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Theorem evlsval2 18036
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 18035 . . 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 2467 . . . . 5  |-  ( Base `  T )  =  (
Base `  T )
12 simp1 996 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  I  e.  _V )
134subrgcrng 17281 . . . . . 6  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  U  e.  CRing
)
14133adant1 1014 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  U  e.  CRing
)
15 simp2 997 . . . . . 6  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  S  e.  CRing
)
16 ovex 6319 . . . . . 6  |-  ( B  ^m  I )  e. 
_V
175pwscrng 17115 . . . . . 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 17278 . . . . . . . . 9  |-  ( R  e.  (SubRing `  S
)  ->  R  C_  B
)
20193ad2ant3 1019 . . . . . . . 8  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  R  C_  B
)
21 resmpt 5328 . . . . . . . 8  |-  ( R 
C_  B  ->  (
( x  e.  B  |->  ( ( B  ^m  I )  X.  {
x } ) )  |`  R )  =  ( x  e.  R  |->  ( ( B  ^m  I
)  X.  { x } ) ) )
2220, 21syl 16 . . . . . . 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 } ) ) )
2322, 8syl6eqr 2526 . . . . . 6  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( (
x  e.  B  |->  ( ( B  ^m  I
)  X.  { x } ) )  |`  R )  =  X )
24 crngring 17058 . . . . . . . . 9  |-  ( S  e.  CRing  ->  S  e.  Ring )
25243ad2ant2 1018 . . . . . . . 8  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  S  e.  Ring )
26 eqid 2467 . . . . . . . . 9  |-  ( x  e.  B  |->  ( ( B  ^m  I )  X.  { x }
) )  =  ( x  e.  B  |->  ( ( B  ^m  I
)  X.  { x } ) )
275, 6, 26pwsdiagrhm 17310 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  ( B  ^m  I )  e. 
_V )  ->  (
x  e.  B  |->  ( ( B  ^m  I
)  X.  { x } ) )  e.  ( S RingHom  T )
)
2825, 16, 27sylancl 662 . . . . . . 7  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( x  e.  B  |->  ( ( B  ^m  I )  X.  { x }
) )  e.  ( S RingHom  T ) )
29 simp3 998 . . . . . . 7  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  R  e.  (SubRing `  S ) )
304resrhm 17306 . . . . . . 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
) )
3128, 29, 30syl2anc 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 ) )
3223, 31eqeltrrd 2556 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  X  e.  ( U RingHom  T ) )
33 fvex 5881 . . . . . . . . . . . 12  |-  ( Base `  S )  e.  _V
346, 33eqeltri 2551 . . . . . . . . . . 11  |-  B  e. 
_V
35 simpl1 999 . . . . . . . . . . 11  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  I  e.  _V )
36 elmapg 7443 . . . . . . . . . . 11  |-  ( ( B  e.  _V  /\  I  e.  _V )  ->  ( g  e.  ( B  ^m  I )  <-> 
g : I --> B ) )
3734, 35, 36sylancr 663 . . . . . . . . . 10  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  (
g  e.  ( B  ^m  I )  <->  g :
I --> B ) )
3837biimpa 484 . . . . . . . . 9  |-  ( ( ( ( I  e. 
_V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) )  /\  x  e.  I )  /\  g  e.  ( B  ^m  I ) )  ->  g : I --> B )
39 simplr 754 . . . . . . . . 9  |-  ( ( ( ( I  e. 
_V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) )  /\  x  e.  I )  /\  g  e.  ( B  ^m  I ) )  ->  x  e.  I
)
4038, 39ffvelrnd 6032 . . . . . . . 8  |-  ( ( ( ( I  e. 
