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Theorem evlseu 17953
Description: For a given interpretation of the variables  G and of the scalars  F, this extends to a homomorphic interpretation of the polynomial ring in exactly one way. (Contributed by Stefan O'Rear, 9-Mar-2015.)
Hypotheses
Ref Expression
evlseu.p  |-  P  =  ( I mPoly  R )
evlseu.c  |-  C  =  ( Base `  S
)
evlseu.a  |-  A  =  (algSc `  P )
evlseu.v  |-  V  =  ( I mVar  R )
evlseu.i  |-  ( ph  ->  I  e.  _V )
evlseu.r  |-  ( ph  ->  R  e.  CRing )
evlseu.s  |-  ( ph  ->  S  e.  CRing )
evlseu.f  |-  ( ph  ->  F  e.  ( R RingHom  S ) )
evlseu.g  |-  ( ph  ->  G : I --> C )
Assertion
Ref Expression
evlseu  |-  ( ph  ->  E! m  e.  ( P RingHom  S ) ( ( m  o.  A )  =  F  /\  (
m  o.  V )  =  G ) )
Distinct variable groups:    A, m    m, F    m, G    m, I    P, m    ph, m    S, m    m, V
Allowed substitution hints:    C( m)    R( m)

Proof of Theorem evlseu
Dummy variables  n  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlseu.p . . . 4  |-  P  =  ( I mPoly  R )
2 eqid 2467 . . . 4  |-  ( Base `  P )  =  (
Base `  P )
3 evlseu.c . . . 4  |-  C  =  ( Base `  S
)
4 eqid 2467 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
5 eqid 2467 . . . 4  |-  { z  e.  ( NN0  ^m  I )  |  ( `' z " NN )  e.  Fin }  =  { z  e.  ( NN0  ^m  I )  |  ( `' z
" NN )  e. 
Fin }
6 eqid 2467 . . . 4  |-  (mulGrp `  S )  =  (mulGrp `  S )
7 eqid 2467 . . . 4  |-  (.g `  (mulGrp `  S ) )  =  (.g `  (mulGrp `  S
) )
8 eqid 2467 . . . 4  |-  ( .r
`  S )  =  ( .r `  S
)
9 evlseu.v . . . 4  |-  V  =  ( I mVar  R )
10 eqid 2467 . . . 4  |-  ( x  e.  ( Base `  P
)  |->  ( S  gsumg  ( y  e.  { z  e.  ( NN0  ^m  I
)  |  ( `' z " NN )  e.  Fin }  |->  ( ( F `  (
x `  y )
) ( .r `  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S
) ) G ) ) ) ) ) )  =  ( x  e.  ( Base `  P
)  |->  ( S  gsumg  ( y  e.  { z  e.  ( NN0  ^m  I
)  |  ( `' z " NN )  e.  Fin }  |->  ( ( F `  (
x `  y )
) ( .r `  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S
) ) G ) ) ) ) ) )
11 evlseu.i . . . 4  |-  ( ph  ->  I  e.  _V )
12 evlseu.r . . . 4  |-  ( ph  ->  R  e.  CRing )
13 evlseu.s . . . 4  |-  ( ph  ->  S  e.  CRing )
14 evlseu.f . . . 4  |-  ( ph  ->  F  e.  ( R RingHom  S ) )
15 evlseu.g . . . 4  |-  ( ph  ->  G : I --> C )
16 evlseu.a . . . 4  |-  A  =  (algSc `  P )
171, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16evlslem1 17952 . . 3  |-  ( ph  ->  ( ( x  e.  ( Base `  P
)  |->  ( S  gsumg  ( y  e.  { z  e.  ( NN0  ^m  I
)  |  ( `' z " NN )  e.  Fin }  |->  ( ( F `  (
x `  y )
) ( .r `  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S
) ) G ) ) ) ) ) )  e.  ( P RingHom  S )  /\  (
( x  e.  (
Base `  P )  |->  ( S  gsumg  ( y  e.  {
z  e.  ( NN0 
^m  I )  |  ( `' z " NN )  e.  Fin } 
|->  ( ( F `  ( x `  y
) ) ( .r
`  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  A )  =  F  /\  ( ( x  e.  ( Base `  P )  |->  ( S 
gsumg  ( y  e.  {
z  e.  ( NN0 
^m  I )  |  ( `' z " NN )  e.  Fin } 
|->  ( ( F `  ( x `  y
) ) ( .r
`  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  V )  =  G ) )
