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Theorem mpfconst 19422
Description: Constants are multivariate polynomial functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypotheses
Ref Expression
mpfconst.b  |-  B  =  ( Base `  S
)
mpfconst.q  |-  Q  =  ran  ( ( I evalSub  S ) `  R
)
mpfconst.i  |-  ( ph  ->  I  e.  V )
mpfconst.s  |-  ( ph  ->  S  e.  CRing )
mpfconst.r  |-  ( ph  ->  R  e.  (SubRing `  S
) )
mpfconst.x  |-  ( ph  ->  X  e.  R )
Assertion
Ref Expression
mpfconst  |-  ( ph  ->  ( ( B  ^m  I )  X.  { X } )  e.  Q
)

Proof of Theorem mpfconst
StepHypRef Expression
1 eqid 2283 . . . 4  |-  ( ( I evalSub  S ) `  R
)  =  ( ( I evalSub  S ) `  R
)
2 eqid 2283 . . . 4  |-  ( I mPoly 
( Ss  R ) )  =  ( I mPoly  ( Ss  R ) )
3 eqid 2283 . . . 4  |-  ( Ss  R )  =  ( Ss  R )
4 mpfconst.b . . . 4  |-  B  =  ( Base `  S
)
5 eqid 2283 . . . 4  |-  (algSc `  ( I mPoly  ( Ss  R
) ) )  =  (algSc `  ( I mPoly  ( Ss  R ) ) )
6 mpfconst.i . . . 4  |-  ( ph  ->  I  e.  V )
7 mpfconst.s . . . 4  |-  ( ph  ->  S  e.  CRing )
8 mpfconst.r . . . 4  |-  ( ph  ->  R  e.  (SubRing `  S
) )
9 mpfconst.x . . . 4  |-  ( ph  ->  X  e.  R )
101, 2, 3, 4, 5, 6, 7, 8, 9evlssca 19406 . . 3  |-  ( ph  ->  ( ( ( I evalSub  S ) `  R
) `  ( (algSc `  ( I mPoly  ( Ss  R ) ) ) `  X ) )  =  ( ( B  ^m  I )  X.  { X } ) )
11 eqid 2283 . . . . . . 7  |-  ( S  ^s  ( B  ^m  I
) )  =  ( S  ^s  ( B  ^m  I
) )
121, 2, 3, 11, 4evlsrhm 19405 . . . . . 6  |-  ( ( I  e.  V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( (
I evalSub  S ) `  R
)  e.  ( ( I mPoly  ( Ss  R ) ) RingHom  ( S  ^s  ( B  ^m  I ) ) ) )
136, 7, 8, 12syl3anc 1182 . . . . 5  |-  ( ph  ->  ( ( I evalSub  S
) `  R )  e.  ( ( I mPoly  ( Ss  R ) ) RingHom  ( S  ^s  ( B  ^m  I
) ) ) )
14 eqid 2283 . . . . . 6  |-  ( Base `  ( I mPoly  ( Ss  R ) ) )  =  ( Base `  (
I mPoly  ( Ss  R ) ) )
15 eqid 2283 . . . . . 6  |-  ( Base `  ( S  ^s  ( B  ^m  I ) ) )  =  ( Base `  ( S  ^s  ( B  ^m  I ) ) )
1614, 15rhmf 15504 . . . . 5  |-  ( ( ( I evalSub  S ) `
 R )  e.  ( ( I mPoly  ( Ss  R ) ) RingHom  ( S  ^s  ( B  ^m  I
) ) )  -> 
( ( I evalSub  S
) `  R ) : ( Base `  (
I mPoly  ( Ss  R ) ) ) --> ( Base `  ( S  ^s  ( B  ^m  I ) ) ) )
17 ffn 5389 . . . . 5  |-  ( ( ( I evalSub  S ) `
 R ) : ( Base `  (
I mPoly  ( Ss  R ) ) ) --> ( Base `  ( S  ^s  ( B  ^m  I ) ) )  ->  ( (
I evalSub  S ) `  R
)  Fn  ( Base `  ( I mPoly  ( Ss  R ) ) ) )
1813, 16, 173syl 18 . . . 4  |-  ( ph  ->  ( ( I evalSub  S
) `  R )  Fn  ( Base `  (
I mPoly  ( Ss  R ) ) ) )
193subrgrng 15548 . . . . . . 7  |-  ( R  e.  (SubRing `  S
)  ->  ( Ss  R
)  e.  Ring )
208, 19syl 15 . . . . . 6  |-  ( ph  ->  ( Ss  R )  e.  Ring )
21 eqid 2283 . . . . . . 7  |-  (Scalar `  ( I mPoly  ( Ss  R
) ) )  =  (Scalar `  ( I mPoly  ( Ss  R ) ) )
222mplrng 16196 . . . . . . 7  |-  ( ( I  e.  V  /\  ( Ss  R )  e.  Ring )  ->  ( I mPoly  ( Ss  R ) )  e. 
Ring )
232mpllmod 16195 . . . . . . 7  |-  ( ( I  e.  V  /\  ( Ss  R )  e.  Ring )  ->  ( I mPoly  ( Ss  R ) )  e. 
