Theorem comcmtr1 494

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

Hypothesis

Ref	Expression
comcmtr1.1	a C b

Assertion

Ref	Expression
comcmtr1	C (a, b) = 1

Proof of Theorem comcmtr1

Step	Hyp	Ref	Expression
1		comcmtr1.1	. . . . 5 a C b
2	1	df-c2 133	. . . 4 a = ((a ∩ b) ∪ (a ∩ b⊥ ))
3	1	comcom3 454	. . . . 5 a⊥ C b
4	3	df-c2 133	. . . 4 a⊥ = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
5	2, 4	2or 72	. . 3 (a ∪ a⊥ ) = (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
6	5	ax-r1 35	. 2 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (a ∪ a⊥ )
7		df-cmtr 134	. 2 C (a, b) = (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
8		df-t 41	. 2 1 = (a ∪ a⊥ )
9	6, 7, 8	3tr1 63	1 C (a, b) = 1

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-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-c2 133  df-cmtr 134

This theorem is referenced by: (None)