18 coeq1 5158 . . . . . . 7  |-  ( m  =  ( x  e.  ( Base `  P
)  |->  ( S  gsumg  ( y  e.  { z  e.  ( NN0  ^m  I
)  |  ( `' z " NN )  e.  Fin }  |->  ( ( F `  (
x `  y )
) ( .r `  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S
) ) G ) ) ) ) ) )  ->  ( m  o.  A )  =  ( ( x  e.  (
Base `  P )  |->  ( S  gsumg  ( y  e.  {
z  e.  ( NN0 
^m  I )  |  ( `' z " NN )  e.  Fin } 
|->  ( ( F `  ( x `  y
) ) ( .r
`  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  A ) )
1918eqeq1d 2469 . . . . . 6  |-  ( m  =  ( x  e.  ( Base `  P
)  |->  ( S  gsumg  ( y  e.  { z  e.  ( NN0  ^m  I
)  |  ( `' z " NN )  e.  Fin }  |->  ( ( F `  (
x `  y )
) ( .r `  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S
) ) G ) ) ) ) ) )  ->  ( (
m  o.  A )  =  F  <->  ( (
x  e.  ( Base `  P )  |->  ( S 
gsumg  ( y  e.  {
z  e.  ( NN0 
^m  I )  |  ( `' z " NN )  e.  Fin } 
|->  ( ( F `  ( x `  y
) ) ( .r
`  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  A )  =  F ) )
20 coeq1 5158 . . . . . . 7  |-  ( m  =  ( x  e.  ( Base `  P
)  |->  ( S  gsumg  ( y  e.  { z  e.  ( NN0  ^m  I
)  |  ( `' z " NN )  e.  Fin }  |->  ( ( F `  (
x `  y )
) ( .r `  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S
) ) G ) ) ) ) ) )  ->  ( m  o.  V )  =  ( ( x  e.  (
Base `  P )  |->  ( S  gsumg  ( y  e.  {
z  e.  ( NN0 
^m  I )  |  ( `' z " NN )  e.  Fin } 
|->  ( ( F `  ( x `  y
) ) ( .r
`  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  V ) )
2120eqeq1d 2469 . . . . . 6  |-  ( m  =  ( x  e.  ( Base `  P
)  |->  ( S  gsumg  ( y  e.  { z  e.  ( NN0  ^m  I
)  |  ( `' z " NN )  e.  Fin }  |->  ( ( F `  (
x `  y )
) ( .r `  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S
) ) G ) ) ) ) ) )  ->  ( (
m  o.  V )  =  G  <->  ( (
x  e.  ( Base `  P )  |->  ( S 
gsumg  ( y  e.  {
z  e.  ( NN0 
^m  I )  |  ( `' z " NN )  e.  Fin } 
|->  ( ( F `  ( x `  y
) ) ( .r
`  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  V )  =  G ) )
2219, 21anbi12d 710 . . . . 5  |-  ( m  =  ( x  e.  ( Base `  P
)  |->  ( S  gsumg  ( y  e.  { z  e.  ( NN0  ^m  I
)  |  ( `' z " NN )  e.  Fin }  |->  ( ( F `  (
x `  y )
) ( .r `  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S
) ) G ) ) ) ) ) )  ->  ( (
( m  o.  A
)  =  F  /\  ( m  o.  V
)  =  G )  <-> 
( ( ( x  e.  ( Base `  P
)  |->  ( S  gsumg  ( y  e.  { z  e.  ( NN0  ^m  I
)  |  ( `' z " NN )  e.  Fin }  |->  ( ( F `  (
x `  y )
) ( .r `  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S
) ) G ) ) ) ) ) )  o.  A )  =  F  /\  (
( x  e.  (
Base `  P )  |->  ( S  gsumg  ( y  e.  {
z  e.  ( NN0 
^m  I )  |  ( `' z " NN )  e.  Fin } 
|->  ( ( F `  ( x `  y
) ) ( .r
`  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  V )  =  G ) ) )
2322rspcev 3214 . . . 4  |-  ( ( ( x  e.  (
Base `  P )  |->  ( S  gsumg  ( y  e.  {
z  e.  ( NN0 
^m  I )  |  ( `' z " NN )  e.  Fin } 
|->  ( ( F `  ( x `  y
) ) ( .r
`  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  e.  ( P RingHom  S
)  /\  ( (
( x  e.  (
Base `  P )  |->  ( S  gsumg  ( y  e.  {
z  e.  ( NN0 
^m  I )  |  ( `' z " NN )  e.  Fin } 
|->  ( ( F `  ( x `  y
) ) ( .r
