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Theorem prdsvscacl 17061
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 5571 . . . 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 14429 . 2  |-  ( ph  ->  ( F  .x.  G
)  =  ( x  e.  I  |->  ( F ( .s `  ( R `  x )
) ( G `  x ) ) ) )
137ffvelrnda 5855 . . . . 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 5707 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  (Scalar `  ( R `  x )
) )  =  (
Base `  S )
)
1716, 4syl6eqr 2493 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  (Scalar `  ( R `  x )
) )  =  K )
1814, 17eleqtrrd 2520 . . . . 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 14422 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( G `  x )  e.  ( Base `  ( R `  x )
) )
25 eqid 2443 . . . . . 6  |-  ( Base `  ( R `  x
) )  =  (
Base `  ( R `  x ) )
26 eqid 2443 . . . . . 6  |-  (Scalar `  ( R `  x ) )  =  (Scalar `  ( R `  x ) )
27 eqid 2443 . . . . . 6  |-  ( .s
`  ( R `  x ) )  =  ( .s `  ( R `  x )
)
28 eqid 2443 . . . . . 6  |-  ( Base `  (Scalar `  ( R `  x ) ) )  =  ( Base `  (Scalar `  ( R `  x
) ) )
2925, 26, 27, 28lmodvscl 16977 . . . . 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 1218 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( F ( .s `  ( R `  x ) ) ( G `  x ) )  e.  ( Base `  ( R `  x )
) )
3130ralrimiva 2811 . . 3  |-  ( ph  ->  A. x  e.  I 
( F ( .s
`  ( R `  x ) ) ( G `  x ) )  e.  ( Base `  ( R `  x
) ) )
321, 2, 5, 6, 9prdsbasmpt 14420 . . 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 2517 1  |-  ( ph  ->  ( F  .x.  G
)  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2727    e. cmpt 4362    Fn wfn 5425   -->wf 5426   ` cfv 5430  (class class class)co 6103   Basecbs 14186  Scalarcsca 14253   .scvsca 14254   X_scprds 14396   Ringcrg 16657   LModclmod 16960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-fz 11450  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-plusg 14263  df-mulr 14264  df-sca 14266  df-vsca 14267  df-ip 14268  df-tset 14269  df-ple 14270  df-ds 14272  df-hom 14274  df-cco 14275  df-prds 14398  df-lmod 16962
This theorem is referenced by:  prdslmodd  17062
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