Theorem u4lemc4 704

Description: Lemma for non-tollens implication study. (Contributed by NM, 24-Dec-1997.)

Hypothesis

Ref	Expression
ulemc3.1	a C b

Assertion

Ref	Expression
u4lemc4	(a →4 b) = (a⊥ ∪ b)

Proof of Theorem u4lemc4

Step	Hyp	Ref	Expression
1		df-i4 47	. 2 (a →4 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
2		ulemc3.1	. . . . . . 7 a C b
3		comid 187	. . . . . . . 8 a C a
4	3	comcom2 183	. . . . . . 7 a C a⊥
5	2, 4	fh2r 474	. . . . . 6 ((a ∪ a⊥ ) ∩ b) = ((a ∩ b) ∪ (a⊥ ∩ b))
6	5	ax-r1 35	. . . . 5 ((a ∩ b) ∪ (a⊥ ∩ b)) = ((a ∪ a⊥ ) ∩ b)
7		ancom 74	. . . . . 6 ((a ∪ a⊥ ) ∩ b) = (b ∩ (a ∪ a⊥ ))
8		df-t 41	. . . . . . . . 9 1 = (a ∪ a⊥ )
9	8	ax-r1 35	. . . . . . . 8 (a ∪ a⊥ ) = 1
10	9	lan 77	. . . . . . 7 (b ∩ (a ∪ a⊥ )) = (b ∩ 1)
11		an1 106	. . . . . . 7 (b ∩ 1) = b
12	10, 11	ax-r2 36	. . . . . 6 (b ∩ (a ∪ a⊥ )) = b
13	7, 12	ax-r2 36	. . . . 5 ((a ∪ a⊥ ) ∩ b) = b
14	6, 13	ax-r2 36	. . . 4 ((a ∩ b) ∪ (a⊥ ∩ b)) = b
15	2	comcom4 455	. . . . . 6 a⊥ C b⊥
16	2	comcom3 454	. . . . . 6 a⊥ C b
17	15, 16	fh2r 474	. . . . 5 ((a⊥ ∪ b) ∩ b⊥ ) = ((a⊥ ∩ b⊥ ) ∪ (b ∩ b⊥ ))
18		dff 101	. . . . . . . 8 0 = (b ∩ b⊥ )
19	18	ax-r1 35	. . . . . . 7 (b ∩ b⊥ ) = 0
20	19	lor 70	. . . . . 6 ((a⊥ ∩ b⊥ ) ∪ (b ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ 0)
21		or0 102	. . . . . 6 ((a⊥ ∩ b⊥ ) ∪ 0) = (a⊥ ∩ b⊥ )
22	20, 21	ax-r2 36	. . . . 5 ((a⊥ ∩ b⊥ ) ∪ (b ∩ b⊥ )) = (a⊥ ∩ b⊥ )
23	17, 22	ax-r2 36	. . . 4 ((a⊥ ∪ b) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
24	14, 23	2or 72	. . 3 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (b ∪ (a⊥ ∩ b⊥ ))
25	16, 15	fh4 472	. . . 4 (b ∪ (a⊥ ∩ b⊥ )) = ((b ∪ a⊥ ) ∩ (b ∪ b⊥ ))
26		ax-a2 31	. . . . . 6 (b ∪ a⊥ ) = (a⊥ ∪ b)
27		df-t 41	. . . . . . 7 1 = (b ∪ b⊥ )
28	27	ax-r1 35	. . . . . 6 (b ∪ b⊥ ) = 1
29	26, 28	2an 79	. . . . 5 ((b ∪ a⊥ ) ∩ (b ∪ b⊥ )) = ((a⊥ ∪ b) ∩ 1)
30		an1 106	. . . . 5 ((a⊥ ∪ b) ∩ 1) = (a⊥ ∪ b)
31	29, 30	ax-r2 36	. . . 4 ((b ∪ a⊥ ) ∩ (b ∪ b⊥ )) = (a⊥ ∪ b)
32	25, 31	ax-r2 36	. . 3 (b ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∪ b)
33	24, 32	ax-r2 36	. 2 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (a⊥ ∪ b)
34	1, 33	ax-r2 36	1 (a →4 b) = (a⊥ ∪ b)

Syntax hints:   = wb 1   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  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:  u4lemle1  713  u4lem2  747  u4lem3  752