`  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  A )  =  F  /\  ( ( x  e.  ( Base `  P )  |->  ( S 
gsumg  ( y  e.  {
z  e.  ( NN0 
^m  I )  |  ( `' z " NN )  e.  Fin } 
|->  ( ( F `  ( x `  y
) ) ( .r
`  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  V )  =  G ) )  ->  E. m  e.  ( P RingHom  S ) ( ( m  o.  A )  =  F  /\  (
m  o.  V )  =  G ) )
24233impb 1192 . . 3  |-  ( ( ( x  e.  (
Base `  P )  |->  ( S  gsumg  ( y  e.  {
z  e.  ( NN0 
^m  I )  |  ( `' z " NN )  e.  Fin } 
|->  ( ( F `  ( x `  y
) ) ( .r
`  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  e.  ( P RingHom  S
)  /\  ( (
x  e.  ( Base `  P )  |->  ( S 
gsumg  ( y  e.  {
z  e.  ( NN0 
^m  I )  |  ( `' z " NN )  e.  Fin } 
|->  ( ( F `  ( x `  y
) ) ( .r
`  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  A )  =  F  /\  ( ( x  e.  ( Base `  P )  |->  ( S 
gsumg  ( y  e.  {
z  e.  ( NN0 
^m  I )  |  ( `' z " NN )  e.  Fin } 
|->  ( ( F `  ( x `  y
) ) ( .r
`  S ) ( (mulGrp `  S )  gsumg  ( y  oF (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  V )  =  G )  ->  E. m  e.  ( P RingHom  S )
( ( m  o.  A )  =  F  /\  ( m  o.  V )  =  G ) )
2517, 24syl 16 . 2  |-  ( ph  ->  E. m  e.  ( P RingHom  S ) ( ( m  o.  A )  =  F  /\  (
m  o.  V )  =  G ) )
26 crngrng 16993 . . . . . . . . . . 11  |-  ( R  e.  CRing  ->  R  e.  Ring )
2712, 26syl 16 . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
28 eqid 2467 . . . . . . . . . . 11  |-  (Scalar `  P )  =  (Scalar `  P )
291mplrng 17882 . . . . . . . . . . 11  |-  ( ( I  e.  _V  /\  R  e.  Ring )  ->  P  e.  Ring )
301mpllmod 17881 . . . . . . . . . . 11  |-  ( ( I  e.  _V  /\  R  e.  Ring )  ->  P  e.  LMod )
31 eqid 2467 . . . . . . . . . . 11  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
3216, 28, 29, 30, 31, 2asclf 17754 . . . . . . . . . 10  |-  ( ( I  e.  _V  /\  R  e.  Ring )  ->  A : ( Base `  (Scalar `  P ) ) --> (
Base `  P )
)
3311, 27, 32syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  A : ( Base `  (Scalar `  P )
) --> ( Base `  P
) )
34 ffun 5731 . . . . . . . . 9  |-  ( A : ( Base `  (Scalar `  P ) ) --> (
Base `  P )  ->  Fun  A )
3533, 34syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  A )
36 funcoeqres 5844 . . . . . . . 8  |-  ( ( Fun  A  /\  (
m  o.  A )  =  F )  -> 
( m  |`  ran  A
)  =  ( F  o.  `' A ) )
3735, 36sylan 471 . . . . . . 7  |-  ( (
ph  /\  ( m  o.  A )  =  F )  ->  ( m  |` 
ran  A )  =  ( F  o.  `' A ) )
381, 9, 2, 11, 27mvrf2 17925 . . . . . . . . 9  |-  ( ph  ->  V : I --> ( Base `  P ) )
39 ffun 5731 . . . . . . . . 9  |-  ( V : I --> ( Base `  P )  ->  Fun  V )
4038, 39syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  V )
41 funcoeqres 5844 . . . . . . . 8  |-  ( ( Fun  V  /\  (
m  o.  V )  =  G )  -> 
( m  |`  ran  V
)  =  ( G  o.  `' V ) )
4240, 41sylan 471 . . . . . . 7  |-  ( (
ph  /\  ( m  o.  V )  =  G )  ->  ( m  |` 
ran  V )  =  ( G  o.  `' V ) )
4337, 42anim12dan 835 . . . . . 6  |-  ( (
ph  /\  ( (
m  o.  A )  =  F  /\  (
m  o.  V )  =  G ) )  ->  ( ( m  |`  ran  A )  =  ( F  o.  `' A )  /\  (
m  |`  ran  V )  =  ( G  o.  `' V ) ) )
