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Theorem ressmplbas2 16510
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 2435 . . . . . . . 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 16474 . . . . . . 7  |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R
) )  ->  C  e.  (SubRing `  ( I mPwSer  R ) ) )
81, 2, 7syl2anc 643 . . . . . 6  |-  ( ph  ->  C  e.  (SubRing `  (
I mPwSer  R ) ) )
9 eqid 2435 . . . . . . 7  |-  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  R ) )
109subrgss 15861 . . . . . 6  |-  ( C  e.  (SubRing `  (
I mPwSer  R ) )  ->  C  C_  ( Base `  (
I mPwSer  R ) ) )
118, 10syl 16 . . . . 5  |-  ( ph  ->  C  C_  ( Base `  ( I mPwSer  R ) ) )
12 df-ss 3326 . . . . 5  |-  ( C 
C_  ( Base `  (
I mPwSer  R ) )  <->  ( C  i^i  ( Base `  (
I mPwSer  R ) ) )  =  C )
1311, 12sylib 189 . . . 4  |-  ( ph  ->  ( C  i^i  ( Base `  ( I mPwSer  R
) ) )  =  C )
14 eqid 2435 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
154, 14subrg0 15867 . . . . . . . . . 10  |-  ( T  e.  (SubRing `  R
)  ->  ( 0g `  R )  =  ( 0g `  H ) )
162, 15syl 16 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  R
)  =  ( 0g
`  H ) )
1716sneqd 3819 . . . . . . . 8  |-  ( ph  ->  { ( 0g `  R ) }  =  { ( 0g `  H ) } )
1817difeq2d 3457 . . . . . . 7  |-  ( ph  ->  ( _V  \  {
( 0g `  R
) } )  =  ( _V  \  {
( 0g `  H
) } ) )
1918imaeq2d 5195 . . . . . 6  |-  ( ph  ->  ( `' f "
( _V  \  {
( 0g `  R
) } ) )  =  ( `' f
" ( _V  \  { ( 0g `  H ) } ) ) )
2019eleq1d 2501 . . . . 5  |-  ( ph  ->  ( ( `' f
" ( _V  \  { ( 0g `  R ) } ) )  e.  Fin  <->  ( `' f " ( _V  \  { ( 0g `  H ) } ) )  e.  Fin )
)
2120abbidv 2549 . . . 4  |-  ( ph  ->  { f  |  ( `' f " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin }  =  { f  |  ( `' f "
( _V  \  {
( 0g `  H
) } ) )  e.  Fin } )
2213, 21ineq12d 3535 . . 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 2440 . 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 2435 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
26 ressmpl.b . . . 4  |-  B  =  ( Base `  U
)
2724, 5, 6, 25, 26mplbas 16485 . . 3  |-  B  =  { f  e.  C  |  ( `' f
" ( _V  \  { ( 0g `  H ) } ) )  e.  Fin }
28 dfrab3 3609 . . 3  |-  { f  e.  C  |  ( `' f " ( _V  \  { ( 0g
`  H ) } ) )  e.  Fin }  =  ( C  i^i  { f  |  ( `' f " ( _V 
\  { ( 0g
`  H ) } ) )  e.  Fin } )
2927, 28eqtri 2455 . 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 16485 . . . . 5  |-  K  =  { f  e.  (
Base `  ( I mPwSer  R ) )  |  ( `' f " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin }
33 dfrab3 3609 . . . . 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 2455 . . . 4  |-  K  =  ( ( Base `  (
I mPwSer  R ) )  i^i 
{ f  |  ( `' f " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin } )
3534ineq2i 3531 . . 3  |-  ( C  i^i  K )  =  ( C  i^i  (
( Base `  ( I mPwSer  R ) )  i^i  {
f  |  ( `' f " ( _V 
\  { ( 0g
`  R ) } ) )  e.  Fin } ) )
36 inass 3543 . . 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 2458 . 2  |-  ( C  i^i  K )  =  ( ( C  i^i  ( Base `  ( I mPwSer  R ) ) )  i^i 
{ f  |  ( `' f " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin } )
3823, 29, 373eqtr4g 2492 1  |-  ( ph  ->  B  =  ( C  i^i  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {cab 2421   {crab 2701   _Vcvv 2948    \ cdif 3309    i^i cin 3311    C_ wss 3312   {csn 3806   `'ccnv 4869   "cima 4873   ` cfv 5446  (class class class)co 6073   Fincfn 7101   Basecbs 13461   ↾s cress 13462   0gc0g 13715  SubRingcsubrg 15856   mPwSer cmps 16398   mPoly cmpl 16400
This theorem is referenced by:  ressmplbas  16511  subrgmpl  16515  ressply1bas2  16614
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-ofr 6298  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-tset 13540  df-0g 13719  df-gsum 13720  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-mulg 14807  df-subg 14933  df-ghm 14996  df-cntz 15108  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-subrg 15858  df-psr 16409  df-mpl 16411
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