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Theorem mpfsubrg 17725
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 2451 . . . . 5  |-  ( ( I evalSub  S ) `  R
)  =  ( ( I evalSub  S ) `  R
)
2 eqid 2451 . . . . 5  |-  ( I mPoly 
( Ss  R ) )  =  ( I mPoly  ( Ss  R ) )
3 eqid 2451 . . . . 5  |-  ( Ss  R )  =  ( Ss  R )
4 eqid 2451 . . . . 5  |-  ( S  ^s  ( ( Base `  S
)  ^m  I )
)  =  ( S  ^s  ( ( Base `  S
)  ^m  I )
)
5 eqid 2451 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
61, 2, 3, 4, 5evlsrhm 17714 . . . 4  |-  ( ( 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 2451 . . . . 5  |-  ( Base `  ( I mPoly  ( Ss  R ) ) )  =  ( Base `  (
I mPoly  ( Ss  R ) ) )
8 eqid 2451 . . . . 5  |-  ( Base `  ( S  ^s  ( (
Base `  S )  ^m  I ) ) )  =  ( Base `  ( S  ^s  ( ( Base `  S
)  ^m  I )
) )
97, 8rhmf 16922 . . . 4  |-  ( ( ( 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 )
) ) )
10 ffn 5657 . . . 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 ) ) ) )
11 fnima 5627 . . . 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
) )
126, 9, 10, 114syl 21 . . 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 ) )
13 mpfaddcl.q . . 3  |-  Q  =  ran  ( ( I evalSub  S ) `  R
)
1412, 13syl6reqr 2511 . 2  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  Q  =  ( ( ( I evalSub  S ) `  R
) " ( Base `  ( I mPoly  ( Ss  R ) ) ) ) )
153subrgrng 16974 . . . . . 6  |-  ( R  e.  (SubRing `  S
)  ->  ( Ss  R
)  e.  Ring )
162mplrng 17638 . . . . . 6  |-  ( ( I  e.  _V  /\  ( Ss  R )  e.  Ring )  ->  ( I mPoly  ( Ss  R ) )  e. 
Ring )
1715, 16sylan2 474 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  (SubRing `  S
) )  ->  (
I mPoly  ( Ss  R ) )  e.  Ring )
18173adant2 1007 . . . 4  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( I mPoly  ( Ss  R ) )  e. 
Ring )
197subrgid 16973 . . . 4  |-  ( ( I mPoly  ( Ss  R ) )  e.  Ring  ->  (
Base `  ( I mPoly  ( Ss  R ) ) )  e.  (SubRing `  (
I mPoly  ( Ss  R ) ) ) )
2018, 19syl 16 . . 3  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( Base `  ( I mPoly  ( Ss  R ) ) )  e.  (SubRing `  ( I mPoly  ( Ss  R ) ) ) )
21 rhmima 17002 . . 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 )
) ) )
226, 20, 21syl2anc 661 . 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 )
) ) )
2314, 22eqeltrd 2539 1  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  Q  e.  (SubRing `  ( S  ^s  (
( Base `  S )  ^m  I ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3068   ran crn 4939   "cima 4941    Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190    ^m cmap 7314   Basecbs 14276   ↾s cress 14277    ^s cpws 14487   Ringcrg 16751   CRingccrg 16752   RingHom crh 16910  SubRingcsubrg 16967   mPoly cmpl 17526   evalSub ces 17693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-ofr 6421  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-sup 7792  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-fz 11539  df-fzo 11650  df-seq 11908  df-hash 12205  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-hom 14364  df-cco 14365  df-0g 14482  df-gsum 14483  df-prds 14488  df-pws 14490  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-mhm 15566  df-submnd 15567  df-grp 15647  df-minusg 15648  df-sbg 15649  df-mulg 15650  df-subg 15780  df-ghm 15847  df-cntz 15937  df-cmn 16383  df-abl 16384  df-mgp 16697  df-ur 16709  df-srg 16713  df-rng 16753  df-cring 16754  df-rnghom 16912  df-subrg 16969  df-lmod 17056  df-lss 17120  df-lsp 17159  df-assa 17490  df-asp 17491  df-ascl 17492  df-psr 17529  df-mvr 17530  df-mpl 17531  df-evls 17695
This theorem is referenced by:  mpff  17726  mpfaddcl  17727  mpfmulcl  17728
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