Theorem u4lemona 610

Description: Lemma for non-tollens implication study.

Assertion

Ref	Expression
u4lemona	((a →4 b) ∪ a⊥ ) = (a⊥ ∪ b)

Proof of Theorem u4lemona

Step	Hyp	Ref	Expression
1		df-i4 46	. . 3 (a →4 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
2	1	ax-r5 37	. 2 ((a →4 b) ∪ a⊥ ) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∪ a⊥ )
3		or32 75	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∪ a⊥ ) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ a⊥ ) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
4		ax-a3 31	. . . . . 6 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ a⊥ ) = ((a ∩ b) ∪ ((a⊥ ∩ b) ∪ a⊥ ))
5		lea 152	. . . . . . . 8 (a⊥ ∩ b) ≤ a⊥
6	5	df-le2 123	. . . . . . 7 ((a⊥ ∩ b) ∪ a⊥ ) = a⊥
7	6	lor 66	. . . . . 6 ((a ∩ b) ∪ ((a⊥ ∩ b) ∪ a⊥ )) = ((a ∩ b) ∪ a⊥ )
8	4, 7	ax-r2 35	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ a⊥ ) = ((a ∩ b) ∪ a⊥ )
9	8	ax-r5 37	. . . 4 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ a⊥ ) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (((a ∩ b) ∪ a⊥ ) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
10		comor1 443	. . . . . . . . 9 (a⊥ ∪ b) C a⊥
11	10	comcom7 442	. . . . . . . 8 (a⊥ ∪ b) C a
12		comor2 444	. . . . . . . 8 (a⊥ ∪ b) C b
13	11, 12	com2an 466	. . . . . . 7 (a⊥ ∪ b) C (a ∩ b)
14	13, 10	com2or 465	. . . . . 6 (a⊥ ∪ b) C ((a ∩ b) ∪ a⊥ )
15	12	comcom2 175	. . . . . 6 (a⊥ ∪ b) C b⊥
16	14, 15	fh4 454	. . . . 5 (((a ∩ b) ∪ a⊥ ) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = ((((a ∩ b) ∪ a⊥ ) ∪ (a⊥ ∪ b)) ∩ (((a ∩ b) ∪ a⊥ ) ∪ b⊥ ))
17		lear 153	. . . . . . . . . 10 (a ∩ b) ≤ b
18		leor 151	. . . . . . . . . 10 b ≤ (a⊥ ∪ b)
19	17, 18	letr 129	. . . . . . . . 9 (a ∩ b) ≤ (a⊥ ∪ b)
20		leo 150	. . . . . . . . 9 a⊥ ≤ (a⊥ ∪ b)
21	19, 20	lel2or 162	. . . . . . . 8 ((a ∩ b) ∪ a⊥ ) ≤ (a⊥ ∪ b)
22	21	df-le2 123	. . . . . . 7 (((a ∩ b) ∪ a⊥ ) ∪ (a⊥ ∪ b)) = (a⊥ ∪ b)
23		ax-a3 31	. . . . . . . 8 (((a ∩ b) ∪ a⊥ ) ∪ b⊥ ) = ((a ∩ b) ∪ (a⊥ ∪ b⊥ ))
24		df-a 39	. . . . . . . . . . . 12 (a ∩ b) = (a⊥ ∪ b⊥ )⊥
25	24	ax-r1 34	. . . . . . . . . . 11 (a⊥ ∪ b⊥ )⊥ = (a ∩ b)
26	25	con3 65	. . . . . . . . . 10 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
27	26	lor 66	. . . . . . . . 9 ((a ∩ b) ∪ (a⊥ ∪ b⊥ )) = ((a ∩ b) ∪ (a ∩ b)⊥ )
28		df-t 40	. . . . . . . . . 10 1 = ((a ∩ b) ∪ (a ∩ b)⊥ )
29	28	ax-r1 34	. . . . . . . . 9 ((a ∩ b) ∪ (a ∩ b)⊥ ) = 1
30	27, 29	ax-r2 35	. . . . . . . 8 ((a ∩ b) ∪ (a⊥ ∪ b⊥ )) = 1
31	23, 30	ax-r2 35	. . . . . . 7 (((a ∩ b) ∪ a⊥ ) ∪ b⊥ ) = 1
32	22, 31	2an 72	. . . . . 6 ((((a ∩ b) ∪ a⊥ ) ∪ (a⊥ ∪ b)) ∩ (((a ∩ b) ∪ a⊥ ) ∪ b⊥ )) = ((a⊥ ∪ b) ∩ 1)
33		an1 98	. . . . . 6 ((a⊥ ∪ b) ∩ 1) = (a⊥ ∪ b)
34	32, 33	ax-r2 35	. . . . 5 ((((a ∩ b) ∪ a⊥ ) ∪ (a⊥ ∪ b)) ∩ (((a ∩ b) ∪ a⊥ ) ∪ b⊥ )) = (a⊥ ∪ b)
35	16, 34	ax-r2 35	. . . 4 (((a ∩ b) ∪ a⊥ ) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (a⊥ ∪ b)
36	9, 35	ax-r2 35	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ a⊥ ) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (a⊥ ∪ b)
37	3, 36	ax-r2 35	. 2 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∪ a⊥ ) = (a⊥ ∪ b)
38	2, 37	ax-r2 35	1 ((a →4 b) ∪ a⊥ ) = (a⊥ ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₄ wi4 16

This theorem is referenced by:  u4lemnaa 625  u4lem5 746

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i4 46  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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