4443ex 434 . . . . 5  |-  ( ph  ->  ( ( ( m  o.  A )  =  F  /\  ( m  o.  V )  =  G )  ->  (
( m  |`  ran  A
)  =  ( F  o.  `' A )  /\  ( m  |`  ran  V )  =  ( G  o.  `' V
) ) ) )
45 resundi 5285 . . . . . 6  |-  ( m  |`  ( ran  A  u.  ran  V ) )  =  ( ( m  |`  ran  A )  u.  (
m  |`  ran  V ) )
46 uneq12 3653 . . . . . 6  |-  ( ( ( m  |`  ran  A
)  =  ( F  o.  `' A )  /\  ( m  |`  ran  V )  =  ( G  o.  `' V
) )  ->  (
( m  |`  ran  A
)  u.  ( m  |`  ran  V ) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V ) ) )
4745, 46syl5eq 2520 . . . . 5  |-  ( ( ( m  |`  ran  A
)  =  ( F  o.  `' A )  /\  ( m  |`  ran  V )  =  ( G  o.  `' V
) )  ->  (
m  |`  ( ran  A  u.  ran  V ) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V ) ) )
4844, 47syl6 33 . . . 4  |-  ( ph  ->  ( ( ( m  o.  A )  =  F  /\  ( m  o.  V )  =  G )  ->  (
m  |`  ( ran  A  u.  ran  V ) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V ) ) ) )
4948ralrimivw 2879 . . 3  |-  ( ph  ->  A. m  e.  ( P RingHom  S ) ( ( ( m  o.  A
)  =  F  /\  ( m  o.  V
)  =  G )  ->  ( m  |`  ( ran  A  u.  ran  V ) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V
) ) ) )
50 eqtr3 2495 . . . . . 6  |-  ( ( ( m  |`  ( ran  A  u.  ran  V
) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V
) )  /\  (
n  |`  ( ran  A  u.  ran  V ) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V ) ) )  ->  ( m  |`  ( ran  A  u.  ran  V ) )  =  ( n  |`  ( ran  A  u.  ran  V ) ) )
51 eqid 2467 . . . . . . . . . . . . 13  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
5251, 11, 12psrassa 17837 . . . . . . . . . . . 12  |-  ( ph  ->  ( I mPwSer  R )  e. AssAlg )
53 eqid 2467 . . . . . . . . . . . . . 14  |-  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  R ) )
5451, 9, 53, 11, 27mvrf 17848 . . . . . . . . . . . . 13  |-  ( ph  ->  V : I --> ( Base `  ( I mPwSer  R ) ) )
55 frn 5735 . . . . . . . . . . . . 13  |-  ( V : I --> ( Base `  ( I mPwSer  R ) )  ->  ran  V  C_  ( Base `  ( I mPwSer  R ) ) )
5654, 55syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ran  V  C_  ( Base `  ( I mPwSer  R
) ) )
57 eqid 2467 . . . . . . . . . . . . 13  |-  (AlgSpan `  (
I mPwSer  R ) )  =  (AlgSpan `  ( I mPwSer  R ) )
58 eqid 2467 . . . . . . . . . . . . 13  |-  (algSc `  ( I mPwSer  R ) )  =  (algSc `  (
I mPwSer  R ) )
59 eqid 2467 . . . . . . . . . . . . 13  |-  (mrCls `  (SubRing `  ( I mPwSer  R
) ) )  =  (mrCls `  (SubRing `  (
I mPwSer  R ) ) )
6057, 58, 59, 53aspval2 17764 . . . . . . . . . . . 12  |-  ( ( ( I mPwSer  R )  e. AssAlg  /\  ran  V  C_  ( Base `  ( I mPwSer  R ) ) )  -> 
( (AlgSpan `  (
I mPwSer  R ) ) `  ran  V )  =  ( (mrCls `  (SubRing `  (
I mPwSer  R ) ) ) `
 ( ran  (algSc `  ( I mPwSer  R ) )  u.  ran  V
) ) )
6152, 56, 60syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( (AlgSpan `  (
I mPwSer  R ) ) `  ran  V )  =  ( (mrCls `  (SubRing `  (
I mPwSer  R ) ) ) `
 ( ran  (algSc `  ( I mPwSer  R ) )  u.  ran  V
) ) )
621, 51, 9, 57, 11, 12mplbas2 17902 . . . . . . . . . . 11  |-  ( ph  ->  ( (AlgSpan `  (
I mPwSer  R ) ) `  ran  V )  =  (
Base `  P )
)
