Theorem wcom0 407

Description: Commutation with 0. Kalmbach 83 p. 20. (Contributed by NM, 13-Oct-1997.)

Assertion

Ref	Expression
wcom0	C (a, 0) = 1

Proof of Theorem wcom0

Step	Hyp	Ref	Expression
1		comm0 178	. . . 4 a C 0
2	1	df-c2 133	. . 3 a = ((a ∩ 0) ∪ (a ∩ 0⊥ ))
3	2	bi1 118	. 2 (a ≡ ((a ∩ 0) ∪ (a ∩ 0⊥ ))) = 1
4	3	wdf-c1 383	1 C (a, 0) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   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-c1 132  df-c2 133  df-cmtr 134

This theorem is referenced by: (None)