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Theorem subrgascl 17568
Description: The scalar injection function in a subring algebra is the same up to a restriction to the subring. (Contributed by Mario Carneiro, 4-Jul-2015.)
Hypotheses
Ref Expression
subrgascl.p  |-  P  =  ( I mPoly  R )
subrgascl.a  |-  A  =  (algSc `  P )
subrgascl.h  |-  H  =  ( Rs  T )
subrgascl.u  |-  U  =  ( I mPoly  H )
subrgascl.i  |-  ( ph  ->  I  e.  W )
subrgascl.r  |-  ( ph  ->  T  e.  (SubRing `  R
) )
subrgascl.c  |-  C  =  (algSc `  U )
Assertion
Ref Expression
subrgascl  |-  ( ph  ->  C  =  ( A  |`  T ) )

Proof of Theorem subrgascl
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrgascl.c . . . 4  |-  C  =  (algSc `  U )
2 eqid 2441 . . . 4  |-  (Scalar `  U )  =  (Scalar `  U )
3 eqid 2441 . . . 4  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
41, 2, 3asclfn 17385 . . 3  |-  C  Fn  ( Base `  (Scalar `  U
) )
5 subrgascl.r . . . . . 6  |-  ( ph  ->  T  e.  (SubRing `  R
) )
6 subrgascl.h . . . . . . 7  |-  H  =  ( Rs  T )
76subrgbas 16854 . . . . . 6  |-  ( T  e.  (SubRing `  R
)  ->  T  =  ( Base `  H )
)
85, 7syl 16 . . . . 5  |-  ( ph  ->  T  =  ( Base `  H ) )
9 subrgascl.u . . . . . . 7  |-  U  =  ( I mPoly  H )
10 subrgascl.i . . . . . . 7  |-  ( ph  ->  I  e.  W )
11 ovex 6115 . . . . . . . . 9  |-  ( Rs  T )  e.  _V
126, 11eqeltri 2511 . . . . . . . 8  |-  H  e. 
_V
1312a1i 11 . . . . . . 7  |-  ( ph  ->  H  e.  _V )
149, 10, 13mplsca 17514 . . . . . 6  |-  ( ph  ->  H  =  (Scalar `  U ) )
1514fveq2d 5692 . . . . 5  |-  ( ph  ->  ( Base `  H
)  =  ( Base `  (Scalar `  U )
) )
168, 15eqtrd 2473 . . . 4  |-  ( ph  ->  T  =  ( Base `  (Scalar `  U )
) )
1716fneq2d 5499 . . 3  |-  ( ph  ->  ( C  Fn  T  <->  C  Fn  ( Base `  (Scalar `  U ) ) ) )
184, 17mpbiri 233 . 2  |-  ( ph  ->  C  Fn  T )
19 subrgascl.a . . . . 5  |-  A  =  (algSc `  P )
20 eqid 2441 . . . . 5  |-  (Scalar `  P )  =  (Scalar `  P )
21 eqid 2441 . . . . 5  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
2219, 20, 21asclfn 17385 . . . 4  |-  A  Fn  ( Base `  (Scalar `  P
) )
23 subrgascl.p . . . . . . 7  |-  P  =  ( I mPoly  R )
24 subrgrcl 16850 . . . . . . . 8  |-  ( T  e.  (SubRing `  R
)  ->  R  e.  Ring )
255, 24syl 16 . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
2623, 10, 25mplsca 17514 . . . . . 6  |-  ( ph  ->  R  =  (Scalar `  P ) )
2726fveq2d 5692 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  P )
) )
2827fneq2d 5499 . . . 4  |-  ( ph  ->  ( A  Fn  ( Base `  R )  <->  A  Fn  ( Base `  (Scalar `  P
) ) ) )
2922, 28mpbiri 233 . . 3  |-  ( ph  ->  A  Fn  ( Base `  R ) )
30 eqid 2441 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
3130subrgss 16846 . . . 4  |-  ( T  e.  (SubRing `  R
)  ->  T  C_  ( Base `  R ) )
325, 31syl 16 . . 3  |-  ( ph  ->  T  C_  ( Base `  R ) )
33 fnssres 5521 . . 3  |-  ( ( A  Fn  ( Base `  R )  /\  T  C_  ( Base `  R
) )  ->  ( A  |`  T )  Fn  T )
3429, 32, 33syl2anc 656 . 2  |-  ( ph  ->  ( A  |`  T )  Fn  T )
35 fvres 5701 . . . 4  |-  ( x  e.  T  ->  (
( A  |`  T ) `
 x )  =  ( A `  x
) )
3635adantl 463 . . 3  |-  ( (
ph  /\  x  e.  T )  ->  (
( A  |`  T ) `
 x )  =  ( A `  x
) )
37 eqid 2441 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  R
)
386, 37subrg0 16852 . . . . . . . 8  |-  ( T  e.  (SubRing `  R
)  ->  ( 0g `  R )  =  ( 0g `  H ) )
