Theorem wlecom 409

Description: Comparable elements commute. Beran 84 2.3(iii) p. 40. (Contributed by NM, 13-Oct-1997.)

Hypothesis

Ref	Expression
wlecom.1	(a ≤2 b) = 1

Assertion

Ref	Expression
wlecom	C (a, b) = 1

Proof of Theorem wlecom

Step	Hyp	Ref	Expression
1		orabs 120	. . . . 5 (a ∪ (a ∩ b⊥ )) = a
2	1	bi1 118	. . . 4 ((a ∪ (a ∩ b⊥ )) ≡ a) = 1
3	2	wr1 197	. . 3 (a ≡ (a ∪ (a ∩ b⊥ ))) = 1
4		wlecom.1	. . . . . 6 (a ≤2 b) = 1
5	4	wdf2le2 386	. . . . 5 ((a ∩ b) ≡ a) = 1
6	5	wr1 197	. . . 4 (a ≡ (a ∩ b)) = 1
7	6	wr5-2v 366	. . 3 ((a ∪ (a ∩ b⊥ )) ≡ ((a ∩ b) ∪ (a ∩ b⊥ ))) = 1
8	3, 7	wr2 371	. 2 (a ≡ ((a ∩ b) ∪ (a ∩ b⊥ ))) = 1
9	8	wdf-c1 383	1 C (a, b) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   ≤₂ wle2 10   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-wom 361

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134

This theorem is referenced by:  wcomorr  412  wcoman1  413  wlem14  430  ska4  433