Theorem woml7 437

Description: Variant of weakly orthomodular law. (Contributed by NM, 14-Nov-1998.)

Assertion

Ref	Expression
woml7	(((a →2 b) ∩ (b →2 a))⊥ ∪ (a ≡ b)) = 1

Proof of Theorem woml7

Step	Hyp	Ref	Expression
1		df-i2 45	. . . . . . . 8 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
2		ax-a2 31	. . . . . . . 8 (b ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ b)
3	1, 2	ax-r2 36	. . . . . . 7 (a →2 b) = ((a⊥ ∩ b⊥ ) ∪ b)
4		df-i2 45	. . . . . . . 8 (b →2 a) = (a ∪ (b⊥ ∩ a⊥ ))
5		ax-a2 31	. . . . . . . 8 (a ∪ (b⊥ ∩ a⊥ )) = ((b⊥ ∩ a⊥ ) ∪ a)
6		ancom 74	. . . . . . . . 9 (b⊥ ∩ a⊥ ) = (a⊥ ∩ b⊥ )
7	6	ax-r5 38	. . . . . . . 8 ((b⊥ ∩ a⊥ ) ∪ a) = ((a⊥ ∩ b⊥ ) ∪ a)
8	4, 5, 7	3tr 65	. . . . . . 7 (b →2 a) = ((a⊥ ∩ b⊥ ) ∪ a)
9	3, 8	2an 79	. . . . . 6 ((a →2 b) ∩ (b →2 a)) = (((a⊥ ∩ b⊥ ) ∪ b) ∩ ((a⊥ ∩ b⊥ ) ∪ a))
10		ancom 74	. . . . . 6 (((a⊥ ∩ b⊥ ) ∪ b) ∩ ((a⊥ ∩ b⊥ ) ∪ a)) = (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))
11	9, 10	ax-r2 36	. . . . 5 ((a →2 b) ∩ (b →2 a)) = (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))
12	11	ax-r4 37	. . . 4 ((a →2 b) ∩ (b →2 a))⊥ = (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥
13		id 59	. . . 4 (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ = (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥
14	12, 13	ax-r2 36	. . 3 ((a →2 b) ∩ (b →2 a))⊥ = (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥
15		dfb 94	. . 3 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
16	14, 15	2or 72	. 2 (((a →2 b) ∩ (b →2 a))⊥ ∪ (a ≡ b)) = ((((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))
17		1b 117	. . 3 (1 ≡ ((((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))) = ((((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))
18	17	ax-r1 35	. 2 ((((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (1 ≡ ((((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))))
19		df-t 41	. . . . 5 1 = (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ )
20		ax-a2 31	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ) = (((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))
21	19, 20	ax-r2 36	. . . 4 1 = (((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))
22	21	bi1 118	. . 3 (1 ≡ (((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))) = 1
23		wa2 192	. . . . . 6 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ≡ ((a⊥ ∩ b⊥ ) ∪ (a ∩ b))) = 1
24		wcoman1 413	. . . . . . . . 9 C ((a⊥ ∩ b⊥ ), a⊥ ) = 1
25	24	wcomcom3 416	. . . . . . . 8 C ((a⊥ ∩ b⊥ )⊥ , a⊥ ) = 1
26	25	wcomcom5 420	. . . . . . 7 C ((a⊥ ∩ b⊥ ), a) = 1
27		ancom 74	. . . . . . . . . . 11 (a⊥ ∩ b⊥ ) = (b⊥ ∩ a⊥ )
28	27	bi1 118	. . . . . . . . . 10 ((a⊥ ∩ b⊥ ) ≡ (b⊥ ∩ a⊥ )) = 1
29		wcoman1 413	. . . . . . . . . 10 C ((b⊥ ∩ a⊥ ), b⊥ ) = 1
30	28, 29	wbctr 410	. . . . . . . . 9 C ((a⊥ ∩ b⊥ ), b⊥ ) = 1
31	30	wcomcom3 416	. . . . . . . 8 C ((a⊥ ∩ b⊥ )⊥ , b⊥ ) = 1
32	31	wcomcom5 420	. . . . . . 7 C ((a⊥ ∩ b⊥ ), b) = 1
33	26, 32	wfh3 425	. . . . . 6 (((a⊥ ∩ b⊥ ) ∪ (a ∩ b)) ≡ (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))) = 1
34	23, 33	wr2 371	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ≡ (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))) = 1
35	34	wr4 199	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ≡ (((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ ) = 1
36	35	wr5-2v 366	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))) ≡ ((((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))) = 1
37	22, 36	wr2 371	. 2 (1 ≡ ((((a⊥ ∩ b⊥ ) ∪ a) ∩ ((a⊥ ∩ b⊥ ) ∪ b))⊥ ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))) = 1
38	16, 18, 37	3tr 65	1 (((a →2 b) ∩ (b →2 a))⊥ ∪ (a ≡ b)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   →₂ wi2 13

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-wom 361

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134

This theorem is referenced by: (None)