Theorem i0cmtrcom 495

Description: Commutator element →₀ commutator implies commutation. (Contributed by NM, 24-Jan-1999.)

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 160	. . . . . 6 (a ∩ b) ≤ a
2		lea 160	. . . . . 6 (a ∩ b⊥ ) ≤ a
3	1, 2	lel2or 170	. . . . 5 ((a ∩ b) ∪ (a ∩ b⊥ )) ≤ a
4	3	df-le2 131	. . . 4 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a) = a
5		df-cmtr 134	. . . . . . . 8 C (a, b) = (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
6	5	lor 70	. . . . . . 7 (a⊥ ∪ C (a, b)) = (a⊥ ∪ (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))))
7	6	ax-r1 35	. . . . . 6 (a⊥ ∪ (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))) = (a⊥ ∪ C (a, b))
8		ax-a2 31	. . . . . . 7 (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))) = (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ )
9		ax-a2 31	. . . . . . . . . 10 (a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ )
10		lea 160	. . . . . . . . . . . 12 (a⊥ ∩ b) ≤ a⊥
11		lea 160	. . . . . . . . . . . 12 (a⊥ ∩ b⊥ ) ≤ a⊥
12	10, 11	lel2or 170	. . . . . . . . . . 11 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ a⊥
13	12	df-le2 131	. . . . . . . . . 10 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ ) = a⊥
14	9, 13	ax-r2 36	. . . . . . . . 9 (a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = a⊥
15	14	lor 70	. . . . . . . 8 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))) = (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ )
16	15	ax-r1 35	. . . . . . 7 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) = (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))))
17		or12 80	. . . . . . 7 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ (a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))) = (a⊥ ∪ (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))))
18	8, 16, 17	3tr 65	. . . . . 6 (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))) = (a⊥ ∪ (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))))
19		df-i0 43	. . . . . 6 (a →0 C (a, b)) = (a⊥ ∪ C (a, b))
20	7, 18, 19	3tr1 63	. . . . 5 (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))) = (a →0 C (a, b))
21		i0cmtrcom.1	. . . . 5 (a →0 C (a, b)) = 1
22	20, 21	ax-r2 36	. . . 4 (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))) = 1
23	4, 22	lem3.1 443	. . 3 ((a ∩ b) ∪ (a ∩ b⊥ )) = a
24	23	ax-r1 35	. 2 a = ((a ∩ b) ∪ (a ∩ b⊥ ))
25	24	df-c1 132	1 a C b

Syntax hints:   = wb 1   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₀ wi0 11   C wcmtr 29

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  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-i0 43  df-le1 130  df-le2 131  df-c1 132  df-cmtr 134

This theorem is referenced by:  3vded3  819