Theorem wcomcom2 415

Description: Commutation equivalence. Kalmbach 83 p. 23. (Contributed by NM, 13-Oct-1997.)

Hypothesis

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

Assertion

Ref	Expression
wcomcom2	C (a, b⊥ ) = 1

Proof of Theorem wcomcom2

Step	Hyp	Ref	Expression
1		wcomcom.1	. . . . 5 C (a, b) = 1
2	1	wdf-c2 384	. . . 4 (a ≡ ((a ∩ b) ∪ (a ∩ b⊥ ))) = 1
3		ax-a1 30	. . . . . . 7 b = b⊥ ⊥
4	3	bi1 118	. . . . . 6 (b ≡ b⊥ ⊥ ) = 1
5	4	wlan 370	. . . . 5 ((a ∩ b) ≡ (a ∩ b⊥ ⊥ )) = 1
6	5	wr5-2v 366	. . . 4 (((a ∩ b) ∪ (a ∩ b⊥ )) ≡ ((a ∩ b⊥ ⊥ ) ∪ (a ∩ b⊥ ))) = 1
7	2, 6	wr2 371	. . 3 (a ≡ ((a ∩ b⊥ ⊥ ) ∪ (a ∩ b⊥ ))) = 1
8		ax-a2 31	. . . 4 ((a ∩ b⊥ ⊥ ) ∪ (a ∩ b⊥ )) = ((a ∩ b⊥ ) ∪ (a ∩ b⊥ ⊥ ))
9	8	bi1 118	. . 3 (((a ∩ b⊥ ⊥ ) ∪ (a ∩ b⊥ )) ≡ ((a ∩ b⊥ ) ∪ (a ∩ b⊥ ⊥ ))) = 1
10	7, 9	wr2 371	. 2 (a ≡ ((a ∩ b⊥ ) ∪ (a ∩ b⊥ ⊥ ))) = 1
11	10	wdf-c1 383	1 C (a, b⊥ ) = 1

Syntax hints:   = wb 1  ^⊥ 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-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:  wcomcom3  416  wcomcom4  417  wfh1  423  wfh2  424  wnbdi  429  ska2  432  ska4  433