6351, 1, 2, 11, 27mplsubrg 17870 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( Base `  P
)  e.  (SubRing `  (
I mPwSer  R ) ) )
641, 51, 2mplval2 17858 . . . . . . . . . . . . . . . 16  |-  P  =  ( ( I mPwSer  R
)s  ( Base `  P
) )
6564subsubrg2 17236 . . . . . . . . . . . . . . 15  |-  ( (
Base `  P )  e.  (SubRing `  ( I mPwSer  R ) )  ->  (SubRing `  P )  =  ( (SubRing `  ( I mPwSer  R ) )  i^i  ~P ( Base `  P )
) )
6663, 65syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  (SubRing `  P )  =  ( (SubRing `  (
I mPwSer  R ) )  i^i 
~P ( Base `  P
) ) )
6766fveq2d 5868 . . . . . . . . . . . . 13  |-  ( ph  ->  (mrCls `  (SubRing `  P
) )  =  (mrCls `  ( (SubRing `  (
I mPwSer  R ) )  i^i 
~P ( Base `  P
) ) ) )
6858, 64ressascl 17761 . . . . . . . . . . . . . . . . 17  |-  ( (
Base `  P )  e.  (SubRing `  ( I mPwSer  R ) )  ->  (algSc `  ( I mPwSer  R ) )  =  (algSc `  P ) )
6963, 68syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  (algSc `  ( I mPwSer  R ) )  =  (algSc `  P ) )
7069, 16syl6reqr 2527 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  =  (algSc `  ( I mPwSer  R ) ) )
7170rneqd 5228 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  A  =  ran  (algSc `  ( I mPwSer  R
) ) )
7271uneq1d 3657 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  A  u.  ran  V )  =  ( ran  (algSc `  (
I mPwSer  R ) )  u. 
ran  V ) )
7367, 72fveq12d 5870 . . . . . . . . . . . 12  |-  ( ph  ->  ( (mrCls `  (SubRing `  P ) ) `  ( ran  A  u.  ran  V ) )  =  ( (mrCls `  ( (SubRing `  ( I mPwSer  R ) )  i^i  ~P ( Base `  P ) ) ) `  ( ran  (algSc `  ( I mPwSer  R ) )  u.  ran  V ) ) )
74 assarng 17737 . . . . . . . . . . . . . 14  |-  ( ( I mPwSer  R )  e. AssAlg  ->  ( I mPwSer  R )  e.  Ring )
7553subrgmre 17233 . . . . . . . . . . . . . 14  |-  ( ( I mPwSer  R )  e. 
Ring  ->  (SubRing `  ( I mPwSer  R ) )  e.  (Moore `  ( Base `  (
I mPwSer  R ) ) ) )
7652, 74, 753syl 20 . . . . . . . . . . . . 13  |-  ( ph  ->  (SubRing `  ( I mPwSer  R ) )  e.  (Moore `  ( Base `  (
I mPwSer  R ) ) ) )
77 frn 5735 . . . . . . . . . . . . . . . 16  |-  ( A : ( Base `  (Scalar `  P ) ) --> (
Base `  P )  ->  ran  A  C_  ( Base `  P ) )
7833, 77syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  A  C_  ( Base `  P ) )
7971, 78eqsstr3d 3539 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  (algSc `  (
I mPwSer  R ) )  C_  ( Base `  P )
)
80 frn 5735 . . . . . . . . . . . . . . 15  |-  ( V : I --> ( Base `  P )  ->  ran  V 
C_  ( Base `  P
) )
8138, 80syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  V  C_  ( Base `  P ) )
8279, 81unssd 3680 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  (algSc `  ( I mPwSer  R ) )  u.  ran  V ) 
C_  ( Base `  P
) )
83 eqid 2467 . . . . . . . . . . . . . 14  |-  (mrCls `  ( (SubRing `  ( I mPwSer  R ) )  i^i  ~P ( Base `  P )
) )  =  (mrCls `  ( (SubRing `  (
I mPwSer  R ) )  i^i 
~P ( Base `  P
) ) )
8459, 83submrc 14876 . . . . . . . . . . . . 13  |-  ( ( (SubRing `  ( I mPwSer  R ) )  e.  (Moore `  ( Base `  (
I mPwSer  R ) ) )  /\  ( Base `  P
)  e.  (SubRing `  (
I mPwSer  R ) )  /\  ( ran  (algSc `  (
I mPwSer  R ) )  u. 
