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Theorem mpfsubrg 19830
Description: Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
Hypothesis
Ref Expression
mpfaddcl.q  |-  Q  =  ran  ( ( I evalSub  S ) `  R
)
Assertion
Ref Expression
mpfsubrg  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  Q  e.  (SubRing `  ( S  ^s  (
( Base `  S )  ^m  I ) ) ) )

Proof of Theorem mpfsubrg
StepHypRef Expression
1 eqid 2389 . . . . . 6  |-  ( ( I evalSub  S ) `  R
)  =  ( ( I evalSub  S ) `  R
)
2 eqid 2389 . . . . . 6  |-  ( I mPoly 
( Ss  R ) )  =  ( I mPoly  ( Ss  R ) )
3 eqid 2389 . . . . . 6  |-  ( Ss  R )  =  ( Ss  R )
4 eqid 2389 . . . . . 6  |-  ( S  ^s  ( ( Base `  S
)  ^m  I )
)  =  ( S  ^s  ( ( Base `  S
)  ^m  I )
)
5 eqid 2389 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
61, 2, 3, 4, 5evlsrhm 19811 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( (
I evalSub  S ) `  R
)  e.  ( ( I mPoly  ( Ss  R ) ) RingHom  ( S  ^s  (
( Base `  S )  ^m  I ) ) ) )
7 eqid 2389 . . . . . 6  |-  ( Base `  ( I mPoly  ( Ss  R ) ) )  =  ( Base `  (
I mPoly  ( Ss  R ) ) )
8 eqid 2389 . . . . . 6  |-  ( Base `  ( S  ^s  ( (
Base `  S )  ^m  I ) ) )  =  ( Base `  ( S  ^s  ( ( Base `  S
)  ^m  I )
) )
97, 8rhmf 15756 . . . . 5  |-  ( ( ( I evalSub  S ) `
 R )  e.  ( ( I mPoly  ( Ss  R ) ) RingHom  ( S  ^s  ( ( Base `  S
)  ^m  I )
) )  ->  (
( I evalSub  S ) `  R ) : (
Base `  ( I mPoly  ( Ss  R ) ) ) --> ( Base `  ( S  ^s  ( ( Base `  S
)  ^m  I )
) ) )
106, 9syl 16 . . . 4  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( (
I evalSub  S ) `  R
) : ( Base `  ( I mPoly  ( Ss  R ) ) ) --> (
Base `  ( S  ^s  ( ( Base `  S
)  ^m  I )
) ) )
11 ffn 5533 . . . 4  |-  ( ( ( I evalSub  S ) `
 R ) : ( Base `  (
I mPoly  ( Ss  R ) ) ) --> ( Base `  ( S  ^s  ( (
Base `  S )  ^m  I ) ) )  ->  ( ( I evalSub  S ) `  R
)  Fn  ( Base `  ( I mPoly  ( Ss  R ) ) ) )
12 fnima 5505 . . . 4  |-  ( ( ( I evalSub  S ) `
 R )  Fn  ( Base `  (
I mPoly  ( Ss  R ) ) )  ->  (
( ( I evalSub  S
) `  R ) " ( Base `  (
I mPoly  ( Ss  R ) ) ) )  =  ran  ( ( I evalSub  S ) `  R
) )
1310, 11, 123syl 19 . . 3  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( (
( I evalSub  S ) `  R ) " ( Base `  ( I mPoly  ( Ss  R ) ) ) )  =  ran  (
( I evalSub  S ) `  R ) )
14 mpfaddcl.q . . 3  |-  Q  =  ran  ( ( I evalSub  S ) `  R
)
1513, 14syl6reqr 2440 . 2  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  Q  =  ( ( ( I evalSub  S ) `  R
) " ( Base `  ( I mPoly  ( Ss  R ) ) ) ) )
163subrgrng 15800 . . . . . 6  |-  ( R  e.  (SubRing `  S
)  ->  ( Ss  R
)  e.  Ring )
172mplrng 16444 . . . . . 6  |-  ( ( I  e.  _V  /\  ( Ss  R )  e.  Ring )  ->  ( I mPoly  ( Ss  R ) )  e. 
Ring )
1816, 17sylan2 461 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  (SubRing `  S
) )  ->  (
I mPoly  ( Ss  R ) )  e.  Ring )
19183adant2 976 . . . 4  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( I mPoly  ( Ss  R ) )  e. 
Ring )
207subrgid 15799 . . . 4  |-  ( ( I mPoly  ( Ss  R ) )  e.  Ring  ->  (
Base `  ( I mPoly  ( Ss  R ) ) )  e.  (SubRing `  (
I mPoly  ( Ss  R ) ) ) )
2119, 20syl 16 . . 3  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( Base `  ( I mPoly  ( Ss  R ) ) )  e.  (SubRing `  ( I mPoly  ( Ss  R ) ) ) )
22 rhmima 15828 . . 3  |-  ( ( ( ( I evalSub  S
) `  R )  e.  ( ( I mPoly  ( Ss  R ) ) RingHom  ( S  ^s  ( ( Base `  S
)  ^m  I )
) )  /\  ( Base `  ( I mPoly  ( Ss  R ) ) )  e.  (SubRing `  (
I mPoly  ( Ss  R ) ) ) )  -> 
( ( ( I evalSub  S ) `  R
) " ( Base `  ( I mPoly  ( Ss  R ) ) ) )  e.  (SubRing `  ( S  ^s  ( ( Base `  S
)  ^m  I )
) ) )
236, 21, 22syl2anc 643 . 2  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( (
( I evalSub  S ) `  R ) " ( Base `  ( I mPoly  ( Ss  R ) ) ) )  e.  (SubRing `  ( S  ^s  ( ( Base `  S
)  ^m  I )
) ) )
2415, 23eqeltrd 2463 1  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  Q  e.  (SubRing `  ( S  ^s  (
( Base `  S )  ^m  I ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2901   ran crn 4821   "cima 4823    Fn wfn 5391   -->wf 5392   ` cfv 5396  (class class class)co 6022    ^m cmap 6956   Basecbs 13398   ↾s cress 13399    ^s cpws 13599   Ringcrg 15589   CRingccrg 15590   RingHom crh 15746  SubRingcsubrg 15793   mPoly cmpl 16337   evalSub ces 16338
This theorem is referenced by:  mpff  19831  mpfaddcl  19832  mpfmulcl  19833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-ofr 6247  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-map 6958  df-pm 6959  df-ixp 7002  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-oi 7414  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-fz 10978  df-fzo 11068  df-seq 11253  df-hash 11548  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-sca 13474  df-vsca 13475  df-tset 13477  df-ple 13478  df-ds 13480  df-hom 13482  df-cco 13483  df-prds 13600  df-pws 13602  df-0g 13656  df-gsum 13657  df-mre 13740  df-mrc 13741  df-acs 13743  df-mnd 14619  df-mhm 14667  df-submnd 14668  df-grp 14741  df-minusg 14742  df-sbg 14743  df-mulg 14744  df-subg 14870  df-ghm 14933  df-cntz 15045  df-cmn 15343  df-abl 15344  df-mgp 15578  df-rng 15592  df-cring 15593  df-ur 15594  df-rnghom 15748  df-subrg 15795  df-lmod 15881  df-lss 15938  df-lsp 15977  df-assa 16301  df-asp 16302  df-ascl 16303  df-psr 16346  df-mvr 16347  df-mpl 16348  df-evls 16349
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