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Theorem ressmplbas2 16215
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 2296 . . . . . . . 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 16179 . . . . . . 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 2296 . . . . . . 7  |-  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  R ) )
109subrgss 15562 . . . . . 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 3179 . . . . 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 2296 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
154, 14subrg0 15568 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  ( 0g `  R )  =  ( 0g `  H ) )
162, 15syl 15 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  H ) )
1716sneqd 3666 . . . . . . . 8  |-  ( ph  ->  { ( 0g `  R ) }  =  { ( 0g `  H ) } )
1817difeq2d 3307 . . . . . . 7  |-  ( ph  ->  ( _V  \  {
( 0g `  R
) } )  =  ( _V  \  {
( 0g `  H
) } ) )
1918imaeq2d 5028 . . . . . 6  |-  ( ph  ->  ( `' f "
( _V  \  {
( 0g `  R
) } ) )  =  ( `' f
" ( _V  \  { ( 0g `  H ) } ) ) )
2019eleq1d 2362 . . . . 5  |-  ( ph  ->  ( ( `' f
" ( _V  \  { ( 0g `  R ) } ) )  e.  Fin  <->  ( `' f " ( _V  \  { ( 0g `  H ) } ) )  e.  Fin )
)
2120abbidv 2410 . . . 4  |-  ( ph  ->  { f  |  ( `' f " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin }  =  { f  |  ( `' f "
( _V  \  {
( 0g `  H
) } ) )  e.  Fin } )
2213, 21ineq12d 3384 . . 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 2301 . 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 2296 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
26 ressmpl.b . . . 4  |-  B  =  ( Base `  U
)
2724, 5, 6, 25, 26mplbas 16190 . . 3  |-  B  =  { f  e.  C  |  ( `' f
" ( _V  \  { ( 0g `  H ) } ) )  e.  Fin }
28 dfrab3 3457 . . 3  |-  { f  e.  C  |  ( `' f " ( _V  \  { ( 0g
`  H ) } ) )  e.  Fin }  =  ( C  i^i  { f  |  ( `' f " ( _V 
\  { ( 0g
`  H ) } ) )  e.  Fin } )
2927, 28eqtri 2316 . 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 16190 . . . . 5  |-  K  =  { f  e.  (
Base `  ( I mPwSer  R ) )  |  ( `' f " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin }
33 dfrab3 3457 . . . . 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 2316 . . . 4  |-  K  =  ( ( Base `  (
I mPwSer  R ) )  i^i 
{ f  |  ( `' f " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin } )
3534ineq2i 3380 . . 3  |-  ( C  i^i  K )  =  ( C  i^i  (
( Base `  ( I mPwSer  R ) )  i^i  {
f  |  ( `' f " ( _V 
\  { ( 0g
`  R ) } ) )  e.  Fin } ) )
36 inass 3392 . . 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 2319 . 2  |-  ( C  i^i  K )  =  ( ( C  i^i  ( Base `  ( I mPwSer  R ) ) )  i^i 
{ f  |  ( `' f " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin } )
3823, 29, 373eqtr4g 2353 1  |-  ( ph  ->  B  =  ( C  i^i  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {cab 2282   {crab 2560   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   {csn 3653   `'ccnv 4704   "cima 4708   ` cfv 5271  (class class class)co 5874   Fincfn 6879   Basecbs 13164   ↾s cress 13165   0gc0g 13416  SubRingcsubrg 15557   mPwSer cmps 16103   mPoly cmpl 16105
This theorem is referenced by:  ressmplbas  16216  subrgmpl  16220  ressply1bas2  16322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-mulg 14508  df-subg 14634  df-ghm 14697  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-subrg 15559  df-psr 16114  df-mpl 16116
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