	. . . . . 6 (a⊥ ∩ (c⊥ ∩ a⊥ )⊥ ) C (a⊥ ∪ (a ∩ b))
6	5	comcom 453	. . . . 5 (a⊥ ∪ (a ∩ b)) C (a⊥ ∩ (c⊥ ∩ a⊥ )⊥ )
7		anor3 90	. . . . 5 (a⊥ ∩ (c⊥ ∩ a⊥ )⊥ ) = (a ∪ (c⊥ ∩ a⊥ ))⊥
8	6, 7	cbtr 182	. . . 4 (a⊥ ∪ (a ∩ b)) C (a ∪ (c⊥ ∩ a⊥ ))⊥
9	8	comcom7 460	. . 3 (a⊥ ∪ (a ∩ b)) C (a ∪ (c⊥ ∩ a⊥ ))
10		df-i2 45	. . . 4 (c →2 a) = (a ∪ (c⊥ ∩ a⊥ ))
11	10	ax-r1 35	. . 3 (a ∪ (c⊥ ∩ a⊥ )) = (c →2 a)
12	9, 11	cbtr 182	. 2 (a⊥ ∪ (a ∩ b)) C (c →2 a)
13	1, 12	bctr 181	1 (a →1 b) C (c →2 a)

This proof depends on syntax axioms:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →₂ wi2 13

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  ax-r3 439

This proof depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is used by:  orbi  842