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

Theorem prdsvscacl 16046
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 5593 . . . 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 13703 . 2  |-  ( ph  ->  ( F  .x.  G
)  =  ( x  e.  I  |->  ( F ( .s `  ( R `  x )
) ( G `  x ) ) ) )
137ffvelrnda 5872 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( R `  x )  e.  LMod )
1410adantr 453 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  F  e.  K )
15 prdsvscacl.sr . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x
) )  =  S )
1615fveq2d 5734 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  (Scalar `  ( R `  x )
) )  =  (
Base `  S )
)
1716, 4syl6eqr 2488 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( Base `  (Scalar `  ( R `  x )
) )  =  K )
1814, 17eleqtrrd 2515 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  F  e.  ( Base `  (Scalar `  ( R `  x
) ) ) )
195adantr 453 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  S  e.  Ring )
206adantr 453 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  I  e.  W )
219adantr 453 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  R  Fn  I )
2211adantr 453 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  G  e.  B )
23 simpr 449 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
241, 2, 19, 20, 21, 22, 23prdsbasprj 13696 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( G `  x )  e.  ( Base `  ( R `  x )
) )
25 eqid 2438 . . . . . 6  |-  ( Base `  ( R `  x
) )  =  (
Base `  ( R `  x ) )
26 eqid 2438 . . . . . 6  |-  (Scalar `  ( R `  x ) )  =  (Scalar `  ( R `  x ) )
27 eqid 2438 . . . . . 6  |-  ( .s
`  ( R `  x ) )  =  ( .s `  ( R `  x )
)
28 eqid 2438 . . . . . 6  |-  ( Base `  (Scalar `  ( R `  x ) ) )  =  ( Base `  (Scalar `  ( R `  x
) ) )
2925, 26, 27, 28lmodvscl 15969 . . . . 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 1185 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( F ( .s `  ( R `  x ) ) ( G `  x ) )  e.  ( Base `  ( R `  x )
) )
3130ralrimiva 2791 . . 3  |-  ( ph  ->  A. x  e.  I 
( F ( .s
`  ( R `  x ) ) ( G `  x ) )  e.  ( Base `  ( R `  x
) ) )
321, 2, 5, 6, 9prdsbasmpt 13694 . . 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 225 . 2  |-  ( ph  ->  ( x  e.  I  |->  ( F ( .s
`  ( R `  x ) ) ( G `  x ) ) )  e.  B
)
3412, 33eqeltrd 2512 1  |-  ( ph  ->  ( F  .x.  G
)  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    e. cmpt 4268    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   Basecbs 13471  Scalarcsca 13534   .scvsca 13535   X_scprds 13671   Ringcrg 15662   LModclmod 15952
This theorem is referenced by:  prdslmodd  16047
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-hom 13555  df-cco 13556  df-prds 13673  df-lmod 15954
  Copyright terms: Public domain W3C validator