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Theorem cnmpt1vsca 21286
Description: Continuity of scalar multiplication; analogue of cnmpt12f 20758 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 21283 . . . . . . . . 9  |-  ( W  e. TopMod  ->  F  e.  TopSp )
52, 4syl 17 . . . . . . . 8  |-  ( ph  ->  F  e.  TopSp )
6 eqid 2471 . . . . . . . . 9  |-  ( Base `  F )  =  (
Base `  F )
7 cnmpt1vsca.k . . . . . . . . 9  |-  K  =  ( TopOpen `  F )
86, 7istps 20028 . . . . . . . 8  |-  ( F  e.  TopSp 
<->  K  e.  (TopOn `  ( Base `  F )
) )
95, 8sylib 201 . . . . . . 7  |-  ( ph  ->  K  e.  (TopOn `  ( Base `  F )
) )
10 cnmpt1vsca.a . . . . . . 7  |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( L  Cn  K ) )
11 cnf2 20342 . . . . . . 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 1292 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  A ) : X --> ( Base `  F )
)
13 eqid 2471 . . . . . . 7  |-  ( x  e.  X  |->  A )  =  ( x  e.  X  |->  A )
1413fmpt 6058 . . . . . 6  |-  ( A. x  e.  X  A  e.  ( Base `  F
)  <->  ( x  e.  X  |->  A ) : X --> ( Base `  F
) )
1512, 14sylibr 217 . . . . 5  |-  ( ph  ->  A. x  e.  X  A  e.  ( Base `  F ) )
1615r19.21bi 2776 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  ( Base `  F
) )
17 tlmtps 21280 . . . . . . . . 9  |-  ( W  e. TopMod  ->  W  e.  TopSp )
182, 17syl 17 . . . . . . . 8  |-  ( ph  ->  W  e.  TopSp )
19 eqid 2471 . . . . . . . . 9  |-  ( Base `  W )  =  (
Base `  W )
20 cnmpt1vsca.j . . . . . . . . 9  |-  J  =  ( TopOpen `  W )
2119, 20istps 20028 . . . . . . . 8  |-  ( W  e.  TopSp 
<->  J  e.  (TopOn `  ( Base `  W )
) )
2218, 21sylib 201 . . . . . . 7  |-  ( ph  ->  J  e.  (TopOn `  ( Base `  W )
) )
23 cnmpt1vsca.b . . . . . . 7  |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( L  Cn  J ) )
24 cnf2 20342 . . . . . . 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 1292 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  B ) : X --> ( Base `  W )
)
26 eqid 2471 . . . . . . 7  |-  ( x  e.  X  |->  B )  =  ( x  e.  X  |->  B )
2726fmpt 6058 . . . . . 6  |-  ( A. x  e.  X  B  e.  ( Base `  W
)  <->  ( x  e.  X  |->  B ) : X --> ( Base `  W
) )
2825, 27sylibr 217 . . . . 5  |-  ( ph  ->  A. x  e.  X  B  e.  ( Base `  W ) )
2928r19.21bi 2776 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  ( Base `  W
) )
30 eqid 2471 . . . . 5  |-  ( .sf `  W )  =  ( .sf `  W )
31 cnmpt1vsca.t . . . . 5  |-  .x.  =  ( .s `  W )
3219, 3, 6, 30, 31scafval 18188 . . . 4  |-  ( ( A  e.  ( Base `  F )  /\  B  e.  ( Base `  W
) )  ->  ( A ( .sf `  W ) B )  =  ( A  .x.  B ) )
3316, 29, 32syl2anc 673 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( A ( .sf `  W ) B )  =  ( A  .x.  B ) )
3433mpteq2dva 4482 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( .sf `  W ) B ) )  =  ( x  e.  X  |->  ( A  .x.  B
) ) )
3530, 20, 3, 7vscacn 21278 . . . 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 20758 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( .sf `  W ) B ) )  e.  ( L  Cn  J
) )
3834, 37eqeltrrd 2550 1  |-  ( ph  ->  ( x  e.  X  |->  ( A  .x.  B
) )  e.  ( L  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756    |-> cmpt 4454   -->wf 5585   ` cfv 5589  (class class class)co 6308   Basecbs 15199  Scalarcsca 15271   .scvsca 15272   TopOpenctopn 15398   .sfcscaf 18170  TopOnctopon 19995   TopSpctps 19996    Cn ccn 20317    tX ctx 20652  TopModctlm 21250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-map 7492  df-slot 15203  df-base 15204  df-topgen 15420  df-scaf 18172  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cn 20320  df-tx 20654  df-tmd 21165  df-tgp 21166  df-trg 21252  df-tlm 21254
This theorem is referenced by:  tlmtgp  21288
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