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Theorem ressmplbas2 16199
Description: The base set of a restricted polynomial algebra consists of power series in the subring which are also polynomials (in the parent ring). (Contributed by Mario Carneiro, 3-Jul-2015.)
Hypotheses
Ref Expression
ressmpl.s  |-  S  =  ( I mPoly  R )
ressmpl.h  |-  H  =  ( Rs  T )
ressmpl.u  |-  U  =  ( I mPoly  H )
ressmpl.b  |-  B  =  ( Base `  U
)
ressmpl.1  |-  ( ph  ->  I  e.  V )
ressmpl.2  |-  ( ph  ->  T  e.  (SubRing `  R
) )
ressmplbas2.w  |-  W  =  ( I mPwSer  H )
ressmplbas2.c  |-  C  =  ( Base `  W
)
ressmplbas2.k  |-  K  =  ( Base `  S
)
Assertion
Ref Expression
ressmplbas2  |-  ( ph  ->  B  =  ( C  i^i  K ) )

Proof of Theorem ressmplbas2
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 ressmpl.1 . . . . . . 7  |-  ( ph  ->  I  e.  V )
2 ressmpl.2 . . . . . . 7  |-  ( ph  ->  T  e.  (SubRing `  R
) )
3 eqid 2283 . . . . . . . 8  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
4 ressmpl.h . . . . . . . 8  |-  H  =  ( Rs  T )
5 ressmplbas2.w . . . . . . . 8  |-  W  =  ( I mPwSer  H )
6 ressmplbas2.c . . . . . . . 8  |-  C  =  ( Base `  W
)
73, 4, 5, 6subrgpsr 16163 . . . . . . 7  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  C  e.  (SubRing `  ( I mPwSer  R ) ) )
81, 2, 7syl2anc 642 . . . . . 6  |-  ( ph  ->  C  e.  (SubRing `  (
I mPwSer  R ) ) )
9 eqid 2283 . . . . . . 7  |-  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  R ) )
109subrgss 15546 . . . . . 6  |-  ( C  e.  (SubRing `  (
I mPwSer  R ) )  ->  C  C_  ( Base `  (
I mPwSer  R ) ) )
118, 10syl 15 . . . . 5  |-  ( ph  ->  C  C_  ( Base `  ( I mPwSer  R ) ) )
12 df-ss 3166 . . . . 5  |-  ( C 
C_  ( Base `  (
I mPwSer  R ) )  <->  ( C  i^i  ( Base `  (
I mPwSer  R ) ) )  =  C )
1311, 12sylib 188 . . . 4  |-  ( ph  ->  ( C  i^i  ( Base `  ( I mPwSer  R
) ) )  =  C )
14 eqid 2283 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
154, 14subrg0 15552 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  ( 0g `  R )  =  ( 0g `  H ) )
162, 15syl 15 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  H ) )
1716sneqd 3653 . . . . . . . 8  |-  ( ph  ->  { ( 0g `  R ) }  =  { ( 0g `  H ) } )
1817difeq2d 3294 . . . . . . 7  |-  ( ph  ->  ( _V  \  {
( 0g `  R
) } )  =  ( _V  \  {
( 0g `  H
) } ) )
1918imaeq2d 5012 . . . . . 6  |-  ( ph  ->  ( `' f "
( _V  \  {
( 0g `  R
) } ) )  =  ( `' f
" ( _V  \  { ( 0g `  H ) } ) ) )
2019eleq1d 2349 . . . . 5  |-  ( ph  ->  ( ( `' f
" ( _V  \  { ( 0g `  R ) } ) )  e.  Fin  <->  ( `' f " ( _V  \  { ( 0g `  H ) } ) )  e.  Fin )
)
2120abbidv 2397 . . . 4  |-  ( ph  ->  { f  |  ( `' f " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin }  =  { f  |  ( `' f "
( _V  \  {
( 0g `  H
) } ) )  e.  Fin } )
2213, 21ineq12d 3371 . . 3  |-  ( ph  ->  ( ( C  i^i  ( Base `  ( I mPwSer  R ) ) )  i^i 
