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Theorem subrgascl 16558
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 2436 . . . 4  |-  (Scalar `  U )  =  (Scalar `  U )
3 eqid 2436 . . . 4  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
41, 2, 3asclfn 16395 . . 3  |-  C  Fn  ( Base `  (Scalar `  U
) )
5 subrgascl.r . . . . . 6  |-  ( ph  ->  T  e.  (SubRing `  R
) )
6 subrgascl.h . . . . . . 7  |-  H  =  ( Rs  T )
76subrgbas 15877 . . . . . 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 6106 . . . . . . . . 9  |-  ( Rs  T )  e.  _V
126, 11eqeltri 2506 . . . . . . . 8  |-  H  e. 
_V
1312a1i 11 . . . . . . 7  |-  ( ph  ->  H  e.  _V )
149, 10, 13mplsca 16508 . . . . . 6  |-  ( ph  ->  H  =  (Scalar `  U ) )
1514fveq2d 5732 . . . . 5  |-  ( ph  ->  ( Base `  H
)  =  ( Base `  (Scalar `  U )
) )
168, 15eqtrd 2468 . . . 4  |-  ( ph  ->  T  =  ( Base `  (Scalar `  U )
) )
1716fneq2d 5537 . . 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 2436 . . . . 5  |-  (Scalar `  P )  =  (Scalar `  P )
21 eqid 2436 . . . . 5  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
2219, 20, 21asclfn 16395 . . . 4  |-  A  Fn  ( Base `  (Scalar `  P
) )
23 subrgascl.p . . . . . . 7  |-  P  =  ( I mPoly  R )
24 subrgrcl 15873 . . . . . . . 8  |-  ( T  e.  (SubRing `  R
)  ->  R  e.  Ring )
255, 24syl 16 . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
2623, 10, 25mplsca 16508 . . . . . 6  |-  ( ph  ->  R  =  (Scalar `  P ) )
2726fveq2d 5732 . . . . 5  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  (Scalar `  P )
) )
2827fneq2d 5537 . . . 4  |-  ( ph  ->  ( A  Fn  ( Base `  R )  <->  A  Fn  ( Base `  (Scalar `  P
) ) ) )
2922, 28mpbiri 225 . . 3  |-  ( ph  ->  A  Fn  ( Base `  R ) )
30 eqid 2436 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
3130subrgss 15869 . . . 4  |-  ( T  e.  (SubRing `  R
)  ->  T  C_  ( Base `  R ) )
325, 31syl 16 . . 3  |-  ( ph  ->  T  C_  ( Base `  R ) )
33 fnssres 5558 . . 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 5745 . . . 4  |-  ( x  e.  T  ->  (
( A  |`  T ) `
 x )  =  ( A `  x
) )
3635adantl 453 . . 3  |-  ( (
ph  /\  x  e.  T )  ->  (
( A  |`  T ) `
 x )  =  ( A `  x
) )
37 eqid 2436 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  R
)
386, 37subrg0 15875 . . . . . . . 8  |-  ( T  e.  (SubRing `  R
)  ->  ( 0g `  R )  =  ( 0g `  H ) )
395, 38syl 16 . . . . . . 7  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  H ) )
4039ifeq2d 3754 . . . . . 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 4296 . . . 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 2436 . . . . 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 3348 . . . . 5  |-  ( (
ph  /\  x  e.  T )  ->  x  e.  ( Base `  R
) )
4723, 43, 37, 30, 19, 44, 45, 46mplascl 16556 . . . 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 2436 . . . . 5  |-  ( 0g
`  H )  =  ( 0g `  H
)
49 eqid 2436 . . . . 5  |-  ( Base `  H )  =  (
Base `  H )
506subrgrng 15871 . . . . . . 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 2503 . . . . . 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 16556 . . . 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 2478 . . 3  |-  ( (
ph  /\  x  e.  T )  ->  ( A `  x )  =  ( C `  x ) )
5736, 56eqtr2d 2469 . 2  |-  ( (
ph  /\  x  e.  T )  ->  ( C `  x )  =  ( ( A  |`  T ) `  x
) )
5818, 34, 57eqfnfvd 5830 1  |-  ( ph  ->  C  =  ( A  |`  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2709   _Vcvv 2956    C_ wss 3320   ifcif 3739   {csn 3814    e. cmpt 4266    X. cxp 4876   `'ccnv 4877    |` cres 4880   "cima 4881    Fn wfn 5449   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   Fincfn 7109   0cc0 8990   NNcn 10000   NN0cn0 10221   Basecbs 13469   ↾s cress 13470  Scalarcsca 13532   0gc0g 13723   Ringcrg 15660  SubRingcsubrg 15864  algSccascl 16371   mPoly cmpl 16408
This theorem is referenced by:  subrgasclcl  16559  subrg1ascl  16652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-ofr 6306  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324  df-hash 11619  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-0g 13727  df-gsum 13728  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-mulg 14815  df-subg 14941  df-ghm 15004  df-cntz 15116  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-subrg 15866  df-ascl 16374  df-psr 16417  df-mpl 16419
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