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Theorem evlsval2 18167
Description: Characterizing properties of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 12-Mar-2015.)
Hypotheses
Ref Expression
evlsval.q  |-  Q  =  ( ( I evalSub  S
) `  R )
evlsval.w  |-  W  =  ( I mPoly  U )
evlsval.v  |-  V  =  ( I mVar  U )
evlsval.u  |-  U  =  ( Ss  R )
evlsval.t  |-  T  =  ( S  ^s  ( B  ^m  I ) )
evlsval.b  |-  B  =  ( Base `  S
)
evlsval.a  |-  A  =  (algSc `  W )
evlsval.x  |-  X  =  ( x  e.  R  |->  ( ( B  ^m  I )  X.  {
x } ) )
evlsval.y  |-  Y  =  ( x  e.  I  |->  ( g  e.  ( B  ^m  I ) 
|->  ( g `  x
) ) )
Assertion
Ref Expression
evlsval2  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( Q  e.  ( W RingHom  T )  /\  ( ( Q  o.  A )  =  X  /\  ( Q  o.  V )  =  Y ) ) )
Distinct variable groups:    g, I, x    x, R    S, g, x    B, g, x    R, g    x, T
Allowed substitution hints:    A( x, g)    Q( x, g)    T( g)    U( x, g)    V( x, g)    W( x, g)    X( x, g)    Y( x, g)

Proof of Theorem evlsval2
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 evlsval.q . . . 4  |-  Q  =  ( ( I evalSub  S
) `  R )
2 evlsval.w . . . 4  |-  W  =  ( I mPoly  U )
3 evlsval.v . . . 4  |-  V  =  ( I mVar  U )
4 evlsval.u . . . 4  |-  U  =  ( Ss  R )
5 evlsval.t . . . 4  |-  T  =  ( S  ^s  ( B  ^m  I ) )
6 evlsval.b . . . 4  |-  B  =  ( Base `  S
)
7 evlsval.a . . . 4  |-  A  =  (algSc `  W )
8 evlsval.x . . . 4  |-  X  =  ( x  e.  R  |->  ( ( B  ^m  I )  X.  {
x } ) )
9 evlsval.y . . . 4  |-  Y  =  ( x  e.  I  |->  ( g  e.  ( B  ^m  I ) 
|->  ( g `  x
) ) )
101, 2, 3, 4, 5, 6, 7, 8, 9evlsval 18166 . . 3  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  Q  =  ( iota_ m  e.  ( W RingHom  T ) ( ( m  o.  A )  =  X  /\  (
m  o.  V )  =  Y ) ) )
11 eqid 2443 . . . . 5  |-  ( Base `  T )  =  (
Base `  T )
12 simp1 997 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  I  e.  _V )
134subrgcrng 17411 . . . . . 6  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  U  e.  CRing
)
14133adant1 1015 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  U  e.  CRing
)
15 simp2 998 . . . . . 6  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  S  e.  CRing
)
16 ovex 6309 . . . . . 6  |-  ( B  ^m  I )  e. 
_V
175pwscrng 17244 . . . . . 6  |-  ( ( S  e.  CRing  /\  ( B  ^m  I )  e. 
_V )  ->  T  e.  CRing )
1815, 16, 17sylancl 662 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  T  e.  CRing
)
196subrgss 17408 . . . . . . . . 9  |-  ( R  e.  (SubRing `  S
)  ->  R  C_  B
)
20193ad2ant3 1020 . . . . . . . 8  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  R  C_  B
)
2120resmptd 5315 . . . . . . 7  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( (
x  e.  B  |->  ( ( B  ^m  I
)  X.  { x } ) )  |`  R )  =  ( x  e.  R  |->  ( ( B  ^m  I
)  X.  { x } ) ) )
2221, 8syl6eqr 2502 . . . . . 6  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( (
x  e.  B  |->  ( ( B  ^m  I
)  X.  { x } ) )  |`  R )  =  X )
23 crngring 17187 . . . . . . . . 9  |-  ( S  e.  CRing  ->  S  e.  Ring )
24233ad2ant2 1019 . . . . . . . 8  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  S  e.  Ring )
25 eqid 2443 . . . . . . . . 9  |-  ( x  e.  B  |->  ( ( B  ^m  I )  X.  { x }
) )  =  ( x  e.  B  |->  ( ( B  ^m  I
)  X.  { x } ) )
265, 6, 25pwsdiagrhm 17440 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  ( B  ^m  I )  e. 