LMod )
24 eqid 2283 . . . . . . 7  |-  ( Base `  (Scalar `  ( I mPoly  ( Ss  R ) ) ) )  =  ( Base `  (Scalar `  ( I mPoly  ( Ss  R ) ) ) )
255, 21, 22, 23, 24, 14asclf 16077 . . . . . 6  |-  ( ( I  e.  V  /\  ( Ss  R )  e.  Ring )  ->  (algSc `  (
I mPoly  ( Ss  R ) ) ) : (
Base `  (Scalar `  (
I mPoly  ( Ss  R ) ) ) ) --> (
Base `  ( I mPoly  ( Ss  R ) ) ) )
266, 20, 25syl2anc 642 . . . . 5  |-  ( ph  ->  (algSc `  ( I mPoly  ( Ss  R ) ) ) : ( Base `  (Scalar `  ( I mPoly  ( Ss  R ) ) ) ) --> ( Base `  (
I mPoly  ( Ss  R ) ) ) )
274subrgss 15546 . . . . . . . 8  |-  ( R  e.  (SubRing `  S
)  ->  R  C_  B
)
283, 4ressbas2 13199 . . . . . . . 8  |-  ( R 
C_  B  ->  R  =  ( Base `  ( Ss  R ) ) )
298, 27, 283syl 18 . . . . . . 7  |-  ( ph  ->  R  =  ( Base `  ( Ss  R ) ) )
30 ovex 5883 . . . . . . . . . 10  |-  ( Ss  R )  e.  _V
3130a1i 10 . . . . . . . . 9  |-  ( ph  ->  ( Ss  R )  e.  _V )
322, 6, 31mplsca 16189 . . . . . . . 8  |-  ( ph  ->  ( Ss  R )  =  (Scalar `  ( I mPoly  ( Ss  R ) ) ) )
3332fveq2d 5529 . . . . . . 7  |-  ( ph  ->  ( Base `  ( Ss  R ) )  =  ( Base `  (Scalar `  ( I mPoly  ( Ss  R ) ) ) ) )
3429, 33eqtrd 2315 . . . . . 6  |-  ( ph  ->  R  =  ( Base `  (Scalar `  ( I mPoly  ( Ss  R ) ) ) ) )
359, 34eleqtrd 2359 . . . . 5  |-  ( ph  ->  X  e.  ( Base `  (Scalar `  ( I mPoly  ( Ss  R ) ) ) ) )
36 ffvelrn 5663 . . . . 5  |-  ( ( (algSc `  ( I mPoly  ( Ss  R ) ) ) : ( Base `  (Scalar `  ( I mPoly  ( Ss  R ) ) ) ) --> ( Base `  (
I mPoly  ( Ss  R ) ) )  /\  X  e.  ( Base `  (Scalar `  ( I mPoly  ( Ss  R ) ) ) ) )  ->  ( (algSc `  ( I mPoly  ( Ss  R ) ) ) `  X )  e.  (
Base `  ( I mPoly  ( Ss  R ) ) ) )
3726, 35, 36syl2anc 642 . . . 4  |-  ( ph  ->  ( (algSc `  (
I mPoly  ( Ss  R ) ) ) `  X
)  e.  ( Base `  ( I mPoly  ( Ss  R ) ) ) )
38 fnfvelrn 5662 . . . 4  |-  ( ( ( ( I evalSub  S
) `  R )  Fn  ( Base `  (
I mPoly  ( Ss  R ) ) )  /\  (
(algSc `  ( I mPoly  ( Ss  R ) ) ) `
 X )  e.  ( Base `  (
I mPoly  ( Ss  R ) ) ) )  -> 
( ( ( I evalSub  S ) `  R
) `  ( (algSc `  ( I mPoly  ( Ss  R ) ) ) `  X ) )  e. 
ran  ( ( I evalSub  S ) `  R
) )
3918, 37, 38syl2anc 642 . . 3  |-  ( ph  ->  ( ( ( I evalSub  S ) `  R
) `  ( (algSc `  ( I mPoly  ( Ss  R ) ) ) `  X ) )  e. 
ran  ( ( I evalSub  S ) `  R
) )
4010, 39eqeltrrd 2358 . 2  |-  ( ph  ->  ( ( B  ^m  I )  X.  { X } )  e.  ran  ( ( I evalSub  S
) `  R )
)
41 mpfconst.q . 2  |-  Q  =  ran  ( ( I evalSub  S ) `  R
)
4240, 41syl6eleqr 2374 1  |-  ( ph  ->  ( ( B  ^m  I )  X.  { X } )  e.  Q
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {csn 3640    X. cxp 4687   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Basecbs 13148   ↾s cress 13149  Scalarcsca 13211    ^s cpws 13347   Ringcrg 15337   CRingccrg 15338   RingHom crh 15494  SubRingcsubrg 15541  algSccascl 16052   mPoly cmpl 16089   evalSub ces 16090
This theorem is referenced by:  mzpmfp  26825
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-sup 7194  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-10 9812  df-n0 9966  df-z 10025  df-dec 10125  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-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  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-sbg 14491  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-cring 15341  df-ur 15342  df-rnghom 15496  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-assa 16053  df-asp 16054  df-ascl 16055  df-psr 16098  df-mvr 16099  df-mpl 16100  df-evls 16101
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