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Theorem prdsvscacl 17741
Description: Pointwise scalar multiplication is closed in products of modules. (Contributed by Stefan O'Rear, 10-Jan-2015.)
Hypotheses
Ref Expression
prdsvscacl.y  |-  Y  =  ( S X_s R )
prdsvscacl.b  |-  B  =  ( Base `  Y
)
prdsvscacl.t  |-  .x.  =  ( .s `  Y )
prdsvscacl.k  |-  K  =  ( Base `  S
)
prdsvscacl.s  |-  ( ph  ->  S  e.  Ring )
prdsvscacl.i  |-  ( ph  ->  I  e.  W )
prdsvscacl.r  |-  ( ph  ->  R : I --> LMod )
prdsvscacl.f  |-  ( ph  ->  F  e.  K )
prdsvscacl.g  |-  ( ph  ->  G  e.  B )
prdsvscacl.sr  |-  ( (
ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x
) )  =  S )
Assertion
Ref Expression
prdsvscacl  |-  ( ph  ->  ( F  .x.  G
)  e.  B )
Distinct variable groups:    x, B    x, F    x, G    x, I    x, K    x, R    x, S    ph, x    x, W    x, Y
Allowed substitution hint:    .x. ( x)

Proof of Theorem prdsvscacl
StepHypRef Expression
1 prdsvscacl.y . . 3  |-  Y  =  ( S X_s R )
2 prdsvscacl.b . . 3  |-  B  =  ( Base `  Y
)
3 prdsvscacl.t . . 3  |-  .x.  =  ( .s `  Y )
4 prdsvscacl.k . . 3  |-  K  =  ( Base `  S
)
5 prdsvscacl.s . . 3  |-  ( ph  ->  S  e.  Ring )
6 prdsvscacl.i . . 3  |-  ( ph  ->  I  e.  W )
7 prdsvscacl.r . . . 4  |-  ( ph  ->  R : I --> LMod )
8 ffn 5737 . . . 4  |-  ( R : I --> LMod  ->  R  Fn  I )
97, 8syl 16 . . 3  |-  ( ph  ->  R  Fn  I )
10 prdsvscacl.f . . 3  |-  ( ph  ->  F  e.  K )
11 prdsvscacl.g . . 3  |-  ( ph  ->  G  e.  B )
121, 2, 3, 4, 5, 6, 9, 10, 11prdsvscaval 14896 . 2  |-  ( ph  ->  ( F  .x.  G
)  =  ( x  e.  I  |->  ( F ( .s `  ( R `  x )
) ( G `  x ) ) ) )
137ffvelrnda 6032 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( R `  x )  e.  LMod )
1410adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  F  e.  K )
15 prdsvscacl.sr . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x
) )  =  S )
1615fveq2d 5876 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  (Scalar `  ( R `  x )
) )  =  (
Base `  S )
)
1716, 4syl6eqr 2516 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  (Scalar `  ( R `  x )
) )  =  K )
1814, 17eleqtrrd 2548 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  F  e.  ( Base `  (Scalar `  ( R `  x
) ) ) )
195adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  S  e.  Ring )
206adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  I  e.  W )
219adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  R  Fn  I )
2211adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  G  e.  B )
23 simpr 461 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
241, 2, 19, 20, 21, 22, 23prdsbasprj 14889 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( G `  x )  e.  ( Base `  ( R `  x )
) )
25 eqid 2457 . . . . . 6  |-  ( Base `  ( R `  x
) )  =  (
Base `  ( R `  x ) )
26 eqid 2457 . . . . . 6  |-  (Scalar `  ( R `  x ) )  =  (Scalar `  ( R `  x ) )
27 eqid 2457 . . . . . 6  |-  ( .s
`  ( R `  x ) )  =  ( .s `  ( R `  x )
)
28 eqid 2457 . . . . . 6  |-  ( Base `  (Scalar `  ( R `  x ) ) )  =  ( Base `  (Scalar `  ( R `  x
) ) )
2925, 26, 27, 28lmodvscl 17656 . . . . 5  |-  ( ( ( R `  x
)  e.  LMod  /\  F  e.  ( Base `  (Scalar `  ( R `  x
) ) )  /\  ( G `  x )  e.  ( Base `  ( R `  x )
) )  ->  ( F ( .s `  ( R `  x ) ) ( G `  x ) )  e.  ( Base `  ( R `  x )
) )
3013, 18, 24, 29syl3anc 1228 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( F ( .s `  ( R `  x ) ) ( G `  x ) )  e.  ( Base `  ( R `  x )
) )
3130ralrimiva 2871 . . 3  |-  ( ph  ->  A. x  e.  I 
( F ( .s
`  ( R `  x ) ) ( G `  x ) )  e.  ( Base `  ( R `  x
) ) )
321, 2, 5, 6, 9prdsbasmpt 14887 . . 3  |-  ( ph  ->  ( ( x  e.  I  |->  ( F ( .s `  ( R `
 x ) ) ( G `  x
) ) )  e.  B  <->  A. x  e.  I 
( F ( .s
`  ( R `  x ) ) ( G `  x ) )  e.  ( Base `  ( R `  x
) ) ) )
3331, 32mpbird 232 . 2  |-  ( ph  ->  ( x  e.  I  |->  ( F ( .s
`  ( R `  x ) ) ( G `  x ) ) )  e.  B
)
3412, 33eqeltrd 2545 1  |-  ( ph  ->  ( F  .x.  G
)  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807    |-> cmpt 4515    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   Basecbs 14644  Scalarcsca 14715   .scvsca 14716   X_scprds 14863   Ringcrg 17325   LModclmod 17639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-plusg 14725  df-mulr 14726  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-hom 14736  df-cco 14737  df-prds 14865  df-lmod 17641
This theorem is referenced by:  prdslmodd  17742
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