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Theorem cnmpt1vsca 19910
Description: Continuity of scalar multiplication; analogue of cnmpt12f 19381 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 19907 . . . . . . . . 9  |-  ( W  e. TopMod  ->  F  e.  TopSp )
52, 4syl 16 . . . . . . . 8  |-  ( ph  ->  F  e.  TopSp )
6 eqid 2454 . . . . . . . . 9  |-  ( Base `  F )  =  (
Base `  F )
7 cnmpt1vsca.k . . . . . . . . 9  |-  K  =  ( TopOpen `  F )
86, 7istps 18683 . . . . . . . 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 18995 . . . . . . 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 1219 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  A ) : X --> ( Base `  F )
)
13 eqid 2454 . . . . . . 7  |-  ( x  e.  X  |->  A )  =  ( x  e.  X  |->  A )
1413fmpt 5976 . . . . . 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 2920 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  ( Base `  F
) )
17 tlmtps 19904 . . . . . . . . 9  |-  ( W  e. TopMod  ->  W  e.  TopSp )
182, 17syl 16 . . . . . . . 8  |-  ( ph  ->  W  e.  TopSp )
19 eqid 2454 . . . . . . . . 9  |-  ( Base `  W )  =  (
Base `  W )
20 cnmpt1vsca.j . . . . . . . . 9  |-  J  =  ( TopOpen `  W )
2119, 20istps 18683 . . . . . . . 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 18995 . . . . . . 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 1219 . . . . . 6  |-  ( ph  ->  ( x  e.  X  |->  B ) : X --> ( Base `  W )
)
26 eqid 2454 . . . . . . 7  |-  ( x  e.  X  |->  B )  =  ( x  e.  X  |->  B )
2726fmpt 5976 . . . . . 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 2920 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  ( Base `  W
) )
30 eqid 2454 . . . . 5  |-  ( .sf `  W )  =  ( .sf `  W )
31 cnmpt1vsca.t . . . . 5  |-  .x.  =  ( .s `  W )
3219, 3, 6, 30, 31scafval 17100 . . . 4  |-  ( ( A  e.  ( Base `  F )  /\  B  e.  ( Base `  W
) )  ->  ( A ( .sf `  W ) B )  =  ( A  .x.  B ) )
3316, 29, 32syl2anc 661 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( A ( .sf `  W ) B )  =  ( A  .x.  B ) )
3433mpteq2dva 4489 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( .sf `  W ) B ) )  =  ( x  e.  X  |->  ( A  .x.  B
) ) )
3530, 20, 3, 7vscacn 19902 . . . 4  |-  ( W  e. TopMod  ->  ( .sf `  W )  e.  ( ( K  tX  J
)  Cn  J ) )
362, 35syl 16 . . 3  |-  ( ph  ->  ( .sf `  W )  e.  ( ( K  tX  J
)  Cn  J ) )
371, 10, 23, 36cnmpt12f 19381 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( A ( .sf `  W ) B ) )  e.  ( L  Cn  J
) )
3834, 37eqeltrrd 2543 1  |-  ( ph  ->  ( x  e.  X  |->  ( A  .x.  B
) )  e.  ( L  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799    |-> cmpt 4461   -->wf 5525   ` cfv 5529  (class class class)co 6203   Basecbs 14296  Scalarcsca 14364   .scvsca 14365   TopOpenctopn 14483   .sfcscaf 17082  TopOnctopon 18641   TopSpctps 18643    Cn ccn 18970    tX ctx 19275  TopModctlm 19874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-map 7329  df-slot 14300  df-base 14301  df-topgen 14505  df-scaf 17084  df-top 18645  df-bases 18647  df-topon 18648  df-topsp 18649  df-cn 18973  df-tx 19277  df-tmd 19785  df-tgp 19786  df-trg 19876  df-tlm 19878
This theorem is referenced by:  tlmtgp  19912
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