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Theorem evlsval2 17606
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 17605 . . 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 988 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  I  e.  _V )
134subrgcrng 16869 . . . . . 6  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  U  e.  CRing
)
14133adant1 1006 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  U  e.  CRing
)
15 simp2 989 . . . . . 6  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  S  e.  CRing
)
16 ovex 6116 . . . . . 6  |-  ( B  ^m  I )  e. 
_V
175pwscrng 16709 . . . . . 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 16866 . . . . . . . . 9  |-  ( R  e.  (SubRing `  S
)  ->  R  C_  B
)
20193ad2ant3 1011 . . . . . . . 8  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  R  C_  B
)
21 resmpt 5156 . . . . . . . 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 2493 . . . . . 6  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( (
x  e.  B  |->  ( ( B  ^m  I
)  X.  { x } ) )  |`  R )  =  X )
24 crngrng 16655 . . . . . . . . 9  |-  ( S  e.  CRing  ->  S  e.  Ring )
25243ad2ant2 1010 . . . . . . . 8  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  S  e.  Ring )
26 eqid 2443 . . . . . . . . 9  |-  ( x  e.  B  |->  ( ( B  ^m  I )  X.  { x }
) )  =  ( x  e.  B  |->  ( ( B  ^m  I
)  X.  { x } ) )
275, 6, 26pwsdiagrhm 16898 . . . . . . . 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 990 . . . . . . 7  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  R  e.  (SubRing `  S ) )
304resrhm 16894 . . . . . . 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 2518 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  X  e.  ( U RingHom  T ) )
33 fvex 5701 . . . . . . . . . . . 12  |-  ( Base `  S )  e.  _V
346, 33eqeltri 2513 . . . . . . . . . . 11  |-  B  e. 
_V
35 simpl1 991 . . . . . . . . . . 11  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  I  e.  _V )
36 elmapg 7227 . . . . . . . . . . 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 5844 . . . . . . . 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 2443 . . . . . . . 8  |-  ( g  e.  ( B  ^m  I )  |->  ( g `
 x ) )  =  ( g  e.  ( B  ^m  I
)  |->  ( g `  x ) )
4240, 41fmptd 5867 . . . . . . 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 992 . . . . . . . 8  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  S  e.  CRing )
445, 6, 11pwselbasb 14426 . . . . . . . 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 5867 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  Y :
I --> ( Base `  T
) )
482, 11, 7, 3, 12, 14, 18, 32, 47evlseu 17602 . . . 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 6066 . . . 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 2517 . 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 4997 . . . . 5  |-  ( m  =  Q  ->  (
m  o.  A )  =  ( Q  o.  A ) )
5352eqeq1d 2451 . . . 4  |-  ( m  =  Q  ->  (
( m  o.  A
)  =  X  <->  ( Q  o.  A )  =  X ) )
54 coeq1 4997 . . . . 5  |-  ( m  =  Q  ->  (
m  o.  V )  =  ( Q  o.  V ) )
5554eqeq1d 2451 . . . 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 3117 . 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 965    = wceq 1369    e. wcel 1756   E!wreu 2717   {crab 2719   _Vcvv 2972    C_ wss 3328   {csn 3877    e. cmpt 4350    X. cxp 4838    |` cres 4842    o. ccom 4844   -->wf 5414   ` cfv 5418   iota_crio 6051  (class class class)co 6091    ^m cmap 7214   Basecbs 14174   ↾s cress 14175    ^s cpws 14385   Ringcrg 16645   CRingccrg 16646   RingHom crh 16804  SubRingcsubrg 16861  algSccascl 17383   mVar cmvr 17419   mPoly cmpl 17420   evalSub ces 17586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-ofr 6321  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-hom 14262  df-cco 14263  df-0g 14380  df-gsum 14381  df-prds 14386  df-pws 14388  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-mhm 15464  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-mulg 15548  df-subg 15678  df-ghm 15745  df-cntz 15835  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-srg 16608  df-rng 16647  df-cring 16648  df-rnghom 16806  df-subrg 16863  df-lmod 16950  df-lss 17014  df-lsp 17053  df-assa 17384  df-asp 17385  df-ascl 17386  df-psr 17423  df-mvr 17424  df-mpl 17425  df-evls 17588
This theorem is referenced by:  evlsrhm  17607  evlssca  17608  evlsvar  17609
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