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Theorem cnmpt1vsca 20880
Description: Continuity of scalar multiplication; analogue of cnmpt12f 20351 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
StepHypRef 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 )
43tlmscatps 20877 . . . . . . . . 9  |-  ( W  e. TopMod  ->  F  e.  TopSp )
52, 4syl 17 . . . . . . . 8  |-  ( ph  ->  F  e.  TopSp )
6 eqid 2402 . . . . . . . . 9  |-  ( Base `  F )  =  (
Base `  F )
7 cnmpt1vsca.k . . . . . . . . 9  |-  K  =  ( TopOpen `  F )
86, 7istps 19621 . . . . . . . 8  |-  ( F  e.  TopSp 
<->  K  e.  (TopOn `  ( Base `  F )
) )
95, 8sylib 196 . . . . . . 7  |-  ( ph  ->  K  e.  (TopOn `  ( Base `  F )
) )
10 cnmpt1vsca.a . . . . . . 7  |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( L  Cn  K ) )
11 cnf2 19935 . . . . . . 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 ) )
121, 9, 10, 11syl3anc 1230 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  A ) : X --> ( Base `  F )
)
13 eqid 2402 . . . . . . 7  |-  ( x  e.  X  |->  A )  =  ( x  e.  X  |->  A )
1413fmpt 5986 . . . . . 6  |-  ( A. x  e.  X  A  e.  ( Base `  F
)  <->  ( x  e.  X  |->  A ) : X --> ( Base `  F
) )
1512, 14sylibr 212 . . . . 5  |-  ( ph  ->  A. x  e.  X  A  e.  ( Base `  F ) )
1615r19.21bi 2772 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  ( Base `  F
) )
17 tlmtps 20874 . . . . . . . . 9  |-  ( W  e. TopMod  ->  W  e.  TopSp )
182, 17syl 17 . . . . . . . 8  |-  ( ph  ->  W  e.  TopSp )
19 eqid 2402 . . . . . . . . 9  |-  ( Base `  W )  =  (
Base `  W )
20 cnmpt1vsca.j . . . . . . . . 9  |-  J  =  ( TopOpen `  W )
2119, 20istps 19621 . . . . . . . 8  |-  ( W  e.  TopSp 
<->  J  e.  (TopOn `  ( Base `  W )
) )
2218, 21sylib 196 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  ( Base `  W )
) )
23 cnmpt1vsca.b . . . . . . 7  |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( L  Cn  J ) )
24 cnf2 19935 . . . . . . 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 ) )
251, 22, 23, 24syl3anc 1230 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  B ) : X --> ( Base `  W )
)
26 eqid 2402 . . . . . . 7  |-  ( x  e.  X  |->  B )  =  ( x  e.  X  |->  B )
2726fmpt 5986 . . . . . 6  |-  ( A. x  e.  X  B  e.  ( Base `  W
)  <->  ( x  e.  X  |->  B ) : X --> ( Base `  W
) )
2825, 27sylibr 212 . . . . 5  |-  ( ph  ->  A. x  e.  X  B  e.  ( Base `  W ) )
2928r19.21bi 2772 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  ( Base `  W
) )
30 eqid 2402 . . . . 5  |-  ( .sf `  W )  =  ( .sf `  W )
31 cnmpt1vsca.t . . . . 5  |-  .x.  =  ( .s `  W )
3219, 3, 6, 30, 31scafval 17743 . . . 4  |-  ( ( A  e.  ( Base `  F )  /\  B  e.  ( Base `  W
) )  ->  ( A ( .sf `  W ) B )  =  ( A  .x.  B ) )
3316, 29, 32syl2anc 659 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( A ( .sf `  W ) B )  =  ( A  .x.  B ) )
3433mpteq2dva 4480 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( .sf `  W ) B ) )  =  ( x  e.  X  |->  ( A  .x.  B
) ) )
3530, 20, 3, 7vscacn 20872 . . . 4  |-  ( W  e. TopMod  ->  ( .sf `  W )  e.  ( ( K  tX  J
)  Cn  J ) )
362, 35syl 17 . . 3  |-  ( ph  ->  ( .sf `  W )  e.  ( ( K  tX  J
)  Cn  J ) )
371, 10, 23, 36cnmpt12f 20351 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( .sf `  W ) B ) )  e.  ( L  Cn  J
) )
3834, 37eqeltrrd 2491 1  |-  ( ph  ->  ( x  e.  X  |->  ( A  .x.  B
) )  e.  ( L  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753    |-> cmpt 4452   -->wf 5521   ` cfv 5525  (class class class)co 6234   Basecbs 14733  Scalarcsca 14804   .scvsca 14805   TopOpenctopn 14928   .sfcscaf 17725  TopOnctopon 19579   TopSpctps 19581    Cn ccn 19910    tX ctx 20245  TopModctlm 20844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739  df-map 7379  df-slot 14737  df-base 14738  df-topgen 14950  df-scaf 17727  df-top 19583  df-bases 19585  df-topon 19586  df-topsp 19587  df-cn 19913  df-tx 20247  df-tmd 20755  df-tgp 20756  df-trg 20846  df-tlm 20848
This theorem is referenced by:  tlmtgp  20882
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