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Theorem prdsvscaval 14857
Description: Scalar multiplication in a structure product is pointwise. (Contributed by Stefan O'Rear, 10-Jan-2015.)
Hypotheses
Ref Expression
prdsbasmpt.y  |-  Y  =  ( S X_s R )
prdsbasmpt.b  |-  B  =  ( Base `  Y
)
prdsvscaval.t  |-  .x.  =  ( .s `  Y )
prdsvscaval.k  |-  K  =  ( Base `  S
)
prdsvscaval.s  |-  ( ph  ->  S  e.  V )
prdsvscaval.i  |-  ( ph  ->  I  e.  W )
prdsvscaval.r  |-  ( ph  ->  R  Fn  I )
prdsvscaval.f  |-  ( ph  ->  F  e.  K )
prdsvscaval.g  |-  ( ph  ->  G  e.  B )
Assertion
Ref Expression
prdsvscaval  |-  ( ph  ->  ( F  .x.  G
)  =  ( x  e.  I  |->  ( F ( .s `  ( R `  x )
) ( G `  x ) ) ) )
Distinct variable groups:    x, B    x, F    x, G    ph, x    x, I    x, V    x, R    x, S    x, W    x, Y
Allowed substitution hints:    .x. ( x)    K( x)

Proof of Theorem prdsvscaval
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsbasmpt.y . . 3  |-  Y  =  ( S X_s R )
2 prdsvscaval.s . . 3  |-  ( ph  ->  S  e.  V )
3 prdsvscaval.r . . . 4  |-  ( ph  ->  R  Fn  I )
4 prdsvscaval.i . . . 4  |-  ( ph  ->  I  e.  W )
5 fnex 6124 . . . 4  |-  ( ( R  Fn  I  /\  I  e.  W )  ->  R  e.  _V )
63, 4, 5syl2anc 661 . . 3  |-  ( ph  ->  R  e.  _V )
7 prdsbasmpt.b . . 3  |-  B  =  ( Base `  Y
)
8 fndm 5670 . . . 4  |-  ( R  Fn  I  ->  dom  R  =  I )
93, 8syl 16 . . 3  |-  ( ph  ->  dom  R  =  I )
10 prdsvscaval.k . . 3  |-  K  =  ( Base `  S
)
11 prdsvscaval.t . . 3  |-  .x.  =  ( .s `  Y )
121, 2, 6, 7, 9, 10, 11prdsvsca 14838 . 2  |-  ( ph  ->  .x.  =  ( y  e.  K ,  z  e.  B  |->  ( x  e.  I  |->  ( y ( .s `  ( R `  x )
) ( z `  x ) ) ) ) )
13 id 22 . . . . 5  |-  ( y  =  F  ->  y  =  F )
14 fveq1 5855 . . . . 5  |-  ( z  =  G  ->  (
z `  x )  =  ( G `  x ) )
1513, 14oveqan12d 6300 . . . 4  |-  ( ( y  =  F  /\  z  =  G )  ->  ( y ( .s
`  ( R `  x ) ) ( z `  x ) )  =  ( F ( .s `  ( R `  x )
) ( G `  x ) ) )
1615adantl 466 . . 3  |-  ( (
ph  /\  ( y  =  F  /\  z  =  G ) )  -> 
( y ( .s
`  ( R `  x ) ) ( z `  x ) )  =  ( F ( .s `  ( R `  x )
) ( G `  x ) ) )
1716mpteq2dv 4524 . 2  |-  ( (
ph  /\  ( y  =  F  /\  z  =  G ) )  -> 
( x  e.  I  |->  ( y ( .s
`  ( R `  x ) ) ( z `  x ) ) )  =  ( x  e.  I  |->  ( F ( .s `  ( R `  x ) ) ( G `  x ) ) ) )
18 prdsvscaval.f . 2  |-  ( ph  ->  F  e.  K )
19 prdsvscaval.g . 2  |-  ( ph  ->  G  e.  B )
20 mptexg 6127 . . 3  |-  ( I  e.  W  ->  (
x  e.  I  |->  ( F ( .s `  ( R `  x ) ) ( G `  x ) ) )  e.  _V )
214, 20syl 16 . 2  |-  ( ph  ->  ( x  e.  I  |->  ( F ( .s
`  ( R `  x ) ) ( G `  x ) ) )  e.  _V )
2212, 17, 18, 19, 21ovmpt2d 6415 1  |-  ( ph  ->  ( F  .x.  G
)  =  ( x  e.  I  |->  ( F ( .s `  ( R `  x )
) ( G `  x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   _Vcvv 3095    |-> cmpt 4495   dom cdm 4989    Fn wfn 5573   ` cfv 5578  (class class class)co 6281   Basecbs 14613   .scvsca 14682   X_scprds 14824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10986  df-uz 11092  df-fz 11683  df-struct 14615  df-ndx 14616  df-slot 14617  df-base 14618  df-plusg 14691  df-mulr 14692  df-sca 14694  df-vsca 14695  df-ip 14696  df-tset 14697  df-ple 14698  df-ds 14700  df-hom 14702  df-cco 14703  df-prds 14826
This theorem is referenced by:  prdsvscafval  14858  pwsvscafval  14872  xpsvsca  14957  prdsvscacl  17592  prdslmodd  17593
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