_V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) )  /\  x  e.  I )  /\  g  e.  ( B  ^m  I ) )  ->  ( g `  x )  e.  B
)
41 eqid 2467 . . . . . . . 8  |-  ( g  e.  ( B  ^m  I )  |->  ( g `
 x ) )  =  ( g  e.  ( B  ^m  I
)  |->  ( g `  x ) )
4240, 41fmptd 6055 . . . . . . 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 )
43 simpl2 1000 . . . . . . . 8  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  S  e.  CRing )
445, 6, 11pwselbasb 14755 . . . . . . . 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 ) )
4543, 16, 44sylancl 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 ) )
4642, 45mpbird 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 ) )
4746, 9fmptd 6055 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  Y :
I --> ( Base `  T
) )
482, 11, 7, 3, 12, 14, 18, 32, 47evlseu 18032 . . . 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 ) )
49 riotacl2 6269 . . . 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 ) } )
5048, 49syl 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 ) } )
5110, 50eqeltrd 2555 . 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 ) } )
52 coeq1 5165 . . . . 5  |-  ( m  =  Q  ->  (
m  o.  A )  =  ( Q  o.  A ) )
5352eqeq1d 2469 . . . 4  |-  ( m  =  Q  ->  (
( m  o.  A
)  =  X  <->  ( Q  o.  A )  =  X ) )
54 coeq1 5165 . . . . 5  |-  ( m  =  Q  ->  (
m  o.  V )  =  ( Q  o.  V ) )
5554eqeq1d 2469 . . . 4  |-  ( m  =  Q  ->  (
( m  o.  V
)  =  Y  <->  ( Q  o.  V )  =  Y ) )
5653, 55anbi12d 710 . . 3  |-  ( m  =  Q  ->  (
( ( m  o.  A )  =  X  /\  ( m  o.  V )  =  Y )  <->  ( ( Q  o.  A )  =  X  /\  ( Q  o.  V )  =  Y ) ) )
5756elrab 3266 . 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 ) ) )
5851, 57sylib 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 973    = wceq 1379    e. wcel 1767   E!wreu 2819   {crab 2821   _Vcvv 3118    C_ wss 3481   {csn 4032    |-> cmpt 4510    X. cxp 5002    |` cres 5006    o. ccom 5008   -->wf 5589   ` cfv 5593   iota_crio 6254  (class class class)co 6294    ^m cmap 7430   Basecbs 14502   ↾s cress 14503    ^s cpws 14714   Ringcrg 17047   CRingccrg 17048   RingHom crh 17210  SubRingcsubrg 17273  algSccascl 17807   mVar cmvr 17848   mPoly cmpl 17849   evalSub ces 18016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-of 6534  df-ofr 6535  df-om 6695  df-1st 6794  df-2nd 6795  df-supp 6912  df-recs 7052  df-rdg 7086  df-1o 7140  df-2o 7141  df-oadd 7144  df-er 7321  df-map 7432  df-pm 7433  df-ixp 7480  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-fsupp 7840  df-sup 7911  df-oi 7945  df-card 8330  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-3 10605  df-4 10606  df-5 10607  df-6 10608  df-7 10609  df-8 10610  df-9 10611  df-10 10612  df-n0 10806  df-z 10875  df-dec 10987  df-uz 11093  df-fz 11683  df-fzo 11803  df-seq 12086  df-hash 12384  df-struct 14504  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-mulr 14581  df-sca 14583  df-vsca 14584  df-ip 14585  df-tset 14586  df-ple 14587  df-ds 14589  df-hom 14591  df-cco 14592  df-0g 14709  df-gsum 14710  df-prds 14715  df-pws 14717  df-mre 14853  df-mrc 14854  df-acs 14856  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-mhm 15819  df-submnd 15820  df-grp 15906  df-minusg 15907  df-sbg 15908  df-mulg 15909  df-subg 16047  df-ghm 16114  df-cntz 16204  df-cmn 16650  df-abl 16651  df-mgp 16991  df-ur 17003  df-srg 17007  df-ring 17049  df-cring 17050  df-rnghom 17213  df-subrg 17275  df-lmod 17362  df-lss 17427  df-lsp 17466  df-assa 17808  df-asp 17809  df-ascl 17810  df-psr 17852  df-mvr 17853  df-mpl 17854  df-evls 18018
This theorem is referenced by:  evlsrhm  18037  evlssca  18038  evlsvar  18039
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