Theorem wcomcom5 420

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

Hypothesis

Ref	Expression
wcomcom5.1	C (a⊥ , b⊥ ) = 1

Assertion

Ref	Expression
wcomcom5	C (a, b) = 1

Proof of Theorem wcomcom5

Step	Hyp	Ref	Expression
1		wcomcom5.1	. . . . 5 C (a⊥ , b⊥ ) = 1
2	1	wcomcom4 417	. . . 4 C (a⊥ ⊥ , b⊥ ⊥ ) = 1
3	2	wdf-c2 384	. . 3 (a⊥ ⊥ ≡ ((a⊥ ⊥ ∩ b⊥ ⊥ ) ∪ (a⊥ ⊥ ∩ b⊥ ⊥ ⊥ ))) = 1
4		ax-a1 30	. . . 4 a = a⊥ ⊥
5	4	bi1 118	. . 3 (a ≡ a⊥ ⊥ ) = 1
6		ax-a1 30	. . . . . 6 b = b⊥ ⊥
7	6	bi1 118	. . . . 5 (b ≡ b⊥ ⊥ ) = 1
8	5, 7	w2an 373	. . . 4 ((a ∩ b) ≡ (a⊥ ⊥ ∩ b⊥ ⊥ )) = 1
9		ax-a1 30	. . . . . 6 b⊥ = b⊥ ⊥ ⊥
10	9	bi1 118	. . . . 5 (b⊥ ≡ b⊥ ⊥ ⊥ ) = 1
11	5, 10	w2an 373	. . . 4 ((a ∩ b⊥ ) ≡ (a⊥ ⊥ ∩ b⊥ ⊥ ⊥ )) = 1
12	8, 11	w2or 372	. . 3 (((a ∩ b) ∪ (a ∩ b⊥ )) ≡ ((a⊥ ⊥ ∩ b⊥ ⊥ ) ∪ (a⊥ ⊥ ∩ b⊥ ⊥ ⊥ ))) = 1
13	3, 5, 12	w3tr1 374	. 2 (a ≡ ((a ∩ b) ∪ (a ∩ b⊥ ))) = 1
14	13	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:  wcomdr  421  wcom2an  428  woml6  436  woml7  437