Theorem u4lemanb 618

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

Assertion

Ref	Expression
u4lemanb	((a →4 b) ∩ b⊥ ) = ((a⊥ ∪ b) ∩ b⊥ )

Proof of Theorem u4lemanb

Step	Hyp	Ref	Expression
1		df-i4 47	. . 3 (a →4 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
2	1	ran 78	. 2 ((a →4 b) ∩ b⊥ ) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∩ b⊥ )
3		comanr2 465	. . . . . 6 b C (a ∩ b)
4	3	comcom3 454	. . . . 5 b⊥ C (a ∩ b)
5		comanr2 465	. . . . . 6 b C (a⊥ ∩ b)
6	5	comcom3 454	. . . . 5 b⊥ C (a⊥ ∩ b)
7	4, 6	com2or 483	. . . 4 b⊥ C ((a ∩ b) ∪ (a⊥ ∩ b))
8		comorr2 463	. . . . . 6 b C (a⊥ ∪ b)
9	8	comcom3 454	. . . . 5 b⊥ C (a⊥ ∪ b)
10		comid 187	. . . . 5 b⊥ C b⊥
11	9, 10	com2an 484	. . . 4 b⊥ C ((a⊥ ∪ b) ∩ b⊥ )
12	7, 11	fh1r 473	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∩ b⊥ ) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∩ b⊥ ) ∪ (((a⊥ ∪ b) ∩ b⊥ ) ∩ b⊥ ))
13		ax-a2 31	. . . 4 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∩ b⊥ ) ∪ (((a⊥ ∪ b) ∩ b⊥ ) ∩ b⊥ )) = ((((a⊥ ∪ b) ∩ b⊥ ) ∩ b⊥ ) ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∩ b⊥ ))
14		anass 76	. . . . . . 7 (((a⊥ ∪ b) ∩ b⊥ ) ∩ b⊥ ) = ((a⊥ ∪ b) ∩ (b⊥ ∩ b⊥ ))
15		anidm 111	. . . . . . . 8 (b⊥ ∩ b⊥ ) = b⊥
16	15	lan 77	. . . . . . 7 ((a⊥ ∪ b) ∩ (b⊥ ∩ b⊥ )) = ((a⊥ ∪ b) ∩ b⊥ )
17	14, 16	ax-r2 36	. . . . . 6 (((a⊥ ∪ b) ∩ b⊥ ) ∩ b⊥ ) = ((a⊥ ∪ b) ∩ b⊥ )
18	4, 6	fh1r 473	. . . . . . 7 (((a ∩ b) ∪ (a⊥ ∩ b)) ∩ b⊥ ) = (((a ∩ b) ∩ b⊥ ) ∪ ((a⊥ ∩ b) ∩ b⊥ ))
19		anass 76	. . . . . . . . . 10 ((a ∩ b) ∩ b⊥ ) = (a ∩ (b ∩ b⊥ ))
20		dff 101	. . . . . . . . . . . . 13 0 = (b ∩ b⊥ )
21	20	lan 77	. . . . . . . . . . . 12 (a ∩ 0) = (a ∩ (b ∩ b⊥ ))
22	21	ax-r1 35	. . . . . . . . . . 11 (a ∩ (b ∩ b⊥ )) = (a ∩ 0)
23		an0 108	. . . . . . . . . . 11 (a ∩ 0) = 0
24	22, 23	ax-r2 36	. . . . . . . . . 10 (a ∩ (b ∩ b⊥ )) = 0
25	19, 24	ax-r2 36	. . . . . . . . 9 ((a ∩ b) ∩ b⊥ ) = 0
26		anass 76	. . . . . . . . . 10 ((a⊥ ∩ b) ∩ b⊥ ) = (a⊥ ∩ (b ∩ b⊥ ))
27	20	lan 77	. . . . . . . . . . . 12 (a⊥ ∩ 0) = (a⊥ ∩ (b ∩ b⊥ ))
28	27	ax-r1 35	. . . . . . . . . . 11 (a⊥ ∩ (b ∩ b⊥ )) = (a⊥ ∩ 0)
29		an0 108	. . . . . . . . . . 11 (a⊥ ∩ 0) = 0
30	28, 29	ax-r2 36	. . . . . . . . . 10 (a⊥ ∩ (b ∩ b⊥ )) = 0
31	26, 30	ax-r2 36	. . . . . . . . 9 ((a⊥ ∩ b) ∩ b⊥ ) = 0
32	25, 31	2or 72	. . . . . . . 8 (((a ∩ b) ∩ b⊥ ) ∪ ((a⊥ ∩ b) ∩ b⊥ )) = (0 ∪ 0)
33		or0 102	. . . . . . . 8 (0 ∪ 0) = 0
34	32, 33	ax-r2 36	. . . . . . 7 (((a ∩ b) ∩ b⊥ ) ∪ ((a⊥ ∩ b) ∩ b⊥ )) = 0
35	18, 34	ax-r2 36	. . . . . 6 (((a ∩ b) ∪ (a⊥ ∩ b)) ∩ b⊥ ) = 0
36	17, 35	2or 72	. . . . 5 ((((a⊥ ∪ b) ∩ b⊥ ) ∩ b⊥ ) ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∩ b⊥ )) = (((a⊥ ∪ b) ∩ b⊥ ) ∪ 0)
37		or0 102	. . . . 5 (((a⊥ ∪ b) ∩ b⊥ ) ∪ 0) = ((a⊥ ∪ b) ∩ b⊥ )
38	36, 37	ax-r2 36	. . . 4 ((((a⊥ ∪ b) ∩ b⊥ ) ∩ b⊥ ) ∪ (((a ∩ b) ∪ (a⊥ ∩ b)) ∩ b⊥ )) = ((a⊥ ∪ b) ∩ b⊥ )
39	13, 38	ax-r2 36	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∩ b⊥ ) ∪ (((a⊥ ∪ b) ∩ b⊥ ) ∩ b⊥ )) = ((a⊥ ∪ b) ∩ b⊥ )
40	12, 39	ax-r2 36	. 2 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∩ b⊥ ) = ((a⊥ ∪ b) ∩ b⊥ )
41	2, 40	ax-r2 36	1 ((a →4 b) ∩ b⊥ ) = ((a⊥ ∪ b) ∩ b⊥ )

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →₄ wi4 15

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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u4lemnob  673  u24lem  770