Theorem u3lemana 607

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

Assertion

Ref	Expression
u3lemana	((a →3 b) ∩ a⊥ ) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))

Proof of Theorem u3lemana

Step	Hyp	Ref	Expression
1		df-i3 46	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
2	1	ran 78	. 2 ((a →3 b) ∩ a⊥ ) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∩ a⊥ )
3		comanr1 464	. . . . 5 a⊥ C (a⊥ ∩ b)
4		comanr1 464	. . . . 5 a⊥ C (a⊥ ∩ b⊥ )
5	3, 4	com2or 483	. . . 4 a⊥ C ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
6		comid 187	. . . . . 6 a C a
7	6	comcom3 454	. . . . 5 a⊥ C a
8		comorr 184	. . . . 5 a⊥ C (a⊥ ∪ b)
9	7, 8	com2an 484	. . . 4 a⊥ C (a ∩ (a⊥ ∪ b))
10	5, 9	fh1r 473	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∩ a⊥ ) = ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ a⊥ ) ∪ ((a ∩ (a⊥ ∪ b)) ∩ a⊥ ))
11		lea 160	. . . . . . 7 (a⊥ ∩ b) ≤ a⊥
12		lea 160	. . . . . . 7 (a⊥ ∩ b⊥ ) ≤ a⊥
13	11, 12	lel2or 170	. . . . . 6 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ a⊥
14	13	df2le2 136	. . . . 5 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ a⊥ ) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
15		an32 83	. . . . . 6 ((a ∩ (a⊥ ∪ b)) ∩ a⊥ ) = ((a ∩ a⊥ ) ∩ (a⊥ ∪ b))
16		ancom 74	. . . . . . 7 ((a ∩ a⊥ ) ∩ (a⊥ ∪ b)) = ((a⊥ ∪ b) ∩ (a ∩ a⊥ ))
17		dff 101	. . . . . . . . . 10 0 = (a ∩ a⊥ )
18	17	ax-r1 35	. . . . . . . . 9 (a ∩ a⊥ ) = 0
19	18	lan 77	. . . . . . . 8 ((a⊥ ∪ b) ∩ (a ∩ a⊥ )) = ((a⊥ ∪ b) ∩ 0)
20		an0 108	. . . . . . . 8 ((a⊥ ∪ b) ∩ 0) = 0
21	19, 20	ax-r2 36	. . . . . . 7 ((a⊥ ∪ b) ∩ (a ∩ a⊥ )) = 0
22	16, 21	ax-r2 36	. . . . . 6 ((a ∩ a⊥ ) ∩ (a⊥ ∪ b)) = 0
23	15, 22	ax-r2 36	. . . . 5 ((a ∩ (a⊥ ∪ b)) ∩ a⊥ ) = 0
24	14, 23	2or 72	. . . 4 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ a⊥ ) ∪ ((a ∩ (a⊥ ∪ b)) ∩ a⊥ )) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ 0)
25		or0 102	. . . 4 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ 0) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
26	24, 25	ax-r2 36	. . 3 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∩ a⊥ ) ∪ ((a ∩ (a⊥ ∪ b)) ∩ a⊥ )) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
27	10, 26	ax-r2 36	. 2 ((((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ∩ a⊥ ) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
28	2, 27	ax-r2 36	1 ((a →3 b) ∩ a⊥ ) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →₃ wi3 14

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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u3lemnoa  662  u3lem13a  789  u3lem13b  790