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Theorem subrgascl 16513
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 2404 . . . 4  |-  (Scalar `  U )  =  (Scalar `  U )
3 eqid 2404 . . . 4  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
41, 2, 3asclfn 16350 . . 3  |-  C  Fn  ( Base `  (Scalar `  U
) )
5 subrgascl.r . . . . . 6  |-  ( ph  ->  T  e.  (SubRing `  R
) )
6 subrgascl.h . . . . . . 7  |-  H  =  ( Rs  T )
76subrgbas 15832 . . . . . 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 6065 . . . . . . . . 9  |-  ( Rs  T )  e.  _V
126, 11eqeltri 2474 . . . . . . . 8  |-  H  e. 
_V
1312a1i 11 . . . . . . 7  |-  ( ph  ->  H  e.  _V )
149, 10, 13mplsca 16463 . . . . . 6  |-  ( ph  ->  H  =  (Scalar `  U ) )
1514fveq2d 5691 . . . . 5  |-  ( ph  ->  ( Base `  H
)  =  ( Base `  (Scalar `  U )
) )
168, 15eqtrd 2436 . . . 4  |-  ( ph  ->  T  =  ( Base `  (Scalar `  U )
) )
1716fneq2d 5496 . . 3  |-  ( ph  ->  ( C  Fn  T  <->  C  Fn  ( Base `  (Scalar `  U ) ) ) )
184, 17mpbiri 225 . 2  |-  ( ph  ->  C  Fn  T )
19 subrgascl.a . . . . 5  |-  A  =  (algSc `  P )
20 eqid 2404 . . . . 5  |-  (Scalar `  P )  =  (Scalar `  P )
21 eqid 2404 . . . . 5  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
2219, 20, 21asclfn 16350 . . . 4  |-  A  Fn  ( Base `  (Scalar `  P
) )
23 subrgascl.p . . . . . . 7  |-  P  =  ( I mPoly  R )
24 subrgrcl 15828 . . . . . . . 8  |-  ( T  e.  (SubRing `  R
)  ->  R  e.  Ring )
255, 24syl 16 . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
2623, 10, 25mplsca 16463 . . . . . 6  |-  ( ph  ->  R  =  (Scalar `  P ) )
2726fveq2d 5691 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  P )
) )
2827fneq2d 5496 . . . 4  |-  ( ph  ->  ( A  Fn  ( Base `  R )  <->  A  Fn  ( Base `  (Scalar `  P
) ) ) )
2922, 28mpbiri 225 . . 3  |-  ( ph  ->  A  Fn  ( Base `  R ) )
30 eqid 2404 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
3130subrgss 15824 . . . 4  |-  ( T  e.  (SubRing `  R
)  ->  T  C_  ( Base `  R ) )
325, 31syl 16 . . 3  |-  ( ph  ->  T  C_  ( Base `  R ) )
33 fnssres 5517 . . 3  |-  ( ( A  Fn  ( Base `  R )  /\  T  C_  ( Base `  R
) )  ->  ( A  |`  T )  Fn  T )
3429, 32, 33syl2anc 643 . 2  |-  ( ph  ->  ( A  |`  T )  Fn  T )
35 fvres 5704 . . . 4  |-  ( x  e.  T  ->  (
( A  |`  T ) `
 x )  =  ( A `  x
) )
3635adantl 453 . . 3  |-  ( (
ph  /\  x  e.  T )  ->  (
( A  |`  T ) `
 x )  =  ( A `  x
) )
37 eqid 2404 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  R
)
386, 37subrg0 15830 . . . . . . . 8  |-  ( T  e.  (SubRing `  R
)  ->  ( 0g `  R )  =  ( 0g `  H ) )
395, 38syl 16 . . . . . . 7  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  H ) )
4039ifeq2d 3714 . . . . . 6  |-  ( ph  ->  if ( y  =  ( I  X.  {
0 } ) ,  x ,  ( 0g
`  R ) )  =  if ( y  =  ( I  X.  { 0 } ) ,  x ,  ( 0g `  H ) ) )
4140adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  if ( y  =  ( I  X.  { 0 } ) ,  x ,  ( 0g `  R ) )  =  if ( y  =  ( I  X.  {
0 } ) ,  x ,  ( 0g
`  H ) ) )
4241mpteq2dv 4256 . . . 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 2404 . . . . 5  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
4410adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  I  e.  W )
4525adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  R  e.  Ring )
4632sselda 3308 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  x  e.  ( Base `  R
) )
4723, 43, 37, 30, 19, 44, 45, 46mplascl 16511 . . . 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 2404 . . . . 5  |-  ( 0g
`  H )  =  ( 0g `  H
)
49 eqid 2404 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
506subrgrng 15826 . . . . . . 7  |-  ( T  e.  (SubRing `  R
)  ->  H  e.  Ring )
515, 50syl 16 . . . . . 6  |-  ( ph  ->  H  e.  Ring )
5251adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  H  e.  Ring )
538eleq2d 2471 . . . . . 6  |-  ( ph  ->  ( x  e.  T  <->  x  e.  ( Base `  H
) ) )
5453biimpa 471 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  x  e.  ( Base `  H
) )
559, 43, 48, 49, 1, 44, 52, 54mplascl 16511 . . . 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 2446 . . 3  |-  ( (
ph  /\  x  e.  T )  ->  ( A `  x )  =  ( C `  x ) )
5736, 56eqtr2d 2437 . 2  |-  ( (
ph  /\  x  e.  T )  ->  ( C `  x )  =  ( ( A  |`  T ) `  x
) )
5818, 34, 57eqfnfvd 5789 1  |-  ( ph  ->  C  =  ( A  |`  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2670   _Vcvv 2916    C_ wss 3280   ifcif 3699   {csn 3774    e. cmpt 4226    X. cxp 4835   `'ccnv 4836    |` cres 4839   "cima 4840    Fn wfn 5408   ` cfv 5413  (class class class)co 6040    ^m cmap 6977   Fincfn 7068   0cc0 8946   NNcn 9956   NN0cn0 10177   Basecbs 13424   ↾s cress 13425  Scalarcsca 13487   0gc0g 13678   Ringcrg 15615  SubRingcsubrg 15819  algSccascl 16326   mPoly cmpl 16363
This theorem is referenced by:  subrgasclcl  16514  subrg1ascl  16607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-ofr 6265  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-mulg 14770  df-subg 14896  df-ghm 14959  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-subrg 15821  df-ascl 16329  df-psr 16372  df-mpl 16374
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