Theorem u2lemana 606

Description: Lemma for Dishkant implication study. (Contributed by NM, 14-Dec-1997.)

Assertion

Ref	Expression
u2lemana	((a →2 b) ∩ a⊥ ) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))

Proof of Theorem u2lemana

Step	Hyp	Ref	Expression
1		df-i2 45	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
2	1	ran 78	. 2 ((a →2 b) ∩ a⊥ ) = ((b ∪ (a⊥ ∩ b⊥ )) ∩ a⊥ )
3		ax-a2 31	. . . 4 (b ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ b)
4	3	ran 78	. . 3 ((b ∪ (a⊥ ∩ b⊥ )) ∩ a⊥ ) = (((a⊥ ∩ b⊥ ) ∪ b) ∩ a⊥ )
5		coman1 185	. . . . 5 (a⊥ ∩ b⊥ ) C a⊥
6		coman2 186	. . . . . 6 (a⊥ ∩ b⊥ ) C b⊥
7	6	comcom7 460	. . . . 5 (a⊥ ∩ b⊥ ) C b
8	5, 7	fh2r 474	. . . 4 (((a⊥ ∩ b⊥ ) ∪ b) ∩ a⊥ ) = (((a⊥ ∩ b⊥ ) ∩ a⊥ ) ∪ (b ∩ a⊥ ))
9		an32 83	. . . . . . 7 ((a⊥ ∩ b⊥ ) ∩ a⊥ ) = ((a⊥ ∩ a⊥ ) ∩ b⊥ )
10		anidm 111	. . . . . . . 8 (a⊥ ∩ a⊥ ) = a⊥
11	10	ran 78	. . . . . . 7 ((a⊥ ∩ a⊥ ) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
12	9, 11	ax-r2 36	. . . . . 6 ((a⊥ ∩ b⊥ ) ∩ a⊥ ) = (a⊥ ∩ b⊥ )
13		ancom 74	. . . . . 6 (b ∩ a⊥ ) = (a⊥ ∩ b)
14	12, 13	2or 72	. . . . 5 (((a⊥ ∩ b⊥ ) ∩ a⊥ ) ∪ (b ∩ a⊥ )) = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b))
15		ax-a2 31	. . . . 5 ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b)) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
16	14, 15	ax-r2 36	. . . 4 (((a⊥ ∩ b⊥ ) ∩ a⊥ ) ∪ (b ∩ a⊥ )) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
17	8, 16	ax-r2 36	. . 3 (((a⊥ ∩ b⊥ ) ∪ b) ∩ a⊥ ) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
18	4, 17	ax-r2 36	. 2 ((b ∪ (a⊥ ∩ b⊥ )) ∩ a⊥ ) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
19	2, 18	ax-r2 36	1 ((a →2 b) ∩ a⊥ ) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u2lemnoa  661