ran  V )  C_  ( Base `  P )
)  ->  ( (mrCls `  ( (SubRing `  (
I mPwSer  R ) )  i^i 
~P ( Base `  P
) ) ) `  ( ran  (algSc `  (
I mPwSer  R ) )  u. 
ran  V ) )  =  ( (mrCls `  (SubRing `  ( I mPwSer  R
) ) ) `  ( ran  (algSc `  (
I mPwSer  R ) )  u. 
ran  V ) ) )
8576, 63, 82, 84syl3anc 1228 . . . . . . . . . . . 12  |-  ( ph  ->  ( (mrCls `  (
(SubRing `  ( I mPwSer  R
) )  i^i  ~P ( Base `  P )
) ) `  ( ran  (algSc `  ( I mPwSer  R ) )  u.  ran  V ) )  =  ( (mrCls `  (SubRing `  (
I mPwSer  R ) ) ) `
 ( ran  (algSc `  ( I mPwSer  R ) )  u.  ran  V
) ) )
8673, 85eqtr2d 2509 . . . . . . . . . . 11  |-  ( ph  ->  ( (mrCls `  (SubRing `  ( I mPwSer  R ) ) ) `  ( ran  (algSc `  ( I mPwSer  R ) )  u.  ran  V ) )  =  ( (mrCls `  (SubRing `  P
) ) `  ( ran  A  u.  ran  V
) ) )
8761, 62, 863eqtr3d 2516 . . . . . . . . . 10  |-  ( ph  ->  ( Base `  P
)  =  ( (mrCls `  (SubRing `  P )
) `  ( ran  A  u.  ran  V ) ) )
8887ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  (
m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S )
) )  /\  ( ran  A  u.  ran  V
)  C_  dom  ( m  i^i  n ) )  ->  ( Base `  P
)  =  ( (mrCls `  (SubRing `  P )
) `  ( ran  A  u.  ran  V ) ) )
8911, 27, 29syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  P  e.  Ring )
902subrgmre 17233 . . . . . . . . . . . 12  |-  ( P  e.  Ring  ->  (SubRing `  P
)  e.  (Moore `  ( Base `  P )
) )
9189, 90syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (SubRing `  P )  e.  (Moore `  ( Base `  P ) ) )
9291ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S )
) )  /\  ( ran  A  u.  ran  V
)  C_  dom  ( m  i^i  n ) )  ->  (SubRing `  P )  e.  (Moore `  ( Base `  P ) ) )
93 simpr 461 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S )
) )  /\  ( ran  A  u.  ran  V
)  C_  dom  ( m  i^i  n ) )  ->  ( ran  A  u.  ran  V )  C_  dom  ( m  i^i  n
) )
94 rhmeql 17239 . . . . . . . . . . 11  |-  ( ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S )
)  ->  dom  ( m  i^i  n )  e.  (SubRing `  P )
)
9594ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S )
) )  /\  ( ran  A  u.  ran  V
)  C_  dom  ( m  i^i  n ) )  ->  dom  ( m  i^i  n )  e.  (SubRing `  P ) )
96 eqid 2467 . . . . . . . . . . 11  |-  (mrCls `  (SubRing `  P ) )  =  (mrCls `  (SubRing `  P ) )
9796mrcsscl 14868 . . . . . . . . . 10  |-  ( ( (SubRing `  P )  e.  (Moore `  ( Base `  P ) )  /\  ( ran  A  u.  ran  V )  C_  dom  ( m  i^i  n )  /\  dom  ( m  i^i  n
)  e.  (SubRing `  P
) )  ->  (
(mrCls `  (SubRing `  P
) ) `  ( ran  A  u.  ran  V
) )  C_  dom  ( m  i^i  n
) )
9892, 93, 95, 97syl3anc 1228 . . . . . . . . 9  |-  ( ( ( ph  /\  (
m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S )
) )  /\  ( ran  A  u.  ran  V
)  C_  dom  ( m  i^i  n ) )  ->  ( (mrCls `  (SubRing `  P ) ) `
 ( ran  A  u.  ran  V ) ) 
C_  dom  ( m  i^i  n ) )
9988, 98eqsstrd 3538 . . . . . . . 8  |-  ( ( ( ph  /\  (
m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S )
) )  /\  ( ran  A  u.  ran  V
)  C_  dom  ( m  i^i  n ) )  ->  ( Base `  P
)  C_  dom  ( m  i^i  n ) )
