MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prdsvscacl Unicode version

Theorem prdsvscacl 15774
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 5427 . . . 4  |-  ( R : I --> LMod  ->  R  Fn  I )
97, 8syl 15 . . 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 13427 . 2  |-  ( ph  ->  ( F  .x.  G
)  =  ( x  e.  I  |->  ( F ( .s `  ( R `  x )
) ( G `  x ) ) ) )
13 ffvelrn 5701 . . . . . 6  |-  ( ( R : I --> LMod  /\  x  e.  I )  ->  ( R `  x )  e.  LMod )
147, 13sylan 457 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( R `  x )  e.  LMod )
1510adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  F  e.  K )
16 prdsvscacl.sr . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x
) )  =  S )
1716fveq2d 5567 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  (Scalar `  ( R `  x )
) )  =  (
Base `  S )
)
1817, 4syl6eqr 2366 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  (Scalar `  ( R `  x )
) )  =  K )
1915, 18eleqtrrd 2393 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  F  e.  ( Base `  (Scalar `  ( R `  x
) ) ) )
205adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  S  e.  Ring )
216adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  I  e.  W )
229adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  R  Fn  I )
2311adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  G  e.  B )
24 simpr 447 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
251, 2, 20, 21, 22, 23, 24prdsbasprj 13420 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( G `  x )  e.  ( Base `  ( R `  x )
) )
26 eqid 2316 . . . . . 6  |-  ( Base `  ( R `  x
) )  =  (
Base `  ( R `  x ) )
27 eqid 2316 . . . . . 6  |-  (Scalar `  ( R `  x ) )  =  (Scalar `  ( R `  x ) )
28 eqid 2316 . . . . . 6  |-  ( .s
`  ( R `  x ) )  =  ( .s `  ( R `  x )
)
29 eqid 2316 . . . . . 6  |-  ( Base `  (Scalar `  ( R `  x ) ) )  =  ( Base `  (Scalar `  ( R `  x
) ) )
3026, 27, 28, 29lmodvscl 15693 . . . . 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 )
) )
3114, 19, 25, 30syl3anc 1182 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( F ( .s `  ( R `  x ) ) ( G `  x ) )  e.  ( Base `  ( R `  x )
) )
3231ralrimiva 2660 . . 3  |-  ( ph  ->  A. x  e.  I 
( F ( .s
`  ( R `  x ) ) ( G `  x ) )  e.  ( Base `  ( R `  x
) ) )
331, 2, 5, 6, 9prdsbasmpt 13418 . . 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
) ) ) )
3432, 33mpbird 223 . 2  |-  ( ph  ->  ( x  e.  I  |->  ( F ( .s
`  ( R `  x ) ) ( G `  x ) ) )  e.  B
)
3512, 34eqeltrd 2390 1  |-  ( ph  ->  ( F  .x.  G
)  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577    e. cmpt 4114    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900   Basecbs 13195  Scalarcsca 13258   .scvsca 13259   X_scprds 13395   Ringcrg 15386   LModclmod 15676
This theorem is referenced by:  prdslmodd  15775
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-fz 10830  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-plusg 13268  df-mulr 13269  df-sca 13271  df-vsca 13272  df-tset 13274  df-ple 13275  df-ds 13277  df-hom 13279  df-cco 13280  df-prds 13397  df-lmod 15678
  Copyright terms: Public domain W3C validator