Theorem wcomd 400

Description: Commutation dual. Kalmbach 83 p. 23.

Hypothesis

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

Assertion

Ref	Expression
wcomd	(a == ((a v b) ^ (a v b_|_))) = 1

Proof of Theorem wcomd

Step	Hyp	Ref	Expression
1		wcomcom.1	. . . . 5 C (a, b) = 1
2	1	wcomcom4 399	. . . 4 C (a_|_, b_|_) = 1
3	2	wdf-c2 366	. . 3 (a_|_ == ((a_|_ ^ b_|_) v (a_|_ ^ b_|__|_))) = 1
4	3	wcon3 201	. 2 (a == ((a_|_ ^ b_|_) v (a_|_ ^ b_|__|_))_|_) = 1
5		oran 79	. . . . 5 ((a_|_ ^ b_|_) v (a_|_ ^ b_|__|_)) = ((a_|_ ^ b_|_)_|_ ^ (a_|_ ^ b_|__|_)_|_)_|_
6	5	bi1 110	. . . 4 (((a_|_ ^ b_|_) v (a_|_ ^ b_|__|_)) == ((a_|_ ^ b_|_)_|_ ^ (a_|_ ^ b_|__|_)_|_)_|_) = 1
7	6	wcon2 200	. . 3 (((a_|_ ^ b_|_) v (a_|_ ^ b_|__|_))_|_ == ((a_|_ ^ b_|_)_|_ ^ (a_|_ ^ b_|__|_)_|_)) = 1
8		oran 79	. . . . . 6 (a v b) = (a_|_ ^ b_|_)_|_
9	8	bi1 110	. . . . 5 ((a v b) == (a_|_ ^ b_|_)_|_) = 1
10		oran 79	. . . . . 6 (a v b_|_) = (a_|_ ^ b_|__|_)_|_
11	10	bi1 110	. . . . 5 ((a v b_|_) == (a_|_ ^ b_|__|_)_|_) = 1
12	9, 11	w2an 355	. . . 4 (((a v b) ^ (a v b_|_)) == ((a_|_ ^ b_|_)_|_ ^ (a_|_ ^ b_|__|_)_|_)) = 1
13	12	wr1 189	. . 3 (((a_|_ ^ b_|_)_|_ ^ (a_|_ ^ b_|__|_)_|_) == ((a v b) ^ (a v b_|_))) = 1
14	7, 13	wr2 353	. 2 (((a_|_ ^ b_|_) v (a_|_ ^ b_|__|_))_|_ == ((a v b) ^ (a v b_|_))) = 1
15	4, 14	wr2 353	1 (a == ((a v b) ^ (a v b_|_))) = 1

Syntax hints:   = wb 1  _|_wn 4   == tb 5   v wo 6   ^ wa 7  1wt 9   C wcmtr 28

This theorem is referenced by:  wcom3ii 401

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

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