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Metamath Proof Explorer

Theorem cncfmpt1f

Description: Composition of continuous functions. -cn-> analogue of cnmpt11f . (Contributed by Mario Carneiro, 3-Sep-2014)

Ref Expression
Hypotheses cncfmpt1f.1 
|- ( ph -> F e. ( CC -cn-> CC ) )
cncfmpt1f.2 
|- ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )
Assertion cncfmpt1f 
|- ( ph -> ( x e. X |-> ( F ` A ) ) e. ( X -cn-> CC ) )

Proof

Step Hyp Ref Expression
1 cncfmpt1f.1 
 |-  ( ph -> F e. ( CC -cn-> CC ) )
2 cncfmpt1f.2 
 |-  ( ph -> ( x e. X |-> A ) e. ( X -cn-> CC ) )
3 cncff 
 |-  ( ( x e. X |-> A ) e. ( X -cn-> CC ) -> ( x e. X |-> A ) : X --> CC )
4 2 3 syl 
 |-  ( ph -> ( x e. X |-> A ) : X --> CC )
5 eqid 
 |-  ( x e. X |-> A ) = ( x e. X |-> A )
6 5 fmpt 
 |-  ( A. x e. X A e. CC <-> ( x e. X |-> A ) : X --> CC )
7 4 6 sylibr 
 |-  ( ph -> A. x e. X A e. CC )
8 eqidd 
 |-  ( ph -> ( x e. X |-> A ) = ( x e. X |-> A ) )
9 cncff 
 |-  ( F e. ( CC -cn-> CC ) -> F : CC --> CC )
10 1 9 syl 
 |-  ( ph -> F : CC --> CC )
11 10 feqmptd 
 |-  ( ph -> F = ( y e. CC |-> ( F ` y ) ) )
12 fveq2 
 |-  ( y = A -> ( F ` y ) = ( F ` A ) )
13 7 8 11 12 fmptcof 
 |-  ( ph -> ( F o. ( x e. X |-> A ) ) = ( x e. X |-> ( F ` A ) ) )
14 2 1 cncfco 
 |-  ( ph -> ( F o. ( x e. X |-> A ) ) e. ( X -cn-> CC ) )
15 13 14 eqeltrrd 
 |-  ( ph -> ( x e. X |-> ( F ` A ) ) e. ( X -cn-> CC ) )