_V )  ->  (
x  e.  B  |->  ( ( B  ^m  I
)  X.  { x } ) )  e.  ( S RingHom  T )
)
2724, 16, 26sylancl 662 . . . . . . 7  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( x  e.  B  |->  ( ( B  ^m  I )  X.  { x }
) )  e.  ( S RingHom  T ) )
28 simp3 999 . . . . . . 7  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  R  e.  (SubRing `  S ) )
294resrhm 17436 . . . . . . 7  |-  ( ( ( x  e.  B  |->  ( ( B  ^m  I )  X.  {
x } ) )  e.  ( S RingHom  T
)  /\  R  e.  (SubRing `  S ) )  ->  ( ( x  e.  B  |->  ( ( B  ^m  I )  X.  { x }
) )  |`  R )  e.  ( U RingHom  T
) )
3027, 28, 29syl2anc 661 . . . . . 6  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( (
x  e.  B  |->  ( ( B  ^m  I
)  X.  { x } ) )  |`  R )  e.  ( U RingHom  T ) )
3122, 30eqeltrrd 2532 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  X  e.  ( U RingHom  T ) )
32 fvex 5866 . . . . . . . . . . . 12  |-  ( Base `  S )  e.  _V
336, 32eqeltri 2527 . . . . . . . . . . 11  |-  B  e. 
_V
34 simpl1 1000 . . . . . . . . . . 11  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  I  e.  _V )
35 elmapg 7435 . . . . . . . . . . 11  |-  ( ( B  e.  _V  /\  I  e.  _V )  ->  ( g  e.  ( B  ^m  I )  <-> 
g : I --> B ) )
3633, 34, 35sylancr 663 . . . . . . . . . 10  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  (
g  e.  ( B  ^m  I )  <->  g :
I --> B ) )
3736biimpa 484 . . . . . . . . 9  |-  ( ( ( ( I  e. 
_V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) )  /\  x  e.  I )  /\  g  e.  ( B  ^m  I ) )  ->  g : I --> B )
38 simplr 755 . . . . . . . . 9  |-  ( ( ( ( I  e. 
_V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) )  /\  x  e.  I )  /\  g  e.  ( B  ^m  I ) )  ->  x  e.  I
)
3937, 38ffvelrnd 6017 . . . . . . . 8  |-  ( ( ( ( I  e. 
_V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) )  /\  x  e.  I )  /\  g  e.  ( B  ^m  I ) )  ->  ( g `  x )  e.  B
)
40 eqid 2443 . . . . . . . 8  |-  ( g  e.  ( B  ^m  I )  |->  ( g `
 x ) )  =  ( g  e.  ( B  ^m  I
)  |->  ( g `  x ) )
4139, 40fmptd 6040 . . . . . . 7  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  (
g  e.  ( B  ^m  I )  |->  ( g `  x ) ) : ( B  ^m  I ) --> B )
42 simpl2 1001 . . . . . . . 8  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  S  e.  CRing )
435, 6, 11pwselbasb 14866 . . . . . . . 8  |-  ( ( S  e.  CRing  /\  ( B  ^m  I )  e. 
_V )  ->  (
( g  e.  ( B  ^m  I ) 
|->  ( g `  x
) )  e.  (
Base `  T )  <->  ( g  e.  ( B  ^m  I )  |->  ( g `  x ) ) : ( B  ^m  I ) --> B ) )
4442, 16, 43sylancl 662 . . . . . . 7  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  (
( g  e.  ( B  ^m  I ) 
|->  ( g `  x
) )  e.  (
Base `  T )  <->  ( g  e.  ( B  ^m  I )  |->  ( g `  x ) ) : ( B  ^m  I ) --> B ) )
4541, 44mpbird 232 . . . . . 6  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  (
g  e.  ( B  ^m  I )  |->  ( g `  x ) )  e.  ( Base `  T ) )
4645, 9fmptd 6040 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  Y :
I --> ( Base `  T
) )
472, 11, 7, 3, 12, 14, 18, 31, 46evlseu 18163 . . . 4  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  E! m  e.  ( W RingHom  T )
( ( m  o.  A )  =  X  /\  ( m  o.  V )  =  Y ) )
48 riotacl2 6256 . . . 4  |-  ( E! m  e.  ( W RingHom  T ) ( ( m  o.  A )  =  X  /\  (
m  o.  V )  =  Y )  -> 
( iota_ m  e.  ( W RingHom  T ) ( ( m  o.  A )  =  X  /\  (
m  o.  V )  =  Y ) )  e.  { m  e.  ( W RingHom  T )  |  ( ( m  o.  A )  =  X  /\  ( m  o.  V )  =  Y ) } )
4947, 48syl 16 . . 3  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( iota_ m  e.  ( W RingHom  T
) ( ( m  o.  A )  =  X  /\  ( m  o.  V )  =  Y ) )  e. 
