Theorem wdf-c2 384

Description: Show that commutator is a 'commutes' analogue for ≡ analogue of =. (Contributed by NM, 27-Jan-2002.)

Hypothesis

Ref	Expression
wdf-c2.1	C (a, b) = 1

Assertion

Ref	Expression
wdf-c2	(a ≡ ((a ∩ b) ∪ (a ∩ b⊥ ))) = 1

Proof of Theorem wdf-c2

Step	Hyp	Ref	Expression
1		le1 146	. 2 (a ≡ ((a ∩ b) ∪ (a ∩ b⊥ ))) ≤ 1
2		lea 160	. . . . 5 (a⊥ ∩ b) ≤ a⊥
3		lea 160	. . . . 5 (a⊥ ∩ b⊥ ) ≤ a⊥
4	2, 3	lel2or 170	. . . 4 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ a⊥
5	4	lelor 166	. . 3 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ≤ (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ )
6		wdf-c2.1	. . . . 5 C (a, b) = 1
7	6	ax-r1 35	. . . 4 1 = C (a, b)
8		df-cmtr 134	. . . 4 C (a, b) = (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
9	7, 8	ax-r2 36	. . 3 1 = (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
10		dfb 94	. . . 4 (a ≡ ((a ∩ b) ∪ (a ∩ b⊥ ))) = ((a ∩ ((a ∩ b) ∪ (a ∩ b⊥ ))) ∪ (a⊥ ∩ ((a ∩ b) ∪ (a ∩ b⊥ ))⊥ ))
11		ancom 74	. . . . . 6 (a ∩ ((a ∩ b) ∪ (a ∩ b⊥ ))) = (((a ∩ b) ∪ (a ∩ b⊥ )) ∩ a)
12		lea 160	. . . . . . . 8 (a ∩ b) ≤ a
13		lea 160	. . . . . . . 8 (a ∩ b⊥ ) ≤ a
14	12, 13	lel2or 170	. . . . . . 7 ((a ∩ b) ∪ (a ∩ b⊥ )) ≤ a
15	14	df2le2 136	. . . . . 6 (((a ∩ b) ∪ (a ∩ b⊥ )) ∩ a) = ((a ∩ b) ∪ (a ∩ b⊥ ))
16	11, 15	ax-r2 36	. . . . 5 (a ∩ ((a ∩ b) ∪ (a ∩ b⊥ ))) = ((a ∩ b) ∪ (a ∩ b⊥ ))
17		anandi 114	. . . . . 6 (a⊥ ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b))) = ((a⊥ ∩ (a⊥ ∪ b⊥ )) ∩ (a⊥ ∩ (a⊥ ∪ b)))
18		oran3 93	. . . . . . . . 9 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
19		oran2 92	. . . . . . . . 9 (a⊥ ∪ b) = (a ∩ b⊥ )⊥
20	18, 19	2an 79	. . . . . . . 8 ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b)) = ((a ∩ b)⊥ ∩ (a ∩ b⊥ )⊥ )
21		anor3 90	. . . . . . . 8 ((a ∩ b)⊥ ∩ (a ∩ b⊥ )⊥ ) = ((a ∩ b) ∪ (a ∩ b⊥ ))⊥
22	20, 21	ax-r2 36	. . . . . . 7 ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b)) = ((a ∩ b) ∪ (a ∩ b⊥ ))⊥
23	22	lan 77	. . . . . 6 (a⊥ ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b))) = (a⊥ ∩ ((a ∩ b) ∪ (a ∩ b⊥ ))⊥ )
24		anabs 121	. . . . . . . 8 (a⊥ ∩ (a⊥ ∪ b⊥ )) = a⊥
25		anabs 121	. . . . . . . 8 (a⊥ ∩ (a⊥ ∪ b)) = a⊥
26	24, 25	2an 79	. . . . . . 7 ((a⊥ ∩ (a⊥ ∪ b⊥ )) ∩ (a⊥ ∩ (a⊥ ∪ b))) = (a⊥ ∩ a⊥ )
27		anidm 111	. . . . . . 7 (a⊥ ∩ a⊥ ) = a⊥
28	26, 27	ax-r2 36	. . . . . 6 ((a⊥ ∩ (a⊥ ∪ b⊥ )) ∩ (a⊥ ∩ (a⊥ ∪ b))) = a⊥
29	17, 23, 28	3tr2 64	. . . . 5 (a⊥ ∩ ((a ∩ b) ∪ (a ∩ b⊥ ))⊥ ) = a⊥
30	16, 29	2or 72	. . . 4 ((a ∩ ((a ∩ b) ∪ (a ∩ b⊥ ))) ∪ (a⊥ ∩ ((a ∩ b) ∪ (a ∩ b⊥ ))⊥ )) = (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ )
31	10, 30	ax-r2 36	. . 3 (a ≡ ((a ∩ b) ∪ (a ∩ b⊥ ))) = (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ )
32	5, 9, 31	le3tr1 140	. 2 1 ≤ (a ≡ ((a ∩ b) ∪ (a ∩ b⊥ )))
33	1, 32	lebi 145	1 (a ≡ ((a ∩ b) ∪ (a ∩ b⊥ ))) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   C wcmtr 29

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

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-cmtr 134

This theorem is referenced by:  wbctr  410  wcbtr  411  wcomcom2  415  wcomd  418  wcomcom5  420  wcom2or  427