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Theorem subrgasclcl 16240
Description: The scalars in a polynomial algebra are in the subring algebra iff the scalar value is in 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
) )
subrgasclcl.b  |-  B  =  ( Base `  U
)
subrgasclcl.k  |-  K  =  ( Base `  R
)
subrgasclcl.x  |-  ( ph  ->  X  e.  K )
Assertion
Ref Expression
subrgasclcl  |-  ( ph  ->  ( ( A `  X )  e.  B  <->  X  e.  T ) )

Proof of Theorem subrgasclcl
Dummy variables  x  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrgascl.i . . . . . 6  |-  ( ph  ->  I  e.  W )
21adantr 451 . . . . 5  |-  ( (
ph  /\  ( A `  X )  e.  B
)  ->  I  e.  W )
3 eqid 2283 . . . . . 6  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
43psrbag0 16235 . . . . 5  |-  ( I  e.  W  ->  (
I  X.  { 0 } )  e.  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } )
52, 4syl 15 . . . 4  |-  ( (
ph  /\  ( A `  X )  e.  B
)  ->  ( I  X.  { 0 } )  e.  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin } )
6 eqid 2283 . . . . . 6  |-  ( I mPwSer  H )  =  ( I mPwSer  H )
7 eqid 2283 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
8 eqid 2283 . . . . . 6  |-  ( Base `  ( I mPwSer  H ) )  =  ( Base `  ( I mPwSer  H ) )
9 subrgascl.p . . . . . . . . 9  |-  P  =  ( I mPoly  R )
10 eqid 2283 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  R
)
11 subrgasclcl.k . . . . . . . . 9  |-  K  =  ( Base `  R
)
12 subrgascl.a . . . . . . . . 9  |-  A  =  (algSc `  P )
13 subrgascl.r . . . . . . . . . 10  |-  ( ph  ->  T  e.  (SubRing `  R
) )
14 subrgrcl 15550 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  R  e.  Ring )
1513, 14syl 15 . . . . . . . . 9  |-  ( ph  ->  R  e.  Ring )
16 subrgasclcl.x . . . . . . . . 9  |-  ( ph  ->  X  e.  K )
179, 3, 10, 11, 12, 1, 15, 16mplascl 16237 . . . . . . . 8  |-  ( ph  ->  ( A `  X
)  =  ( x  e.  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin }  |->  if ( x  =  ( I  X.  { 0 } ) ,  X ,  ( 0g `  R ) ) ) )
1817adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( A `  X )  e.  B
)  ->  ( A `  X )  =  ( x  e.  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  |->  if ( x  =  ( I  X.  { 0 } ) ,  X ,  ( 0g `  R ) ) ) )
19 subrgascl.u . . . . . . . . . 10  |-  U  =  ( I mPoly  H )
20 subrgasclcl.b . . . . . . . . . 10  |-  B  =  ( Base `  U
)
21 subrgascl.h . . . . . . . . . . . 12  |-  H  =  ( Rs  T )
2221subrgrng 15548 . . . . . . . . . . 11  |-  ( T  e.  (SubRing `  R
)  ->  H  e.  Ring )
2313, 22syl 15 . . . . . . . . . 10  |-  ( ph  ->  H  e.  Ring )
246, 19, 20, 1, 23mplsubrg 16184 . . . . . . . . 9  |-  ( ph  ->  B  e.  (SubRing `  (
I mPwSer  H ) ) )
258subrgss 15546 . . . . . . . . 9  |-  ( B  e.  (SubRing `  (
I mPwSer  H ) )  ->  B  C_  ( Base `  (
I mPwSer  H ) ) )
2624, 25syl 15 . . . . . . . 8  |-  ( ph  ->  B  C_  ( Base `  ( I mPwSer  H ) ) )
2726sselda 3180 . . . . . . 7  |-  ( (
ph  /\  ( A `  X )  e.  B
)  ->  ( A `  X )  e.  (
Base `  ( I mPwSer  H ) ) )
2818, 27eqeltrrd 2358 . . . . . 6  |-  ( (
ph  /\  ( A `  X )  e.  B
)  ->  ( x  e.  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  |->  if ( x  =  ( I  X.  { 0 } ) ,  X , 
( 0g `  R
) ) )  e.  ( Base `  (
I mPwSer  H ) ) )
296, 7, 3, 8, 28psrelbas 16125 . . . . 5  |-  ( (
ph  /\  ( A `  X )  e.  B
)  ->  ( x  e.  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  |->  if ( x  =  ( I  X.  { 0 } ) ,  X , 
( 0g `  R
) ) ) : { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } --> ( Base `  H
) )
30 eqid 2283 . . . . . 6  |-  ( x  e.  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin }  |->  if ( x  =  ( I  X.  { 0 } ) ,  X ,  ( 0g `  R ) ) )  =  ( x  e. 
{ f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  |->  if ( x  =  ( I  X.  { 0 } ) ,  X , 
( 0g `  R
) ) )
3130fmpt 5681 . . . . 5  |-  ( A. x  e.  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin } if ( x  =  (
I  X.  { 0 } ) ,  X ,  ( 0g `  R ) )  e.  ( Base `  H
)  <->  ( x  e. 