{ m  e.  ( W RingHom  T )  |  ( ( m  o.  A
)  =  X  /\  ( m  o.  V
)  =  Y ) } )
5010, 49eqeltrd 2531 . 2  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  Q  e.  { m  e.  ( W RingHom  T )  |  ( ( m  o.  A
)  =  X  /\  ( m  o.  V
)  =  Y ) } )
51 coeq1 5150 . . . . 5  |-  ( m  =  Q  ->  (
m  o.  A )  =  ( Q  o.  A ) )
5251eqeq1d 2445 . . . 4  |-  ( m  =  Q  ->  (
( m  o.  A
)  =  X  <->  ( Q  o.  A )  =  X ) )
53 coeq1 5150 . . . . 5  |-  ( m  =  Q  ->  (
m  o.  V )  =  ( Q  o.  V ) )
5453eqeq1d 2445 . . . 4  |-  ( m  =  Q  ->  (
( m  o.  V
)  =  Y  <->  ( Q  o.  V )  =  Y ) )
5552, 54anbi12d 710 . . 3  |-  ( m  =  Q  ->  (
( ( m  o.  A )  =  X  /\  ( m  o.  V )  =  Y )  <->  ( ( Q  o.  A )  =  X  /\  ( Q  o.  V )  =  Y ) ) )
5655elrab 3243 . 2  |-  ( Q  e.  { m  e.  ( W RingHom  T )  |  ( ( m  o.  A )  =  X  /\  ( m  o.  V )  =  Y ) }  <->  ( Q  e.  ( W RingHom  T )  /\  ( ( Q  o.  A )  =  X  /\  ( Q  o.  V )  =  Y ) ) )
5750, 56sylib 196 1  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( Q  e.  ( W RingHom  T )  /\  ( ( Q  o.  A )  =  X  /\  ( Q  o.  V )  =  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   E!wreu 2795   {crab 2797   _Vcvv 3095    C_ wss 3461   {csn 4014    |-> cmpt 4495    X. cxp 4987    |` cres 4991    o. ccom 4993   -->wf 5574   ` cfv 5578   iota_crio 6241  (class class class)co 6281    ^m cmap 7422   Basecbs 14613   ↾s cress 14614    ^s cpws 14825   Ringcrg 17176   CRingccrg 17177   RingHom crh 17339  SubRingcsubrg 17403  algSccascl 17938   mVar cmvr 17979   mPoly cmpl 17980   evalSub ces 18147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-ofr 6526  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10986  df-uz 11092  df-fz 11683  df-fzo 11806  df-seq 12089  df-hash 12387  df-struct 14615  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-plusg 14691  df-mulr 14692  df-sca 14694  df-vsca 14695  df-ip 14696  df-tset 14697  df-ple 14698  df-ds 14700  df-hom 14702  df-cco 14703  df-0g 14820  df-gsum 14821  df-prds 14826  df-pws 14828  df-mre 14964  df-mrc 14965  df-acs 14967  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-mhm 15944  df-submnd 15945  df-grp 16035  df-minusg 16036  df-sbg 16037  df-mulg 16038  df-subg 16176  df-ghm 16243  df-cntz 16333  df-cmn 16778  df-abl 16779  df-mgp 17120  df-ur 17132  df-srg 17136  df-ring 17178  df-cring 17179  df-rnghom 17342  df-subrg 17405  df-lmod 17492  df-lss 17557  df-lsp 17596  df-assa 17939  df-asp 17940  df-ascl 17941  df-psr 17983  df-mvr 17984  df-mpl 17985  df-evls 18149
This theorem is referenced by:  evlsrhm  18168  evlssca  18169  evlsvar  18170
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