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Theorem evlsval2 18731
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 18730 . . 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 2422 . . . . 5  |-  ( Base `  T )  =  (
Base `  T )
12 simp1 1005 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  I  e.  _V )
134subrgcrng 18000 . . . . . 6  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  U  e.  CRing
)
14133adant1 1023 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  U  e.  CRing
)
15 simp2 1006 . . . . . 6  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  S  e.  CRing
)
16 ovex 6330 . . . . . 6  |-  ( B  ^m  I )  e. 
_V
175pwscrng 17833 . . . . . 6  |-  ( ( S  e.  CRing  /\  ( B  ^m  I )  e. 
_V )  ->  T  e.  CRing )
1815, 16, 17sylancl 666 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  T  e.  CRing
)
196subrgss 17997 . . . . . . . . 9  |-  ( R  e.  (SubRing `  S
)  ->  R  C_  B
)
20193ad2ant3 1028 . . . . . . . 8  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  R  C_  B
)
2120resmptd 5172 . . . . . . 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 2481 . . . . . 6  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( (
x  e.  B  |->  ( ( B  ^m  I
)  X.  { x } ) )  |`  R )  =  X )
23 crngring 17779 . . . . . . . . 9  |-  ( S  e.  CRing  ->  S  e.  Ring )
24233ad2ant2 1027 . . . . . . . 8  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  S  e.  Ring )
25 eqid 2422 . . . . . . . . 9  |-  ( x  e.  B  |->  ( ( B  ^m  I )  X.  { x }
) )  =  ( x  e.  B  |->  ( ( B  ^m  I
)  X.  { x } ) )
265, 6, 25pwsdiagrhm 18029 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  ( B  ^m  I )  e. 
_V )  ->  (
x  e.  B  |->  ( ( B  ^m  I
)  X.  { x } ) )  e.  ( S RingHom  T )
)
2724, 16, 26sylancl 666 . . . . . . 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 1007 . . . . . . 7  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  R  e.  (SubRing `  S ) )
294resrhm 18025 . . . . . . 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 665 . . . . . 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 2511 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  X  e.  ( U RingHom  T ) )
32 fvex 5888 . . . . . . . . . . . 12  |-  ( Base `  S )  e.  _V
336, 32eqeltri 2506 . . . . . . . . . . 11  |-  B  e. 
_V
34 simpl1 1008 . . . . . . . . . . 11  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  I  e.  _V )
35 elmapg 7490 . . . . . . . . . . 11  |-  ( ( B  e.  _V  /\  I  e.  _V )  ->  ( g  e.  ( B  ^m  I )  <-> 
g : I --> B ) )
3633, 34, 35sylancr 667 . . . . . . . . . 10  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  (
g  e.  ( B  ^m  I )  <->  g :
I --> B ) )
3736biimpa 486 . . . . . . . . 9  |-  ( ( ( ( I  e. 
_V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) )  /\  x  e.  I )  /\  g  e.  ( B  ^m  I ) )  ->  g : I --> B )
38 simplr 760 . . . . . . . . 9  |-  ( ( ( ( I  e. 
_V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) )  /\  x  e.  I )  /\  g  e.  ( B  ^m  I ) )  ->  x  e.  I
)
3937, 38ffvelrnd 6035 . . . . . . . 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 2422 . . . . . . . 8  |-  ( g  e.  ( B  ^m  I )  |->  ( g `
 x ) )  =  ( g  e.  ( B  ^m  I
)  |->  ( g `  x ) )
4139, 40fmptd 6058 . . . . . . 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 1009 . . . . . . . 8  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  S  e.  CRing )
435, 6, 11pwselbasb 15374 . . . . . . . 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 666 . . . . . . 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 235 . . . . . 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 6058 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  Y :
I --> ( Base `  T
) )
472, 11, 7, 3, 12, 14, 18, 31, 46evlseu 18727 . . . 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 6277 . . . 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 17 . . 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 2510 . 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 5008 . . . . 5  |-  ( m  =  Q  ->  (
m  o.  A )  =  ( Q  o.  A ) )
5251eqeq1d 2424 . . . 4  |-  ( m  =  Q  ->  (
( m  o.  A
)  =  X  <->  ( Q  o.  A )  =  X ) )
53 coeq1 5008 . . . . 5  |-  ( m  =  Q  ->  (
m  o.  V )  =  ( Q  o.  V ) )
5453eqeq1d 2424 . . . 4  |-  ( m  =  Q  ->  (
( m  o.  V
)  =  Y  <->  ( Q  o.  V )  =  Y ) )
5552, 54anbi12d 715 . . 3  |-  ( m  =  Q  ->  (
( ( m  o.  A )  =  X  /\  ( m  o.  V )  =  Y )  <->  ( ( Q  o.  A )  =  X  /\  ( Q  o.  V )  =  Y ) ) )
5655elrab 3229 . 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 199 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   E!wreu 2777   {crab 2779   _Vcvv 3081    C_ wss 3436   {csn 3996    |-> cmpt 4479    X. cxp 4848    |` cres 4852    o. ccom 4854   -->wf 5594   ` cfv 5598   iota_crio 6263  (class class class)co 6302    ^m cmap 7477   Basecbs 15109   ↾s cress 15110    ^s cpws 15333   Ringcrg 17768   CRingccrg 17769   RingHom crh 17928  SubRingcsubrg 17992  algSccascl 18523   mVar cmvr 18564   mPoly cmpl 18565   evalSub ces 18715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-ofr 6543  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-sup 7959  df-oi 8028  df-card 8375  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-fz 11786  df-fzo 11917  df-seq 12214  df-hash 12516  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-sca 15194  df-vsca 15195  df-ip 15196  df-tset 15197  df-ple 15198  df-ds 15200  df-hom 15202  df-cco 15203  df-0g 15328  df-gsum 15329  df-prds 15334  df-pws 15336  df-mre 15480  df-mrc 15481  df-acs 15483  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-mhm 16570  df-submnd 16571  df-grp 16661  df-minusg 16662  df-sbg 16663  df-mulg 16664  df-subg 16802  df-ghm 16869  df-cntz 16959  df-cmn 17420  df-abl 17421  df-mgp 17712  df-ur 17724  df-srg 17728  df-ring 17770  df-cring 17771  df-rnghom 17931  df-subrg 17994  df-lmod 18081  df-lss 18144  df-lsp 18183  df-assa 18524  df-asp 18525  df-ascl 18526  df-psr 18568  df-mvr 18569  df-mpl 18570  df-evls 18717
This theorem is referenced by:  evlsrhm  18732  evlssca  18733  evlsvar  18734
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