{ f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  |->  if ( x  =  ( I  X.  { 0 } ) ,  X , 
( 0g `  R
) ) ) : { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } --> ( Base `  H
) )
3229, 31sylibr 203 . . . 4  |-  ( (
ph  /\  ( A `  X )  e.  B
)  ->  A. x  e.  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } if ( x  =  ( I  X.  { 0 } ) ,  X ,  ( 0g `  R ) )  e.  ( Base `  H ) )
33 iftrue 3571 . . . . . 6  |-  ( x  =  ( I  X.  { 0 } )  ->  if ( x  =  ( I  X.  { 0 } ) ,  X ,  ( 0g `  R ) )  =  X )
3433eleq1d 2349 . . . . 5  |-  ( x  =  ( I  X.  { 0 } )  ->  ( if ( x  =  ( I  X.  { 0 } ) ,  X , 
( 0g `  R
) )  e.  (
Base `  H )  <->  X  e.  ( Base `  H
) ) )
3534rspcv 2880 . . . 4  |-  ( ( I  X.  { 0 } )  e.  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin }  ->  ( A. x  e.  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin } if ( x  =  ( I  X.  { 0 } ) ,  X ,  ( 0g `  R ) )  e.  ( Base `  H )  ->  X  e.  ( Base `  H
) ) )
365, 32, 35sylc 56 . . 3  |-  ( (
ph  /\  ( A `  X )  e.  B
)  ->  X  e.  ( Base `  H )
)
3721subrgbas 15554 . . . . 5  |-  ( T  e.  (SubRing `  R
)  ->  T  =  ( Base `  H )
)
3813, 37syl 15 . . . 4  |-  ( ph  ->  T  =  ( Base `  H ) )
3938adantr 451 . . 3  |-  ( (
ph  /\  ( A `  X )  e.  B
)  ->  T  =  ( Base `  H )
)
4036, 39eleqtrrd 2360 . 2  |-  ( (
ph  /\  ( A `  X )  e.  B
)  ->  X  e.  T )
41 eqid 2283 . . . . . 6  |-  (algSc `  U )  =  (algSc `  U )
429, 12, 21, 19, 1, 13, 41subrgascl 16239 . . . . 5  |-  ( ph  ->  (algSc `  U )  =  ( A  |`  T ) )
4342fveq1d 5527 . . . 4  |-  ( ph  ->  ( (algSc `  U
) `  X )  =  ( ( A  |`  T ) `  X
) )
44 fvres 5542 . . . 4  |-  ( X  e.  T  ->  (
( A  |`  T ) `
 X )  =  ( A `  X
) )
4543, 44sylan9eq 2335 . . 3  |-  ( (
ph  /\  X  e.  T )  ->  (
(algSc `  U ) `  X )  =  ( A `  X ) )
46 eqid 2283 . . . . . . 7  |-  (Scalar `  U )  =  (Scalar `  U )
4719mplrng 16196 . . . . . . 7  |-  ( ( I  e.  W  /\  H  e.  Ring )  ->  U  e.  Ring )
4819mpllmod 16195 . . . . . . 7  |-  ( ( I  e.  W  /\  H  e.  Ring )  ->  U  e.  LMod )
49 eqid 2283 . . . . . . 7  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
5041, 46, 47, 48, 49, 20asclf 16077 . . . . . 6  |-  ( ( I  e.  W  /\  H  e.  Ring )  -> 
(algSc `  U ) : ( Base `  (Scalar `  U ) ) --> B )
511, 23, 50syl2anc 642 . . . . 5  |-  ( ph  ->  (algSc `  U ) : ( Base `  (Scalar `  U ) ) --> B )
5251adantr 451 . . . 4  |-  ( (
ph  /\  X  e.  T )  ->  (algSc `  U ) : (
Base `  (Scalar `  U
) ) --> B )
5319, 1, 23mplsca 16189 . . . . . . . 8  |-  ( ph  ->  H  =  (Scalar `  U ) )
5453fveq2d 5529 . . . . . . 7  |-  ( ph  ->  ( Base `  H
)  =  ( Base `  (Scalar `  U )
) )
5538, 54eqtrd 2315 . . . . . 6  |-  ( ph  ->  T  =  ( Base `  (Scalar `  U )
) )
5655eleq2d 2350 . . . . 5  |-  ( ph  ->  ( X  e.  T  <->  X  e.  ( Base `  (Scalar `  U ) ) ) )
5756biimpa 470 . . . 4  |-  ( (
ph  /\  X  e.  T )  ->  X  e.  ( Base `  (Scalar `  U ) ) )
58 ffvelrn 5663 . . . 4  |-  ( ( (algSc `  U ) : ( Base `  (Scalar `  U ) ) --> B  /\  X  e.  (
Base `  (Scalar `  U
) ) )  -> 
( (algSc `  U
) `  X )  e.  B )
5952, 57, 58syl2anc 642 . . 3  |-  ( (
ph  /\  X  e.  T )  ->  (
(algSc `  U ) `  X )  e.  B
)
6045, 59eqeltrrd 2358 . 2  |-  ( (
ph  /\  X  e.  T )  ->  ( A `  X )  e.  B )
6140, 60impbida 805 1  |-  ( ph  ->  ( ( A `  X )  e.  B  <->  X  e.  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    C_ wss 3152   ifcif 3565   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688    |` cres 4691   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Fincfn 6863   0cc0 8737   NNcn 9746   NN0cn0 9965   Basecbs 13148   ↾s cress 13149  Scalarcsca 13211   0gc0g 13400   Ringcrg 15337  SubRingcsubrg 15541  algSccascl 16052   mPwSer cmps 16087   mPoly cmpl 16089
This theorem is referenced by:  subrg1asclcl  16337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543  df-lmod 15629  df-lss 15690  df-ascl 16055  df-psr 16098  df-mpl 16100
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