10099ex 434 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  ( ( ran 
A  u.  ran  V
)  C_  dom  ( m  i^i  n )  -> 
( Base `  P )  C_ 
dom  ( m  i^i  n ) ) )
101 simprl 755 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  m  e.  ( P RingHom  S ) )
1022, 3rhmf 17156 . . . . . . . . 9  |-  ( m  e.  ( P RingHom  S
)  ->  m :
( Base `  P ) --> C )
103 ffn 5729 . . . . . . . . 9  |-  ( m : ( Base `  P
) --> C  ->  m  Fn  ( Base `  P
) )
104101, 102, 1033syl 20 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  m  Fn  ( Base `  P ) )
105 simprr 756 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  n  e.  ( P RingHom  S ) )
1062, 3rhmf 17156 . . . . . . . . 9  |-  ( n  e.  ( P RingHom  S
)  ->  n :
( Base `  P ) --> C )
107 ffn 5729 . . . . . . . . 9  |-  ( n : ( Base `  P
) --> C  ->  n  Fn  ( Base `  P
) )
108105, 106, 1073syl 20 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  n  Fn  ( Base `  P ) )
10978adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  ran  A  C_  ( Base `  P ) )
11081adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  ran  V  C_  ( Base `  P ) )
111109, 110unssd 3680 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  ( ran  A  u.  ran  V )  C_  ( Base `  P )
)
112 fnreseql 5989 . . . . . . . 8  |-  ( ( m  Fn  ( Base `  P )  /\  n  Fn  ( Base `  P
)  /\  ( ran  A  u.  ran  V ) 
C_  ( Base `  P
) )  ->  (
( m  |`  ( ran  A  u.  ran  V
) )  =  ( n  |`  ( ran  A  u.  ran  V ) )  <->  ( ran  A  u.  ran  V )  C_  dom  ( m  i^i  n
) ) )
113104, 108, 111, 112syl3anc 1228 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  ( ( m  |`  ( ran  A  u.  ran  V ) )  =  ( n  |`  ( ran  A  u.  ran  V
) )  <->  ( ran  A  u.  ran  V ) 
C_  dom  ( m  i^i  n ) ) )
114 fneqeql2 5988 . . . . . . . 8  |-  ( ( m  Fn  ( Base `  P )  /\  n  Fn  ( Base `  P
) )  ->  (
m  =  n  <->  ( Base `  P )  C_  dom  ( m  i^i  n
) ) )
115104, 108, 114syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  ( m  =  n  <->  ( Base `  P
)  C_  dom  ( m  i^i  n ) ) )
116100, 113, 1153imtr4d 268 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  ( ( m  |`  ( ran  A  u.  ran  V ) )  =  ( n  |`  ( ran  A  u.  ran  V
) )  ->  m  =  n ) )
11750, 116syl5 32 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  ( ( ( m  |`  ( ran  A  u.  ran  V ) )  =  ( ( F  o.  `' A
)  u.  ( G  o.  `' V ) )  /\  ( n  |`  ( ran  A  u.  ran  V ) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V
) ) )  ->  m  =  n )
)
118117ralrimivva 2885 . . . 4  |-  ( ph  ->  A. m  e.  ( P RingHom  S ) A. n  e.  ( P RingHom  S )
( ( ( m  |`  ( ran  A  u.  ran  V ) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V
) )  /\  (
n  |`  ( ran  A  u.  ran  V ) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V ) ) )  ->  m  =  n ) )
119 reseq1 5265 . . . . . 6  |-  ( m  =  n  ->  (
m  |`  ( ran  A  u.  ran  V ) )  =  ( n  |`  ( ran  A  u.  ran  V ) ) )
120119eqeq1d 2469 . . . . 5  |-  ( m  =  n  ->  (
( m  |`  ( ran  A  u.  ran  V
) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V
) )  <->  ( n  |`  ( ran  A  u.  ran  V ) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V
