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Theorem cnmpt2vsca 21140
Description: Continuity of scalar multiplication; analogue of cnmpt22f 20621 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 ) )
cnmpt2vsca.m  |-  ( ph  ->  M  e.  (TopOn `  Y ) )
cnmpt2vsca.a  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( L  tX  M
)  Cn  K ) )
cnmpt2vsca.b  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( L  tX  M
)  Cn  J ) )
Assertion
Ref Expression
cnmpt2vsca  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A  .x.  B
) )  e.  ( ( L  tX  M
)  Cn  J ) )
Distinct variable groups:    x, y, F    x, J, y    x, K, y    x, L    ph, x, y    x, W, y    x, X, y    x, Y, y
Allowed substitution hints:    A( x, y)    B( x, y)    .x. ( x, y)    L( y)    M( x, y)

Proof of Theorem cnmpt2vsca
StepHypRef Expression
1 cnmpt1vsca.l . . . . . . . . . 10  |-  ( ph  ->  L  e.  (TopOn `  X ) )
2 cnmpt2vsca.m . . . . . . . . . 10  |-  ( ph  ->  M  e.  (TopOn `  Y ) )
3 txtopon 20537 . . . . . . . . . 10  |-  ( ( L  e.  (TopOn `  X )  /\  M  e.  (TopOn `  Y )
)  ->  ( L  tX  M )  e.  (TopOn `  ( X  X.  Y
) ) )
41, 2, 3syl2anc 665 . . . . . . . . 9  |-  ( ph  ->  ( L  tX  M
)  e.  (TopOn `  ( X  X.  Y
) ) )
5 cnmpt1vsca.w . . . . . . . . . . 11  |-  ( ph  ->  W  e. TopMod )
6 tlmtrg.f . . . . . . . . . . . 12  |-  F  =  (Scalar `  W )
76tlmscatps 21136 . . . . . . . . . . 11  |-  ( W  e. TopMod  ->  F  e.  TopSp )
85, 7syl 17 . . . . . . . . . 10  |-  ( ph  ->  F  e.  TopSp )
9 eqid 2429 . . . . . . . . . . 11  |-  ( Base `  F )  =  (
Base `  F )
10 cnmpt1vsca.k . . . . . . . . . . 11  |-  K  =  ( TopOpen `  F )
119, 10istps 19882 . . . . . . . . . 10  |-  ( F  e.  TopSp 
<->  K  e.  (TopOn `  ( Base `  F )
) )
128, 11sylib 199 . . . . . . . . 9  |-  ( ph  ->  K  e.  (TopOn `  ( Base `  F )
) )
13 cnmpt2vsca.a . . . . . . . . 9  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( L  tX  M
)  Cn  K ) )
14 cnf2 20196 . . . . . . . . 9  |-  ( ( ( L  tX  M
)  e.  (TopOn `  ( X  X.  Y
) )  /\  K  e.  (TopOn `  ( Base `  F ) )  /\  ( x  e.  X ,  y  e.  Y  |->  A )  e.  ( ( L  tX  M
)  Cn  K ) )  ->  ( x  e.  X ,  y  e.  Y  |->  A ) : ( X  X.  Y
) --> ( Base `  F
) )
154, 12, 13, 14syl3anc 1264 . . . . . . . 8  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  A ) : ( X  X.  Y ) --> ( Base `  F
) )
16 eqid 2429 . . . . . . . . 9  |-  ( x  e.  X ,  y  e.  Y  |->  A )  =  ( x  e.  X ,  y  e.  Y  |->  A )
1716fmpt2 6874 . . . . . . . 8  |-  ( A. x  e.  X  A. y  e.  Y  A  e.  ( Base `  F
)  <->  ( x  e.  X ,  y  e.  Y  |->  A ) : ( X  X.  Y
) --> ( Base `  F
) )
1815, 17sylibr 215 . . . . . . 7  |-  ( ph  ->  A. x  e.  X  A. y  e.  Y  A  e.  ( Base `  F ) )
1918r19.21bi 2801 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  A. y  e.  Y  A  e.  ( Base `  F )
)
2019r19.21bi 2801 . . . . 5  |-  ( ( ( ph  /\  x  e.  X )  /\  y  e.  Y )  ->  A  e.  ( Base `  F
) )
21 tlmtps 21133 . . . . . . . . . . 11  |-  ( W  e. TopMod  ->  W  e.  TopSp )
225, 21syl 17 . . . . . . . . . 10  |-  ( ph  ->  W  e.  TopSp )
23 eqid 2429 . . . . . . . . . . 11  |-  ( Base `  W )  =  (
