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Metamath Proof Explorer

Theorem cnmpt1vsca

Description: Continuity of scalar multiplication; analogue of cnmpt12f which cannot be used directly because .s is not a function. (Contributed by Mario Carneiro, 5-Oct-2015)

Ref Expression
Hypotheses tlmtrg.f 
|- F = ( Scalar ` W )
cnmpt1vsca.t 
|- .x. = ( .s ` W )
cnmpt1vsca.j 
|- J = ( TopOpen ` W )
cnmpt1vsca.k 
|- K = ( TopOpen ` F )
cnmpt1vsca.w 
|- ( ph -> W e. TopMod )
cnmpt1vsca.l 
|- ( ph -> L e. ( TopOn ` X ) )
cnmpt1vsca.a 
|- ( ph -> ( x e. X |-> A ) e. ( L Cn K ) )
cnmpt1vsca.b 
|- ( ph -> ( x e. X |-> B ) e. ( L Cn J ) )
Assertion cnmpt1vsca 
|- ( ph -> ( x e. X |-> ( A .x. B ) ) e. ( L Cn J ) )

Proof

Step Hyp Ref Expression
1 tlmtrg.f 
 |-  F = ( Scalar ` W )
2 cnmpt1vsca.t 
 |-  .x. = ( .s ` W )
3 cnmpt1vsca.j 
 |-  J = ( TopOpen ` W )
4 cnmpt1vsca.k 
 |-  K = ( TopOpen ` F )
5 cnmpt1vsca.w 
 |-  ( ph -> W e. TopMod )
6 cnmpt1vsca.l 
 |-  ( ph -> L e. ( TopOn ` X ) )
7 cnmpt1vsca.a 
 |-  ( ph -> ( x e. X |-> A ) e. ( L Cn K ) )
8 cnmpt1vsca.b 
 |-  ( ph -> ( x e. X |-> B ) e. ( L Cn J ) )
9 1 tlmscatps 
 |-  ( W e. TopMod -> F e. TopSp )
10 5 9 syl 
 |-  ( ph -> F e. TopSp )
11 eqid 
 |-  ( Base ` F ) = ( Base ` F )
12 11 4 istps 
 |-  ( F e. TopSp <-> K e. ( TopOn ` ( Base ` F ) ) )
13 10 12 sylib 
 |-  ( ph -> K e. ( TopOn ` ( Base ` F ) ) )
14 cnf2 
 |-  ( ( L e. ( TopOn ` X ) /\ K e. ( TopOn ` ( Base ` F ) ) /\ ( x e. X |-> A ) e. ( L Cn K ) ) -> ( x e. X |-> A ) : X --> ( Base ` F ) )
15 6 13 7 14 syl3anc 
 |-  ( ph -> ( x e. X |-> A ) : X --> ( Base ` F ) )
16 15 fvmptelcdm 
 |-  ( ( ph /\ x e. X ) -> A e. ( Base ` F ) )
17 tlmtps 
 |-  ( W e. TopMod -> W e. TopSp )
18 5 17 syl 
 |-  ( ph -> W e. TopSp )
19 eqid 
 |-  ( Base ` W ) = ( Base ` W )
20 19 3 istps 
 |-  ( W e. TopSp <-> J e. ( TopOn ` ( Base ` W ) ) )
21 18 20 sylib 
 |-  ( ph -> J e. ( TopOn ` ( Base ` W ) ) )
22 cnf2 
 |-  ( ( L e. ( TopOn ` X ) /\ J e. ( TopOn ` ( Base ` W ) ) /\ ( x e. X |-> B ) e. ( L Cn J ) ) -> ( x e. X |-> B ) : X --> ( Base ` W ) )
23 6 21 8 22 syl3anc 
 |-  ( ph -> ( x e. X |-> B ) : X --> ( Base ` W ) )
24 23 fvmptelcdm 
 |-  ( ( ph /\ x e. X ) -> B e. ( Base ` W ) )
25 eqid 
 |-  ( .sf ` W ) = ( .sf ` W )
26 19 1 11 25 2 scafval 
 |-  ( ( A e. ( Base ` F ) /\ B e. ( Base ` W ) ) -> ( A ( .sf ` W ) B ) = ( A .x. B ) )
27 16 24 26 syl2anc 
 |-  ( ( ph /\ x e. X ) -> ( A ( .sf ` W ) B ) = ( A .x. B ) )
28 27 mpteq2dva 
 |-  ( ph -> ( x e. X |-> ( A ( .sf ` W ) B ) ) = ( x e. X |-> ( A .x. B ) ) )
29 25 3 1 4 vscacn 
 |-  ( W e. TopMod -> ( .sf ` W ) e. ( ( K tX J ) Cn J ) )
30 5 29 syl 
 |-  ( ph -> ( .sf ` W ) e. ( ( K tX J ) Cn J ) )
31 6 7 8 30 cnmpt12f 
 |-  ( ph -> ( x e. X |-> ( A ( .sf ` W ) B ) ) e. ( L Cn J ) )
32 28 31 eqeltrrd 
 |-  ( ph -> ( x e. X |-> ( A .x. B ) ) e. ( L Cn J ) )