Theorem wdf-c1 365

Description: Show that commutator is a 'commutes' analogue for ≡ analogue of =.

Hypothesis

Ref	Expression
wdf-c1.1	(a ≡ ((a ∩ b) ∪ (a ∩ b⊥ ))) = 1

Assertion

Ref	Expression
wdf-c1	C (a, b) = 1

Proof of Theorem wdf-c1

Step	Hyp	Ref	Expression
1		cmtrcom 182	. 2 C (a, b) = C (b, a)
2		df-cmtr 126	. 2 C (b, a) = (((b ∩ a) ∪ (b ∩ a⊥ )) ∪ ((b⊥ ∩ a) ∪ (b⊥ ∩ a⊥ )))
3		df-t 40	. . . . 5 1 = (b ∪ b⊥ )
4	3	bi1 110	. . . 4 (1 ≡ (b ∪ b⊥ )) = 1
5		wdf-c1.1	. . . . . 6 (a ≡ ((a ∩ b) ∪ (a ∩ b⊥ ))) = 1
6	5	wcomlem 364	. . . . 5 (b ≡ ((b ∩ a) ∪ (b ∩ a⊥ ))) = 1
7		ax-a1 29	. . . . . . . . . . 11 b = b⊥ ⊥
8	7	lan 70	. . . . . . . . . 10 (a ∩ b) = (a ∩ b⊥ ⊥ )
9	8	ax-r5 37	. . . . . . . . 9 ((a ∩ b) ∪ (a ∩ b⊥ )) = ((a ∩ b⊥ ⊥ ) ∪ (a ∩ b⊥ ))
10		ax-a2 30	. . . . . . . . 9 ((a ∩ b⊥ ⊥ ) ∪ (a ∩ b⊥ )) = ((a ∩ b⊥ ) ∪ (a ∩ b⊥ ⊥ ))
11	9, 10	ax-r2 35	. . . . . . . 8 ((a ∩ b) ∪ (a ∩ b⊥ )) = ((a ∩ b⊥ ) ∪ (a ∩ b⊥ ⊥ ))
12	11	bi1 110	. . . . . . 7 (((a ∩ b) ∪ (a ∩ b⊥ )) ≡ ((a ∩ b⊥ ) ∪ (a ∩ b⊥ ⊥ ))) = 1
13	5, 12	wr2 353	. . . . . 6 (a ≡ ((a ∩ b⊥ ) ∪ (a ∩ b⊥ ⊥ ))) = 1
14	13	wcomlem 364	. . . . 5 (b⊥ ≡ ((b⊥ ∩ a) ∪ (b⊥ ∩ a⊥ ))) = 1
15	6, 14	w2or 354	. . . 4 ((b ∪ b⊥ ) ≡ (((b ∩ a) ∪ (b ∩ a⊥ )) ∪ ((b⊥ ∩ a) ∪ (b⊥ ∩ a⊥ )))) = 1
16	4, 15	wr2 353	. . 3 (1 ≡ (((b ∩ a) ∪ (b ∩ a⊥ )) ∪ ((b⊥ ∩ a) ∪ (b⊥ ∩ a⊥ )))) = 1
17	16	wr3 190	. 2 (((b ∩ a) ∪ (b ∩ a⊥ )) ∪ ((b⊥ ∩ a) ∪ (b⊥ ∩ a⊥ ))) = 1
18	1, 2, 17	3tr 62	1 C (a, b) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 9   C wcmtr 28

This theorem is referenced by:  wcom0 389  wcom1 390  wlecom 391  wbctr 392  wcbtr 393  wcomcom2 397  wcomcom5 402  wcomdr 403  wcom3i 404  wcom2or 409

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