) ) ) )
121120rmo4 3296 . . . 4  |-  ( E* m  e.  ( P RingHom  S ) ( m  |`  ( ran  A  u.  ran  V ) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V
) )  <->  A. m  e.  ( P RingHom  S ) A. n  e.  ( P RingHom  S ) ( ( ( m  |`  ( ran  A  u.  ran  V
) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V
) )  /\  (
n  |`  ( ran  A  u.  ran  V ) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V ) ) )  ->  m  =  n ) )
122118, 121sylibr 212 . . 3  |-  ( ph  ->  E* m  e.  ( P RingHom  S ) ( m  |`  ( ran  A  u.  ran  V ) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V
) ) )
123 rmoim 3303 . . 3  |-  ( A. m  e.  ( P RingHom  S ) ( ( ( m  o.  A )  =  F  /\  (
m  o.  V )  =  G )  -> 
( m  |`  ( ran  A  u.  ran  V
) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V
) ) )  -> 
( E* m  e.  ( P RingHom  S )
( m  |`  ( ran  A  u.  ran  V
) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V
) )  ->  E* m  e.  ( P RingHom  S ) ( ( m  o.  A )  =  F  /\  ( m  o.  V )  =  G ) ) )
12449, 122, 123sylc 60 . 2  |-  ( ph  ->  E* m  e.  ( P RingHom  S ) ( ( m  o.  A )  =  F  /\  (
m  o.  V )  =  G ) )
125 reu5 3077 . 2  |-  ( E! m  e.  ( P RingHom  S ) ( ( m  o.  A )  =  F  /\  (
m  o.  V )  =  G )  <->  ( E. m  e.  ( P RingHom  S ) ( ( m  o.  A )  =  F  /\  ( m  o.  V )  =  G )  /\  E* m  e.  ( P RingHom  S ) ( ( m  o.  A )  =  F  /\  ( m  o.  V )  =  G ) ) )
12625, 124, 125sylanbrc 664 1  |-  ( ph  ->  E! m  e.  ( P RingHom  S ) ( ( m  o.  A )  =  F  /\  (
m  o.  V )  =  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   E!wreu 2816   E*wrmo 2817   {crab 2818   _Vcvv 3113    u. cun 3474    i^i cin 3475    C_ wss 3476   ~Pcpw 4010    |-> cmpt 4505   `'ccnv 4998   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002    o. ccom 5003   Fun wfun 5580    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282    oFcof 6520    ^m cmap 7417   Fincfn 7513   NNcn 10532   NN0cn0 10791   Basecbs 14483   .rcmulr 14549  Scalarcsca 14551    gsumg cgsu 14689  Moorecmre 14830  mrClscmrc 14831  .gcmg 15724  mulGrpcmgp 16928   Ringcrg 16983   CRingccrg 16984   RingHom crh 17142  SubRingcsubrg 17205  AssAlgcasa 17726  AlgSpancasp 17727  algSccascl 17728   mPwSer cmps 17768   mVar cmvr 17769   mPoly cmpl 17770
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-ofr 6523  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-seq 12071  df-hash 12368  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-sca 14564  df-vsca 14565  df-tset 14567  df-0g 14690  df-gsum 14691  df-mre 14834  df-mrc 14835  df-acs 14837  df-mnd 15725  df-mhm 15774  df-submnd 15775  df-grp 15855  df-minusg 15856  df-sbg 15857  df-mulg 15858  df-subg 15990  df-ghm 16057  df-cntz 16147  df-cmn 16593  df-abl 16594  df-mgp 16929  df-ur 16941  df-srg 16945  df-rng 16985  df-cring 16986  df-rnghom 17145  df-subrg 17207  df-lmod 17294  df-lss 17359  df-lsp 17398  df-assa 17729  df-asp 17730  df-ascl 17731  df-psr 17773  df-mvr 17774  df-mpl 17775
This theorem is referenced by:  evlsval2  17957
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