Metamath Proof Explorer


Theorem divccncf

Description: Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007)

Ref Expression
Hypothesis divccncf.1 F = x x A
Assertion divccncf A A 0 F : cn

Proof

Step Hyp Ref Expression
1 divccncf.1 F = x x A
2 divrec2 x A A 0 x A = 1 A x
3 2 3expb x A A 0 x A = 1 A x
4 3 ancoms A A 0 x x A = 1 A x
5 4 mpteq2dva A A 0 x x A = x 1 A x
6 1 5 eqtrid A A 0 F = x 1 A x
7 reccl A A 0 1 A
8 eqid x 1 A x = x 1 A x
9 8 mulc1cncf 1 A x 1 A x : cn
10 7 9 syl A A 0 x 1 A x : cn
11 6 10 eqeltrd A A 0 F : cn