Theorem u4lem1 737

Description: Lemma for unified implication study. (Contributed by NM, 16-Dec-1997.)

Assertion

Ref	Expression
u4lem1	((a →4 b) →4 a) = ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))

Proof of Theorem u4lem1

Step	Hyp	Ref	Expression
1		df-i4 47	. 2 ((a →4 b) →4 a) = ((((a →4 b) ∩ a) ∪ ((a →4 b)⊥ ∩ a)) ∪ (((a →4 b)⊥ ∪ a) ∩ a⊥ ))
2		u4lemaa 603	. . . . 5 ((a →4 b) ∩ a) = (a ∩ b)
3		u4lemnaa 643	. . . . 5 ((a →4 b)⊥ ∩ a) = (a ∩ b⊥ )
4	2, 3	2or 72	. . . 4 (((a →4 b) ∩ a) ∪ ((a →4 b)⊥ ∩ a)) = ((a ∩ b) ∪ (a ∩ b⊥ ))
5		u4lemnoa 663	. . . . 5 ((a →4 b)⊥ ∪ a) = ((a ∪ b) ∩ (a ∪ b⊥ ))
6	5	ran 78	. . . 4 (((a →4 b)⊥ ∪ a) ∩ a⊥ ) = (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ )
7	4, 6	2or 72	. . 3 ((((a →4 b) ∩ a) ∪ ((a →4 b)⊥ ∩ a)) ∪ (((a →4 b)⊥ ∪ a) ∩ a⊥ )) = (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ ))
8		ancom 74	. . . . 5 (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ ) = (a⊥ ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))
9	8	lor 70	. . . 4 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ )) = (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (a⊥ ∩ ((a ∪ b) ∩ (a ∪ b⊥ ))))
10		comanr1 464	. . . . . . . 8 a C (a ∩ b)
11		comanr1 464	. . . . . . . 8 a C (a ∩ b⊥ )
12	10, 11	com2or 483	. . . . . . 7 a C ((a ∩ b) ∪ (a ∩ b⊥ ))
13	12	comcom3 454	. . . . . 6 a⊥ C ((a ∩ b) ∪ (a ∩ b⊥ ))
14		comorr 184	. . . . . . . 8 a C (a ∪ b)
15		comorr 184	. . . . . . . 8 a C (a ∪ b⊥ )
16	14, 15	com2an 484	. . . . . . 7 a C ((a ∪ b) ∩ (a ∪ b⊥ ))
17	16	comcom3 454	. . . . . 6 a⊥ C ((a ∪ b) ∩ (a ∪ b⊥ ))
18	13, 17	fh4 472	. . . . 5 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (a⊥ ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))) = ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) ∩ (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a ∪ b) ∩ (a ∪ b⊥ ))))
19		comor1 461	. . . . . . . . . . 11 (a ∪ b) C a
20		comor2 462	. . . . . . . . . . 11 (a ∪ b) C b
21	19, 20	com2an 484	. . . . . . . . . 10 (a ∪ b) C (a ∩ b)
22	20	comcom2 183	. . . . . . . . . . 11 (a ∪ b) C b⊥
23	19, 22	com2an 484	. . . . . . . . . 10 (a ∪ b) C (a ∩ b⊥ )
24	21, 23	com2or 483	. . . . . . . . 9 (a ∪ b) C ((a ∩ b) ∪ (a ∩ b⊥ ))
25	19, 22	com2or 483	. . . . . . . . 9 (a ∪ b) C (a ∪ b⊥ )
26	24, 25	fh4 472	. . . . . . . 8 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a ∪ b) ∩ (a ∪ b⊥ ))) = ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (a ∪ b)) ∩ (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (a ∪ b⊥ )))
27		lea 160	. . . . . . . . . . . 12 (a ∩ b) ≤ a
28		lea 160	. . . . . . . . . . . 12 (a ∩ b⊥ ) ≤ a
29	27, 28	lel2or 170	. . . . . . . . . . 11 ((a ∩ b) ∪ (a ∩ b⊥ )) ≤ a
30		leo 158	. . . . . . . . . . 11 a ≤ (a ∪ b)
31	29, 30	letr 137	. . . . . . . . . 10 ((a ∩ b) ∪ (a ∩ b⊥ )) ≤ (a ∪ b)
32	31	df-le2 131	. . . . . . . . 9 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (a ∪ b)) = (a ∪ b)
33		leo 158	. . . . . . . . . . 11 a ≤ (a ∪ b⊥ )
34	29, 33	letr 137	. . . . . . . . . 10 ((a ∩ b) ∪ (a ∩ b⊥ )) ≤ (a ∪ b⊥ )
35	34	df-le2 131	. . . . . . . . 9 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (a ∪ b⊥ )) = (a ∪ b⊥ )
36	32, 35	2an 79	. . . . . . . 8 ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (a ∪ b)) ∩ (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (a ∪ b⊥ ))) = ((a ∪ b) ∩ (a ∪ b⊥ ))
37	26, 36	ax-r2 36	. . . . . . 7 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a ∪ b) ∩ (a ∪ b⊥ ))) = ((a ∪ b) ∩ (a ∪ b⊥ ))
38	37	lan 77	. . . . . 6 ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) ∩ (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a ∪ b) ∩ (a ∪ b⊥ )))) = ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))
39		id 59	. . . . . 6 ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) ∩ ((a ∪ b) ∩ (a ∪ b⊥ ))) = ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))
40	38, 39	ax-r2 36	. . . . 5 ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) ∩ (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a ∪ b) ∩ (a ∪ b⊥ )))) = ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))
41	18, 40	ax-r2 36	. . . 4 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (a⊥ ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))) = ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))
42	9, 41	ax-r2 36	. . 3 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ a⊥ )) = ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))
43	7, 42	ax-r2 36	. 2 ((((a →4 b) ∩ a) ∪ ((a →4 b)⊥ ∩ a)) ∪ (((a →4 b)⊥ ∪ a) ∩ a⊥ )) = ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))
44	1, 43	ax-r2 36	1 ((a →4 b) →4 a) = ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →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:  u4lem1n  742