Theorem wlecom 391

Description: Comparable elements commute. Beran 84 2.3(iii) p. 40.

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		a5b 112	. . . . 5 (a ∪ (a ∩ b⊥ )) = a
2	1	bi1 110	. . . 4 ((a ∪ (a ∩ b⊥ )) ≡ a) = 1
3	2	wr1 189	. . 3 (a ≡ (a ∪ (a ∩ b⊥ ))) = 1
4		wlecom.1	. . . . . 6 (a ≤2 b) = 1
5	4	wdf2le2 368	. . . . 5 ((a ∩ b) ≡ a) = 1
6	5	wr1 189	. . . 4 (a ≡ (a ∩ b)) = 1
7	6	wr5-2v 348	. . 3 ((a ∪ (a ∩ b⊥ )) ≡ ((a ∩ b) ∪ (a ∩ b⊥ ))) = 1
8	3, 7	wr2 353	. 2 (a ≡ ((a ∩ b) ∪ (a ∩ b⊥ ))) = 1
9	8	wdf-c1 365	1 C (a, b) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   ≤₂ wle2 11   C wcmtr 28

This theorem is referenced by:  wcomorr 394  wcoman1 395  wlem14 412  ska4 415

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-wom 343

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le 121  df-le1 122  df-le2 123  df-cmtr 126

metamath.org