Theorem wdf-c2 366

Description: Show that commutator is a 'commutes' analogue for == analogue of =.

Hypothesis

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

Assertion

Ref	Expression
wdf-c2	(a == ((a ^ b) v (a ^ b_|_))) = 1

Proof of Theorem wdf-c2

Step	Hyp	Ref	Expression
1		le1 138	. 2 (a == ((a ^ b) v (a ^ b_|_))) =< 1
2		lea 152	. . . . 5 (a_|_ ^ b) =< a_|_
3		lea 152	. . . . 5 (a_|_ ^ b_|_) =< a_|_
4	2, 3	lel2or 162	. . . 4 ((a_|_ ^ b) v (a_|_ ^ b_|_)) =< a_|_
5	4	lelor 158	. . 3 (((a ^ b) v (a ^ b_|_)) v ((a_|_ ^ b) v (a_|_ ^ b_|_))) =< (((a ^ b) v (a ^ b_|_)) v a_|_)
6		wdf-c2.1	. . . . 5 C (a, b) = 1
7	6	ax-r1 34	. . . 4 1 = C (a, b)
8		df-cmtr 126	. . . 4 C (a, b) = (((a ^ b) v (a ^ b_|_)) v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
9	7, 8	ax-r2 35	. . 3 1 = (((a ^ b) v (a ^ b_|_)) v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
10		dfb 86	. . . 4 (a == ((a ^ b) v (a ^ b_|_))) = ((a ^ ((a ^ b) v (a ^ b_|_))) v (a_|_ ^ ((a ^ b) v (a ^ b_|_))_|_))
11		ancom 68	. . . . . 6 (a ^ ((a ^ b) v (a ^ b_|_))) = (((a ^ b) v (a ^ b_|_)) ^ a)
12		lea 152	. . . . . . . 8 (a ^ b) =< a
13		lea 152	. . . . . . . 8 (a ^ b_|_) =< a
14	12, 13	lel2or 162	. . . . . . 7 ((a ^ b) v (a ^ b_|_)) =< a
15	14	df2le2 128	. . . . . 6 (((a ^ b) v (a ^ b_|_)) ^ a) = ((a ^ b) v (a ^ b_|_))
16	11, 15	ax-r2 35	. . . . 5 (a ^ ((a ^ b) v (a ^ b_|_))) = ((a ^ b) v (a ^ b_|_))
17		anandi 106	. . . . . 6 (a_|_ ^ ((a_|_ v b_|_) ^ (a_|_ v b))) = ((a_|_ ^ (a_|_ v b_|_)) ^ (a_|_ ^ (a_|_ v b)))
18		oran3 85	. . . . . . . . 9 (a_|_ v b_|_) = (a ^ b)_|_
19		oran2 84	. . . . . . . . 9 (a_|_ v b) = (a ^ b_|_)_|_
20	18, 19	2an 72	. . . . . . . 8 ((a_|_ v b_|_) ^ (a_|_ v b)) = ((a ^ b)_|_ ^ (a ^ b_|_)_|_)
21		anor3 82	. . . . . . . 8 ((a ^ b)_|_ ^ (a ^ b_|_)_|_) = ((a ^ b) v (a ^ b_|_))_|_
22	20, 21	ax-r2 35	. . . . . . 7 ((a_|_ v b_|_) ^ (a_|_ v b)) = ((a ^ b) v (a ^ b_|_))_|_
23	22	lan 70	. . . . . 6 (a_|_ ^ ((a_|_ v b_|_) ^ (a_|_ v b))) = (a_|_ ^ ((a ^ b) v (a ^ b_|_))_|_)
24		a5c 113	. . . . . . . 8 (a_|_ ^ (a_|_ v b_|_)) = a_|_
25		a5c 113	. . . . . . . 8 (a_|_ ^ (a_|_ v b)) = a_|_
26	24, 25	2an 72	. . . . . . 7 ((a_|_ ^ (a_|_ v b_|_)) ^ (a_|_ ^ (a_|_ v b))) = (a_|_ ^ a_|_)
27		anidm 103	. . . . . . 7 (a_|_ ^ a_|_) = a_|_
28	26, 27	ax-r2 35	. . . . . 6 ((a_|_ ^ (a_|_ v b_|_)) ^ (a_|_ ^ (a_|_ v b))) = a_|_
29	17, 23, 28	3tr2 61	. . . . 5 (a_|_ ^ ((a ^ b) v (a ^ b_|_))_|_) = a_|_
30	16, 29	2or 67	. . . 4 ((a ^ ((a ^ b) v (a ^ b_|_))) v (a_|_ ^ ((a ^ b) v (a ^ b_|_))_|_)) = (((a ^ b) v (a ^ b_|_)) v a_|_)
31	10, 30	ax-r2 35	. . 3 (a == ((a ^ b) v (a ^ b_|_))) = (((a ^ b) v (a ^ b_|_)) v a_|_)
32	5, 9, 31	le3tr1 132	. 2 1 =< (a == ((a ^ b) v (a ^ b_|_)))
33	1, 32	lebi 137	1 (a == ((a ^ b) v (a ^ b_|_))) = 1

Syntax hints:   = wb 1  _|_wn 4   == tb 5   v wo 6   ^ wa 7  1wt 9   C wcmtr 28

This theorem is referenced by:  wbctr 392  wcbtr 393  wcomcom2 397  wcomd 400  wcomcom5 402  wcom2or 409

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

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123  df-cmtr 126

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