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Theorem prdsvscafval 13589
Description: Scalar multiplication of a single coordinate in a structure product. (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 )
prdsvscafval.j  |-  ( ph  ->  J  e.  I )
Assertion
Ref Expression
prdsvscafval  |-  ( ph  ->  ( ( F  .x.  G ) `  J
)  =  ( F ( .s `  ( R `  J )
) ( G `  J ) ) )

Proof of Theorem prdsvscafval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 prdsbasmpt.y . . 3  |-  Y  =  ( S X_s R )
2 prdsbasmpt.b . . 3  |-  B  =  ( Base `  Y
)
3 prdsvscaval.t . . 3  |-  .x.  =  ( .s `  Y )
4 prdsvscaval.k . . 3  |-  K  =  ( Base `  S
)
5 prdsvscaval.s . . 3  |-  ( ph  ->  S  e.  V )
6 prdsvscaval.i . . 3  |-  ( ph  ->  I  e.  W )
7 prdsvscaval.r . . 3  |-  ( ph  ->  R  Fn  I )
8 prdsvscaval.f . . 3  |-  ( ph  ->  F  e.  K )
9 prdsvscaval.g . . 3  |-  ( ph  ->  G  e.  B )
101, 2, 3, 4, 5, 6, 7, 8, 9prdsvscaval 13588 . 2  |-  ( ph  ->  ( F  .x.  G
)  =  ( x  e.  I  |->  ( F ( .s `  ( R `  x )
) ( G `  x ) ) ) )
11 fveq2 5632 . . . . 5  |-  ( x  =  J  ->  ( R `  x )  =  ( R `  J ) )
1211fveq2d 5636 . . . 4  |-  ( x  =  J  ->  ( .s `  ( R `  x ) )  =  ( .s `  ( R `  J )
) )
13 eqidd 2367 . . . 4  |-  ( x  =  J  ->  F  =  F )
14 fveq2 5632 . . . 4  |-  ( x  =  J  ->  ( G `  x )  =  ( G `  J ) )
1512, 13, 14oveq123d 6002 . . 3  |-  ( x  =  J  ->  ( F ( .s `  ( R `  x ) ) ( G `  x ) )  =  ( F ( .s
`  ( R `  J ) ) ( G `  J ) ) )
1615adantl 452 . 2  |-  ( (
ph  /\  x  =  J )  ->  ( F ( .s `  ( R `  x ) ) ( G `  x ) )  =  ( F ( .s
`  ( R `  J ) ) ( G `  J ) ) )
17 prdsvscafval.j . 2  |-  ( ph  ->  J  e.  I )
18 ovex 6006 . . 3  |-  ( F ( .s `  ( R `  J )
) ( G `  J ) )  e. 
_V
1918a1i 10 . 2  |-  ( ph  ->  ( F ( .s
`  ( R `  J ) ) ( G `  J ) )  e.  _V )
2010, 16, 17, 19fvmptd 5713 1  |-  ( ph  ->  ( ( F  .x.  G ) `  J
)  =  ( F ( .s `  ( R `  J )
) ( G `  J ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. wcel 1715   _Vcvv 2873    Fn wfn 5353   ` cfv 5358  (class class class)co 5981   Basecbs 13356   .scvsca 13420   X_scprds 13556
This theorem is referenced by:  prdslmodd  15936  dsmmlss  26716
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-map 6917  df-ixp 6961  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-fz 10936  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-plusg 13429  df-mulr 13430  df-sca 13432  df-vsca 13433  df-tset 13435  df-ple 13436  df-ds 13438  df-hom 13440  df-cco 13441  df-prds 13558
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