Theorem i0cmtrcom 477

Description: Commutator element ->0 commutator implies commutation.

Hypothesis

Ref	Expression
i0cmtrcom.1	(a ->0 C (a, b)) = 1

Assertion

Ref	Expression
i0cmtrcom	a C b

Proof of Theorem i0cmtrcom

Step	Hyp	Ref	Expression
1		lea 152	. . . . . 6 (a ^ b) =< a
2		lea 152	. . . . . 6 (a ^ b_|_) =< a
3	1, 2	lel2or 162	. . . . 5 ((a ^ b) v (a ^ b_|_)) =< a
4	3	df-le2 123	. . . 4 (((a ^ b) v (a ^ b_|_)) v a) = a
5		df-cmtr 126	. . . . . . . 8 C (a, b) = (((a ^ b) v (a ^ b_|_)) v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
6	5	lor 66	. . . . . . 7 (a_|_ v C (a, b)) = (a_|_ v (((a ^ b) v (a ^ b_|_)) v ((a_|_ ^ b) v (a_|_ ^ b_|_))))
7	6	ax-r1 34	. . . . . 6 (a_|_ v (((a ^ b) v (a ^ b_|_)) v ((a_|_ ^ b) v (a_|_ ^ b_|_)))) = (a_|_ v C (a, b))
8		ax-a2 30	. . . . . . 7 (a_|_ v ((a ^ b) v (a ^ b_|_))) = (((a ^ b) v (a ^ b_|_)) v a_|_)
9		ax-a2 30	. . . . . . . . . 10 (a_|_ v ((a_|_ ^ b) v (a_|_ ^ b_|_))) = (((a_|_ ^ b) v (a_|_ ^ b_|_)) v a_|_)
10		lea 152	. . . . . . . . . . . 12 (a_|_ ^ b) =< a_|_
11		lea 152	. . . . . . . . . . . 12 (a_|_ ^ b_|_) =< a_|_
12	10, 11	lel2or 162	. . . . . . . . . . 11 ((a_|_ ^ b) v (a_|_ ^ b_|_)) =< a_|_
13	12	df-le2 123	. . . . . . . . . 10 (((a_|_ ^ b) v (a_|_ ^ b_|_)) v a_|_) = a_|_
14	9, 13	ax-r2 35	. . . . . . . . 9 (a_|_ v ((a_|_ ^ b) v (a_|_ ^ b_|_))) = a_|_
15	14	lor 66	. . . . . . . 8 (((a ^ b) v (a ^ b_|_)) v (a_|_ v ((a_|_ ^ b) v (a_|_ ^ b_|_)))) = (((a ^ b) v (a ^ b_|_)) v a_|_)
16	15	ax-r1 34	. . . . . . 7 (((a ^ b) v (a ^ b_|_)) v a_|_) = (((a ^ b) v (a ^ b_|_)) v (a_|_ v ((a_|_ ^ b) v (a_|_ ^ b_|_))))
17		or12 73	. . . . . . 7 (((a ^ b) v (a ^ b_|_)) v (a_|_ v ((a_|_ ^ b) v (a_|_ ^ b_|_)))) = (a_|_ v (((a ^ b) v (a ^ b_|_)) v ((a_|_ ^ b) v (a_|_ ^ b_|_))))
18	8, 16, 17	3tr 62	. . . . . 6 (a_|_ v ((a ^ b) v (a ^ b_|_))) = (a_|_ v (((a ^ b) v (a ^ b_|_)) v ((a_|_ ^ b) v (a_|_ ^ b_|_))))
19		df-i0 42	. . . . . 6 (a ->0 C (a, b)) = (a_|_ v C (a, b))
20	7, 18, 19	3tr1 60	. . . . 5 (a_|_ v ((a ^ b) v (a ^ b_|_))) = (a ->0 C (a, b))
21		i0cmtrcom.1	. . . . 5 (a ->0 C (a, b)) = 1
22	20, 21	ax-r2 35	. . . 4 (a_|_ v ((a ^ b) v (a ^ b_|_))) = 1
23	4, 22	lem3.1 425	. . 3 ((a ^ b) v (a ^ b_|_)) = a
24	23	ax-r1 34	. 2 a = ((a ^ b) v (a ^ b_|_))
25	24	df-c1 124	1 a C b

Syntax hints:   = wb 1   C wc 3  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->0 wi0 12   C wcmtr 28

This theorem is referenced by:  3vded3 801

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  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-i0 42  df-le1 122  df-le2 123  df-c1 124  df-cmtr 126

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