{ f  |  ( `' f " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin } )  =  ( C  i^i  { f  |  ( `' f "
( _V  \  {
( 0g `  H
) } ) )  e.  Fin } ) )
2322eqcomd 2288 . 2  |-  ( ph  ->  ( C  i^i  {
f  |  ( `' f " ( _V 
\  { ( 0g
`  H ) } ) )  e.  Fin } )  =  ( ( C  i^i  ( Base `  ( I mPwSer  R ) ) )  i^i  {
f  |  ( `' f " ( _V 
\  { ( 0g
`  R ) } ) )  e.  Fin } ) )
24 ressmpl.u . . . 4  |-  U  =  ( I mPoly  H )
25 eqid 2283 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
26 ressmpl.b . . . 4  |-  B  =  ( Base `  U
)
2724, 5, 6, 25, 26mplbas 16174 . . 3  |-  B  =  { f  e.  C  |  ( `' f
" ( _V  \  { ( 0g `  H ) } ) )  e.  Fin }
28 dfrab3 3444 . . 3  |-  { f  e.  C  |  ( `' f " ( _V  \  { ( 0g
`  H ) } ) )  e.  Fin }  =  ( C  i^i  { f  |  ( `' f " ( _V 
\  { ( 0g
`  H ) } ) )  e.  Fin } )
2927, 28eqtri 2303 . 2  |-  B  =  ( C  i^i  {
f  |  ( `' f " ( _V 
\  { ( 0g
`  H ) } ) )  e.  Fin } )
30 ressmpl.s . . . . . 6  |-  S  =  ( I mPoly  R )
31 ressmplbas2.k . . . . . 6  |-  K  =  ( Base `  S
)
3230, 3, 9, 14, 31mplbas 16174 . . . . 5  |-  K  =  { f  e.  (
Base `  ( I mPwSer  R ) )  |  ( `' f " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin }
33 dfrab3 3444 . . . . 5  |-  { f  e.  ( Base `  (
I mPwSer  R ) )  |  ( `' f "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin }  =  ( ( Base `  (
I mPwSer  R ) )  i^i 
{ f  |  ( `' f " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin } )
3432, 33eqtri 2303 . . . 4  |-  K  =  ( ( Base `  (
I mPwSer  R ) )  i^i 
{ f  |  ( `' f " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin } )
3534ineq2i 3367 . . 3  |-  ( C  i^i  K )  =  ( C  i^i  (
( Base `  ( I mPwSer  R ) )  i^i  {
f  |  ( `' f " ( _V 
\  { ( 0g
`  R ) } ) )  e.  Fin } ) )
36 inass 3379 . . 3  |-  ( ( C  i^i  ( Base `  ( I mPwSer  R ) ) )  i^i  {
f  |  ( `' f " ( _V 
\  { ( 0g
`  R ) } ) )  e.  Fin } )  =  ( C  i^i  ( ( Base `  ( I mPwSer  R ) )  i^i  { f  |  ( `' f
" ( _V  \  { ( 0g `  R ) } ) )  e.  Fin }
) )
3735, 36eqtr4i 2306 . 2  |-  ( C  i^i  K )  =  ( ( C  i^i  ( Base `  ( I mPwSer  R ) ) )  i^i 
{ f  |  ( `' f " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin } )
3823, 29, 373eqtr4g 2340 1  |-  ( ph  ->  B  =  ( C  i^i  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {cab 2269   {crab 2547   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   {csn 3640   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858   Fincfn 6863   Basecbs 13148   ↾s cress 13149   0gc0g 13400  SubRingcsubrg 15541   mPwSer cmps 16087   mPoly cmpl 16089
This theorem is referenced by:  ressmplbas  16200  subrgmpl  16204  ressply1bas2  16306
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-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-psr 16098  df-mpl 16100
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