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Theorem evlsval2 19420
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 19419 . . 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 2296 . . . . 5  |-  ( Base `  T )  =  (
Base `  T )
12 simp1 955 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  I  e.  _V )
134subrgcrng 15565 . . . . . 6  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  U  e.  CRing
)
14133adant1 973 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  U  e.  CRing
)
15 simp2 956 . . . . . 6  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  S  e.  CRing
)
16 ovex 5899 . . . . . 6  |-  ( B  ^m  I )  e. 
_V
175pwscrng 15416 . . . . . 6  |-  ( ( S  e.  CRing  /\  ( B  ^m  I )  e. 
_V )  ->  T  e.  CRing )
1815, 16, 17sylancl 643 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  T  e.  CRing
)
196subrgss 15562 . . . . . . . . 9  |-  ( R  e.  (SubRing `  S
)  ->  R  C_  B
)
20193ad2ant3 978 . . . . . . . 8  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  R  C_  B
)
21 resmpt 5016 . . . . . . . 8  |-  ( R 
C_  B  ->  (
( x  e.  B  |->  ( ( B  ^m  I )  X.  {
x } ) )  |`  R )  =  ( x  e.  R  |->  ( ( B  ^m  I
)  X.  { x } ) ) )
2220, 21syl 15 . . . . . . 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 2346 . . . . . 6  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( (
x  e.  B  |->  ( ( B  ^m  I
)  X.  { x } ) )  |`  R )  =  X )
24 crngrng 15367 . . . . . . . . 9  |-  ( S  e.  CRing  ->  S  e.  Ring )
25243ad2ant2 977 . . . . . . . 8  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  S  e.  Ring )
26 eqid 2296 . . . . . . . . 9  |-  ( x  e.  B  |->  ( ( B  ^m  I )  X.  { x }
) )  =  ( x  e.  B  |->  ( ( B  ^m  I
)  X.  { x } ) )
275, 6, 26pwsdiagrhm 15594 . . . . . . . 8  |-  ( ( S  e.  Ring  /\  ( B  ^m  I )  e. 
_V )  ->  (
x  e.  B  |->  ( ( B  ^m  I
)  X.  { x } ) )  e.  ( S RingHom  T )
)
2825, 16, 27sylancl 643 . . . . . . 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 957 . . . . . . 7  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  R  e.  (SubRing `  S ) )
304resrhm 15590 . . . . . . 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 642 . . . . . 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 2371 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  X  e.  ( U RingHom  T ) )
33 fvex 5555 . . . . . . . . . . . 12  |-  ( Base `  S )  e.  _V
346, 33eqeltri 2366 . . . . . . . . . . 11  |-  B  e. 
_V
35 simpl1 958 . . . . . . . . . . 11  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  I  e.  _V )
36 elmapg 6801 . . . . . . . . . . 11  |-  ( ( B  e.  _V  /\  I  e.  _V )  ->  ( g  e.  ( B  ^m  I )  <-> 
g : I --> B ) )
3734, 35, 36sylancr 644 . . . . . . . . . 10  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  (
g  e.  ( B  ^m  I )  <->  g :
I --> B ) )
3837biimpa 470 . . . . . . . . 9  |-  ( ( ( ( I  e. 
_V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) )  /\  x  e.  I )  /\  g  e.  ( B  ^m  I ) )  ->  g : I --> B )
39 simplr 731 . . . . . . . . 9  |-  ( ( ( ( I  e. 
_V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) )  /\  x  e.  I )  /\  g  e.  ( B  ^m  I ) )  ->  x  e.  I
)
40 ffvelrn 5679 . . . . . . . . 9  |-  ( ( g : I --> B  /\  x  e.  I )  ->  ( g `  x
)  e.  B )
4138, 39, 40syl2anc 642 . . . . . . . 8  |-  ( ( ( ( I  e. 
_V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S ) )  /\  x  e.  I )  /\  g  e.  ( B  ^m  I ) )  ->  ( g `  x )  e.  B
)
42 eqid 2296 . . . . . . . 8  |-  ( g  e.  ( B  ^m  I )  |->  ( g `
 x ) )  =  ( g  e.  ( B  ^m  I
)  |->  ( g `  x ) )
4341, 42fmptd 5700 . . . . . . 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 )
44 simpl2 959 . . . . . . . 8  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  S  e.  CRing )
455, 6, 11pwselbasb 13403 . . . . . . . 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 ) )
4644, 16, 45sylancl 643 . . . . . . 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 ) )
4743, 46mpbird 223 . . . . . 6  |-  ( ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  /\  x  e.  I )  ->  (
g  e.  ( B  ^m  I )  |->  ( g `  x ) )  e.  ( Base `  T ) )
4847, 9fmptd 5700 . . . . 5  |-  ( ( I  e.  _V  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  Y :
I --> ( Base `  T
) )
492, 11, 7, 3, 12, 14, 18, 32, 48evlseu 19416 . . . 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 ) )
50 riotacl2 6334 . . . 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 ) } )
5149, 50syl 15 . . 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 ) } )
5210, 51eqeltrd 2370 . 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 ) } )
53 coeq1 4857 . . . . 5  |-  ( m  =  Q  ->  (
m  o.  A )  =  ( Q  o.  A ) )
5453eqeq1d 2304 . . . 4  |-  ( m  =  Q  ->  (
( m  o.  A
)  =  X  <->  ( Q  o.  A )  =  X ) )
55 coeq1 4857 . . . . 5  |-  ( m  =  Q  ->  (
m  o.  V )  =  ( Q  o.  V ) )
5655eqeq1d 2304 . . . 4  |-  ( m  =  Q  ->  (
( m  o.  V
)  =  Y  <->  ( Q  o.  V )  =  Y ) )
5754, 56anbi12d 691 . . 3  |-  ( m  =  Q  ->  (
( ( m  o.  A )  =  X  /\  ( m  o.  V )  =  Y )  <->  ( ( Q  o.  A )  =  X  /\  ( Q  o.  V )  =  Y ) ) )
5857elrab 2936 . 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 ) ) )
5952, 58sylib 188 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E!wreu 2558   {crab 2560   _Vcvv 2801    C_ wss 3165   {csn 3653    e. cmpt 4093    X. cxp 4703    |` cres 4707    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   iota_crio 6313    ^m cmap 6788   Basecbs 13164   ↾s cress 13165    ^s cpws 13363   Ringcrg 15353   CRingccrg 15354   RingHom crh 15510  SubRingcsubrg 15557  algSccascl 16068   mVar cmvr 16104   mPoly cmpl 16105   evalSub ces 16106
This theorem is referenced by:  evlsrhm  19421  evlssca  19422  evlsvar  19423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-pws 13366  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-ghm 14697  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-rnghom 15512  df-subrg 15559  df-lmod 15645  df-lss 15706  df-lsp 15745  df-assa 16069  df-asp 16070  df-ascl 16071  df-psr 16114  df-mvr 16115  df-mpl 16116  df-evls 16117
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