Theorem cmtr1com 493

Description: Commutator equal to 1 commutes. Theorem 2.11 of Beran, p. 86. (Contributed by NM, 24-Jan-1999.)

Hypothesis

Ref	Expression
cmtr1com.1	C (a, b) = 1

Assertion

Ref	Expression
cmtr1com	a C b

Proof of Theorem cmtr1com

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		le1 146	. . . . 5 (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))) ≤ 1
6		df-cmtr 134	. . . . . . 7 C (a, b) = (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
7		cmtr1com.1	. . . . . . 7 C (a, b) = 1
8		ax-a2 31	. . . . . . 7 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
9	6, 7, 8	3tr2 64	. . . . . 6 1 = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
10		lea 160	. . . . . . . 8 (a⊥ ∩ b) ≤ a⊥
11		lea 160	. . . . . . . 8 (a⊥ ∩ b⊥ ) ≤ a⊥
12	10, 11	lel2or 170	. . . . . . 7 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ a⊥
13	12	leror 152	. . . . . 6 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))) ≤ (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
14	9, 13	bltr 138	. . . . 5 1 ≤ (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
15	5, 14	lebi 145	. . . 4 (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))) = 1
16	4, 15	lem3.1 443	. . 3 ((a ∩ b) ∪ (a ∩ b⊥ )) = a
17	16	ax-r1 35	. 2 a = ((a ∩ b) ∪ (a ∩ b⊥ ))
18	17	df-c1 132	1 a C b

Syntax hints:   = wb 1   C wc 3  ^⊥ wn 4   ∪ 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  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-c1 132  df-cmtr 134

This theorem is referenced by: (None)