Theorem u4lemona 628

Description: Lemma for non-tollens implication study. (Contributed by NM, 15-Dec-1997.)

Assertion

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

Proof of Theorem u4lemona

Step	Hyp	Ref	Expression
1		df-i4 47	. . 3 (a →4 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
2	1	ax-r5 38	. 2 ((a →4 b) ∪ a⊥ ) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∪ a⊥ )
3		or32 82	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∪ a⊥ ) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ a⊥ ) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
4		ax-a3 32	. . . . . 6 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ a⊥ ) = ((a ∩ b) ∪ ((a⊥ ∩ b) ∪ a⊥ ))
5		lea 160	. . . . . . . 8 (a⊥ ∩ b) ≤ a⊥
6	5	df-le2 131	. . . . . . 7 ((a⊥ ∩ b) ∪ a⊥ ) = a⊥
7	6	lor 70	. . . . . 6 ((a ∩ b) ∪ ((a⊥ ∩ b) ∪ a⊥ )) = ((a ∩ b) ∪ a⊥ )
8	4, 7	ax-r2 36	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ a⊥ ) = ((a ∩ b) ∪ a⊥ )
9	8	ax-r5 38	. . . 4 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ a⊥ ) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (((a ∩ b) ∪ a⊥ ) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
10		comor1 461	. . . . . . . . 9 (a⊥ ∪ b) C a⊥
11	10	comcom7 460	. . . . . . . 8 (a⊥ ∪ b) C a
12		comor2 462	. . . . . . . 8 (a⊥ ∪ b) C b
13	11, 12	com2an 484	. . . . . . 7 (a⊥ ∪ b) C (a ∩ b)
14	13, 10	com2or 483	. . . . . 6 (a⊥ ∪ b) C ((a ∩ b) ∪ a⊥ )
15	12	comcom2 183	. . . . . 6 (a⊥ ∪ b) C b⊥
16	14, 15	fh4 472	. . . . 5 (((a ∩ b) ∪ a⊥ ) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = ((((a ∩ b) ∪ a⊥ ) ∪ (a⊥ ∪ b)) ∩ (((a ∩ b) ∪ a⊥ ) ∪ b⊥ ))
17		lear 161	. . . . . . . . . 10 (a ∩ b) ≤ b
18		leor 159	. . . . . . . . . 10 b ≤ (a⊥ ∪ b)
19	17, 18	letr 137	. . . . . . . . 9 (a ∩ b) ≤ (a⊥ ∪ b)
20		leo 158	. . . . . . . . 9 a⊥ ≤ (a⊥ ∪ b)
21	19, 20	lel2or 170	. . . . . . . 8 ((a ∩ b) ∪ a⊥ ) ≤ (a⊥ ∪ b)
22	21	df-le2 131	. . . . . . 7 (((a ∩ b) ∪ a⊥ ) ∪ (a⊥ ∪ b)) = (a⊥ ∪ b)
23		ax-a3 32	. . . . . . . 8 (((a ∩ b) ∪ a⊥ ) ∪ b⊥ ) = ((a ∩ b) ∪ (a⊥ ∪ b⊥ ))
24		df-a 40	. . . . . . . . . . . 12 (a ∩ b) = (a⊥ ∪ b⊥ )⊥
25	24	ax-r1 35	. . . . . . . . . . 11 (a⊥ ∪ b⊥ )⊥ = (a ∩ b)
26	25	con3 68	. . . . . . . . . 10 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
27	26	lor 70	. . . . . . . . 9 ((a ∩ b) ∪ (a⊥ ∪ b⊥ )) = ((a ∩ b) ∪ (a ∩ b)⊥ )
28		df-t 41	. . . . . . . . . 10 1 = ((a ∩ b) ∪ (a ∩ b)⊥ )
29	28	ax-r1 35	. . . . . . . . 9 ((a ∩ b) ∪ (a ∩ b)⊥ ) = 1
30	27, 29	ax-r2 36	. . . . . . . 8 ((a ∩ b) ∪ (a⊥ ∪ b⊥ )) = 1
31	23, 30	ax-r2 36	. . . . . . 7 (((a ∩ b) ∪ a⊥ ) ∪ b⊥ ) = 1
32	22, 31	2an 79	. . . . . 6 ((((a ∩ b) ∪ a⊥ ) ∪ (a⊥ ∪ b)) ∩ (((a ∩ b) ∪ a⊥ ) ∪ b⊥ )) = ((a⊥ ∪ b) ∩ 1)
33		an1 106	. . . . . 6 ((a⊥ ∪ b) ∩ 1) = (a⊥ ∪ b)
34	32, 33	ax-r2 36	. . . . 5 ((((a ∩ b) ∪ a⊥ ) ∪ (a⊥ ∪ b)) ∩ (((a ∩ b) ∪ a⊥ ) ∪ b⊥ )) = (a⊥ ∪ b)
35	16, 34	ax-r2 36	. . . 4 (((a ∩ b) ∪ a⊥ ) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (a⊥ ∪ b)
36	9, 35	ax-r2 36	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ a⊥ ) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (a⊥ ∪ b)
37	3, 36	ax-r2 36	. 2 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∪ a⊥ ) = (a⊥ ∪ b)
38	2, 37	ax-r2 36	1 ((a →4 b) ∪ a⊥ ) = (a⊥ ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →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:  u4lemnaa  643  u4lem5  764