Base `  W )
24 cnmpt1vsca.j . . . . . . . . . . 11  |-  J  =  ( TopOpen `  W )
2523, 24istps 19882 . . . . . . . . . 10  |-  ( W  e.  TopSp 
<->  J  e.  (TopOn `  ( Base `  W )
) )
2622, 25sylib 199 . . . . . . . . 9  |-  ( ph  ->  J  e.  (TopOn `  ( Base `  W )
) )
27 cnmpt2vsca.b . . . . . . . . 9  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( L  tX  M
)  Cn  J ) )
28 cnf2 20196 . . . . . . . . 9  |-  ( ( ( L  tX  M
)  e.  (TopOn `  ( X  X.  Y
) )  /\  J  e.  (TopOn `  ( Base `  W ) )  /\  ( x  e.  X ,  y  e.  Y  |->  B )  e.  ( ( L  tX  M
)  Cn  J ) )  ->  ( x  e.  X ,  y  e.  Y  |->  B ) : ( X  X.  Y
) --> ( Base `  W
) )
294, 26, 27, 28syl3anc 1264 . . . . . . . 8  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  B ) : ( X  X.  Y ) --> ( Base `  W
) )
30 eqid 2429 . . . . . . . . 9  |-  ( x  e.  X ,  y  e.  Y  |->  B )  =  ( x  e.  X ,  y  e.  Y  |->  B )
3130fmpt2 6874 . . . . . . . 8  |-  ( A. x  e.  X  A. y  e.  Y  B  e.  ( Base `  W
)  <->  ( x  e.  X ,  y  e.  Y  |->  B ) : ( X  X.  Y
) --> ( Base `  W
) )
3229, 31sylibr 215 . . . . . . 7  |-  ( ph  ->  A. x  e.  X  A. y  e.  Y  B  e.  ( Base `  W ) )
3332r19.21bi 2801 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  A. y  e.  Y  B  e.  ( Base `  W )
)
3433r19.21bi 2801 . . . . 5  |-  ( ( ( ph  /\  x  e.  X )  /\  y  e.  Y )  ->  B  e.  ( Base `  W
) )
35 eqid 2429 . . . . . 6  |-  ( .sf `  W )  =  ( .sf `  W )
36 cnmpt1vsca.t . . . . . 6  |-  .x.  =  ( .s `  W )
3723, 6, 9, 35, 36scafval 18045 . . . . 5  |-  ( ( A  e.  ( Base `  F )  /\  B  e.  ( Base `  W
) )  ->  ( A ( .sf `  W ) B )  =  ( A  .x.  B ) )
3820, 34, 37syl2anc 665 . . . 4  |-  ( ( ( ph  /\  x  e.  X )  /\  y  e.  Y )  ->  ( A ( .sf `  W ) B )  =  ( A  .x.  B ) )
39383impa 1200 . . 3  |-  ( (
ph  /\  x  e.  X  /\  y  e.  Y
)  ->  ( A
( .sf `  W ) B )  =  ( A  .x.  B ) )
4039mpt2eq3dva 6369 . 2  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A ( .sf `  W ) B ) )  =  ( x  e.  X ,  y  e.  Y  |->  ( A  .x.  B
) ) )
4135, 24, 6, 10vscacn 21131 . . . 4  |-  ( W  e. TopMod  ->  ( .sf `  W )  e.  ( ( K  tX  J
)  Cn  J ) )
425, 41syl 17 . . 3  |-  ( ph  ->  ( .sf `  W )  e.  ( ( K  tX  J
)  Cn  J ) )
431, 2, 13, 27, 42cnmpt22f 20621 . 2  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A ( .sf `  W ) B ) )  e.  ( ( L  tX  M )  Cn  J
) )
4440, 43eqeltrrd 2518 1  |-  ( ph  ->  ( x  e.  X ,  y  e.  Y  |->  ( A  .x.  B
) )  e.  ( ( L  tX  M
)  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782    X. cxp 4852   -->wf 5597   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   Basecbs 15084  Scalarcsca 15155   .scvsca 15156   TopOpenctopn 15279   .sfcscaf 18027  TopOnctopon 19849   TopSpctps 19850    Cn ccn 20171    tX ctx 20506  TopModctlm 21103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7482  df-slot 15088  df-base 15089  df-topgen 15301  df-scaf 18029  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cn 20174  df-tx 20508  df-tmd 21018  df-tgp 21019  df-trg 21105  df-tlm 21107
This theorem is referenced by: (None)
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