395, 38syl 16 . . . . . . 7  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  H ) )
4039ifeq2d 3805 . . . . . 6  |-  ( ph  ->  if ( y  =  ( I  X.  {
0 } ) ,  x ,  ( 0g
`  R ) )  =  if ( y  =  ( I  X.  { 0 } ) ,  x ,  ( 0g `  H ) ) )
4140adantr 462 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  if ( y  =  ( I  X.  { 0 } ) ,  x ,  ( 0g `  R ) )  =  if ( y  =  ( I  X.  {
0 } ) ,  x ,  ( 0g
`  H ) ) )
4241mpteq2dv 4376 . . . 4  |-  ( (
ph  /\  x  e.  T )  ->  (
y  e.  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  |->  if ( y  =  ( I  X.  { 0 } ) ,  x ,  ( 0g `  R ) ) )  =  ( y  e. 
{ f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  |->  if ( y  =  ( I  X.  { 0 } ) ,  x ,  ( 0g `  H
) ) ) )
43 eqid 2441 . . . . 5  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
4410adantr 462 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  I  e.  W )
4525adantr 462 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  R  e.  Ring )
4632sselda 3353 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  x  e.  ( Base `  R
) )
4723, 43, 37, 30, 19, 44, 45, 46mplascl 17566 . . . 4  |-  ( (
ph  /\  x  e.  T )  ->  ( A `  x )  =  ( y  e. 
{ f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  |->  if ( y  =  ( I  X.  { 0 } ) ,  x ,  ( 0g `  R
) ) ) )
48 eqid 2441 . . . . 5  |-  ( 0g
`  H )  =  ( 0g `  H
)
49 eqid 2441 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
506subrgrng 16848 . . . . . . 7  |-  ( T  e.  (SubRing `  R
)  ->  H  e.  Ring )
515, 50syl 16 . . . . . 6  |-  ( ph  ->  H  e.  Ring )
5251adantr 462 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  H  e.  Ring )
538eleq2d 2508 . . . . . 6  |-  ( ph  ->  ( x  e.  T  <->  x  e.  ( Base `  H
) ) )
5453biimpa 481 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  x  e.  ( Base `  H
) )
559, 43, 48, 49, 1, 44, 52, 54mplascl 17566 . . . 4  |-  ( (
ph  /\  x  e.  T )  ->  ( C `  x )  =  ( y  e. 
{ f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  |->  if ( y  =  ( I  X.  { 0 } ) ,  x ,  ( 0g `  H
) ) ) )
5642, 47, 553eqtr4d 2483 . . 3  |-  ( (
ph  /\  x  e.  T )  ->  ( A `  x )  =  ( C `  x ) )
5736, 56eqtr2d 2474 . 2  |-  ( (
ph  /\  x  e.  T )  ->  ( C `  x )  =  ( ( A  |`  T ) `  x
) )
5818, 34, 57eqfnfvd 5797 1  |-  ( ph  ->  C  =  ( A  |`  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   {crab 2717   _Vcvv 2970    C_ wss 3325   ifcif 3788   {csn 3874    e. cmpt 4347    X. cxp 4834   `'ccnv 4835    |` cres 4838   "cima 4839    Fn wfn 5410   ` cfv 5415  (class class class)co 6090    ^m cmap 7210   Fincfn 7306   0cc0 9278   NNcn 10318   NN0cn0 10575   Basecbs 14170   ↾s cress 14171  Scalarcsca 14237   0gc0g 14374   Ringcrg 16635  SubRingcsubrg 16841  algSccascl 17361   mPoly cmpl 17398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-ofr 6320  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-seq 11803  df-hash 12100  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-tset 14253  df-0g 14376  df-gsum 14377  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-mulg 15541  df-subg 15671  df-ghm 15738  df-cntz 15828  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-subrg 16843  df-ascl 17364  df-psr 17407  df-mpl 17409
This theorem is referenced by:  subrgasclcl  17569  subrg1ascl  17667
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