Theorem cnmpt1vsca 21997

Description: Continuity of scalar multiplication; analogue of cnmpt12f 21469 which cannot be used directly because .s is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
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

Ref	Expression
cnmpt1vsca	|- ( ph -> ( x e. X |-> ( A .x. B ) ) e. ( L Cn J ) )

Distinct variable groups:   x, F   x, J   x, K   x, L   ph, x   x, W   x, X

Allowed substitution hints:   A( x)   B( x)   .x. ( x)

Proof of Theorem cnmpt1vsca

Step	Hyp	Ref	Expression
1		cnmpt1vsca.l	. . . . . . 7 |- ( ph -> L e. (TopOn ` X ) )
2		cnmpt1vsca.w	. . . . . . . . 9 |- ( ph -> W e. TopMod )
3		tlmtrg.f	. . . . . . . . . 10 |- F = (Scalar ` W )
4	3	tlmscatps 21994	. . . . . . . . 9 |- ( W e. TopMod -> F e. TopSp )
5	2, 4	syl 17	. . . . . . . 8 |- ( ph -> F e. TopSp )
6		eqid 2622	. . . . . . . . 9 |- ( Base ` F ) = ( Base ` F )
7		cnmpt1vsca.k	. . . . . . . . 9 |- K = ( TopOpen ` F )
8	6, 7	istps 20738	. . . . . . . 8 |- ( F e. TopSp <-> K e. (TopOn ` ( Base ` F ) ) )
9	5, 8	sylib 208	. . . . . . 7 |- ( ph -> K e. (TopOn ` ( Base ` F ) ) )
10		cnmpt1vsca.a	. . . . . . 7 |- ( ph -> ( x e. X |-> A ) e. ( L Cn K ) )
11		cnf2 21053	. . . . . . 7 |- ( ( L e. (TopOn ` X ) /\ K e. (TopOn ` ( Base ` F ) ) /\ ( x e. X |-> A ) e. ( L Cn K ) ) -> ( x e. X |-> A ) : X --> ( Base ` F ) )
12	1, 9, 10, 11	syl3anc 1326	. . . . . 6 |- ( ph -> ( x e. X |-> A ) : X --> ( Base ` F ) )
13		eqid 2622	. . . . . . 7 |- ( x e. X |-> A ) = ( x e. X |-> A )
14	13	fmpt 6381	. . . . . 6 |- ( A. x e. X A e. ( Base ` F ) <-> ( x e. X |-> A ) : X --> ( Base ` F ) )
15	12, 14	sylibr 224	. . . . 5 |- ( ph -> A. x e. X A e. ( Base ` F ) )
16	15	r19.21bi 2932	. . . 4 |- ( ( ph /\ x e. X ) -> A e. ( Base ` F ) )
17		tlmtps 21991	. . . . . . . . 9 |- ( W e. TopMod -> W e. TopSp )
18	2, 17	syl 17	. . . . . . . 8 |- ( ph -> W e. TopSp )
19		eqid 2622	. . . . . . . . 9 |- ( Base ` W ) = ( Base ` W )
20		cnmpt1vsca.j	. . . . . . . . 9 |- J = ( TopOpen ` W )
21	19, 20	istps 20738	. . . . . . . 8 |- ( W e. TopSp <-> J e. (TopOn ` ( Base ` W ) ) )
22	18, 21	sylib 208	. . . . . . 7 |- ( ph -> J e. (TopOn ` ( Base ` W ) ) )
23		cnmpt1vsca.b	. . . . . . 7 |- ( ph -> ( x e. X |-> B ) e. ( L Cn J ) )
24		cnf2 21053	. . . . . . 7 |- ( ( L e. (TopOn ` X ) /\ J e. (TopOn ` ( Base ` W ) ) /\ ( x e. X |-> B ) e. ( L Cn J ) ) -> ( x e. X |-> B ) : X --> ( Base ` W ) )
25	1, 22, 23, 24	syl3anc 1326	. . . . . 6 |- ( ph -> ( x e. X |-> B ) : X --> ( Base ` W ) )
26		eqid 2622	. . . . . . 7 |- ( x e. X |-> B ) = ( x e. X |-> B )
27	26	fmpt 6381	. . . . . 6 |- ( A. x e. X B e. ( Base ` W ) <-> ( x e. X |-> B ) : X --> ( Base ` W ) )
28	25, 27	sylibr 224	. . . . 5 |- ( ph -> A. x e. X B e. ( Base ` W ) )
29	28	r19.21bi 2932	. . . 4 |- ( ( ph /\ x e. X ) -> B e. ( Base ` W ) )
30		eqid 2622	. . . . 5 |- ( .sf ` W ) = ( .sf ` W )
31		cnmpt1vsca.t	. . . . 5 |- .x. = ( .s ` W )
32	19, 3, 6, 30, 31	scafval 18882	. . . 4 |- ( ( A e. ( Base ` F ) /\ B e. ( Base ` W ) ) -> ( A ( .sf ` W ) B ) = ( A .x. B ) )
33	16, 29, 32	syl2anc 693	. . 3 |- ( ( ph /\ x e. X ) -> ( A ( .sf ` W ) B ) = ( A .x. B ) )
34	33	mpteq2dva 4744	. 2 |- ( ph -> ( x e. X |-> ( A ( .sf ` W ) B ) ) = ( x e. X |-> ( A .x. B ) ) )
35	30, 20, 3, 7	vscacn 21989	. . . 4 |- ( W e. TopMod -> ( .sf ` W ) e. ( ( K tX J ) Cn J ) )
36	2, 35	syl 17	. . 3 |- ( ph -> ( .sf ` W ) e. ( ( K tX J ) Cn J ) )
37	1, 10, 23, 36	cnmpt12f 21469	. 2 |- ( ph -> ( x e. X |-> ( A ( .sf ` W ) B ) ) e. ( L Cn J ) )
38	34, 37	eqeltrrd 2702	1 |- ( ph -> ( x e. X |-> ( A .x. B ) ) e. ( L Cn J ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   TopOpenctopn 16082   .sfcscaf 18864  TopOnctopon 20715   TopSpctps 20736   Cn ccn 21028   tX ctx 21363  TopModctlm 21961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-slot 15861  df-base 15863  df-topgen 16104  df-scaf 18866  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-tx 21365  df-tmd 21876  df-tgp 21877  df-trg 21963  df-tlm 21965

This theorem is referenced by:  tlmtgp  21999