Metamath Proof Explorer

Theorem cnmpt1ip

Description: Continuity of inner product; analogue of cnmpt12f which cannot be used directly because .i is not a function. (Contributed by Mario Carneiro, 13-Oct-2015)

Ref Expression
Hypotheses cnmpt1ip.j 
|- J = ( TopOpen ` W )
cnmpt1ip.c 
|- C = ( TopOpen ` CCfld )
cnmpt1ip.h 
|- ., = ( .i ` W )
cnmpt1ip.r 
|- ( ph -> W e. CPreHil )
cnmpt1ip.k 
|- ( ph -> K e. ( TopOn ` X ) )
cnmpt1ip.a 
|- ( ph -> ( x e. X |-> A ) e. ( K Cn J ) )
cnmpt1ip.b 
|- ( ph -> ( x e. X |-> B ) e. ( K Cn J ) )
Assertion cnmpt1ip 
|- ( ph -> ( x e. X |-> ( A ., B ) ) e. ( K Cn C ) )

Proof

Step Hyp Ref Expression
1 cnmpt1ip.j 
 |-  J = ( TopOpen ` W )
2 cnmpt1ip.c 
 |-  C = ( TopOpen ` CCfld )
3 cnmpt1ip.h 
 |-  ., = ( .i ` W )
4 cnmpt1ip.r 
 |-  ( ph -> W e. CPreHil )
5 cnmpt1ip.k 
 |-  ( ph -> K e. ( TopOn ` X ) )
6 cnmpt1ip.a 
 |-  ( ph -> ( x e. X |-> A ) e. ( K Cn J ) )
7 cnmpt1ip.b 
 |-  ( ph -> ( x e. X |-> B ) e. ( K Cn J ) )
8 cphngp 
 |-  ( W e. CPreHil -> W e. NrmGrp )
9 ngptps 
 |-  ( W e. NrmGrp -> W e. TopSp )
10 4 8 9 3syl 
 |-  ( ph -> W e. TopSp )
11 eqid 
 |-  ( Base ` W ) = ( Base ` W )
12 11 1 istps 
 |-  ( W e. TopSp <-> J e. ( TopOn ` ( Base ` W ) ) )
13 10 12 sylib 
 |-  ( ph -> J e. ( TopOn ` ( Base ` W ) ) )
14 cnf2 
 |-  ( ( K e. ( TopOn ` X ) /\ J e. ( TopOn ` ( Base ` W ) ) /\ ( x e. X |-> A ) e. ( K Cn J ) ) -> ( x e. X |-> A ) : X --> ( Base ` W ) )
15 5 13 6 14 syl3anc 
 |-  ( ph -> ( x e. X |-> A ) : X --> ( Base ` W ) )
16 15 fvmptelrn 
 |-  ( ( ph /\ x e. X ) -> A e. ( Base ` W ) )
17 cnf2 
 |-  ( ( K e. ( TopOn ` X ) /\ J e. ( TopOn ` ( Base ` W ) ) /\ ( x e. X |-> B ) e. ( K Cn J ) ) -> ( x e. X |-> B ) : X --> ( Base ` W ) )
18 5 13 7 17 syl3anc 
 |-  ( ph -> ( x e. X |-> B ) : X --> ( Base ` W ) )
19 18 fvmptelrn 
 |-  ( ( ph /\ x e. X ) -> B e. ( Base ` W ) )
20 eqid 
 |-  ( .if ` W ) = ( .if ` W )
21 11 3 20 ipfval 
 |-  ( ( A e. ( Base ` W ) /\ B e. ( Base ` W ) ) -> ( A ( .if ` W ) B ) = ( A ., B ) )
22 16 19 21 syl2anc 
 |-  ( ( ph /\ x e. X ) -> ( A ( .if ` W ) B ) = ( A ., B ) )
23 22 mpteq2dva 
 |-  ( ph -> ( x e. X |-> ( A ( .if ` W ) B ) ) = ( x e. X |-> ( A ., B ) ) )
24 20 1 2 ipcn 
 |-  ( W e. CPreHil -> ( .if ` W ) e. ( ( J tX J ) Cn C ) )
25 4 24 syl 
 |-  ( ph -> ( .if ` W ) e. ( ( J tX J ) Cn C ) )
26 5 6 7 25 cnmpt12f 
 |-  ( ph -> ( x e. X |-> ( A ( .if ` W ) B ) ) e. ( K Cn C ) )
27 23 26 eqeltrrd 
 |-  ( ph -> ( x e. X |-> ( A ., B ) ) e. ( K Cn C ) )