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Theorem evlseu 19942
Description: For a given intepretation 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 2438 . . . 4  |-  ( Base `  P )  =  (
Base `  P )
3 evlseu.c . . . 4  |-  C  =  ( Base `  S
)
4 eqid 2438 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
5 eqid 2438 . . . 4  |-  { z  e.  ( NN0  ^m  I )  |  ( `' z " NN )  e.  Fin }  =  { z  e.  ( NN0  ^m  I )  |  ( `' z
" NN )  e. 
Fin }
6 eqid 2438 . . . 4  |-  (mulGrp `  S )  =  (mulGrp `  S )
7 eqid 2438 . . . 4  |-  (.g `  (mulGrp `  S ) )  =  (.g `  (mulGrp `  S
) )
8 eqid 2438 . . . 4  |-  ( .r
`  S )  =  ( .r `  S
)
9 evlseu.v . . . 4  |-  V  =  ( I mVar  R )
10 eqid 2438 . . . 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  o F (.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  o F (.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 19941 . . 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  o F (.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  o F (.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  o F (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  V )  =  G ) )
18 coeq1 5033 . . . . . . 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  o F (.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  o F (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  A ) )
1918eqeq1d 2446 . . . . . 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  o F (.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  o F (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  A )  =  F ) )
20 coeq1 5033 . . . . . . 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  o F (.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  o F (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  V ) )
2120eqeq1d 2446 . . . . . 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  o F (.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  o F (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  V )  =  G ) )
2219, 21anbi12d 693 . . . . 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  o F (.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  o F (.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  o F (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  V )  =  G ) ) )
2322rspcev 3054 . . . 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  o F (.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  o F (.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  o F (.g `  (mulGrp `  S )
) G ) ) ) ) ) )  o.  V )  =  G ) )  ->  E. m  e.  ( P RingHom  S ) ( ( m  o.  A )  =  F  /\  (
m  o.  V )  =  G ) )
24233impb 1150 . . 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  o F (.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  o F (.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  o F (.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 15679 . . . . . . . . . . 11  |-  ( R  e.  CRing  ->  R  e.  Ring )
2712, 26syl 16 . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
28 eqid 2438 . . . . . . . . . . 11  |-  (Scalar `  P )  =  (Scalar `  P )
291mplrng 16520 . . . . . . . . . . 11  |-  ( ( I  e.  _V  /\  R  e.  Ring )  ->  P  e.  Ring )
301mpllmod 16519 . . . . . . . . . . 11  |-  ( ( I  e.  _V  /\  R  e.  Ring )  ->  P  e.  LMod )
31 eqid 2438 . . . . . . . . . . 11  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
3216, 28, 29, 30, 31, 2asclf 16401 . . . . . . . . . 10  |-  ( ( I  e.  _V  /\  R  e.  Ring )  ->  A : ( Base `  (Scalar `  P ) ) --> (
Base `  P )
)
3311, 27, 32syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  A : ( Base `  (Scalar `  P )
) --> ( Base `  P
) )
34 ffun 5596 . . . . . . . . 9  |-  ( A : ( Base `  (Scalar `  P ) ) --> (
Base `  P )  ->  Fun  A )
3533, 34syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  A )
36 funcoeqres 5709 . . . . . . . 8  |-  ( ( Fun  A  /\  (
m  o.  A )  =  F )  -> 
( m  |`  ran  A
)  =  ( F  o.  `' A ) )
3735, 36sylan 459 . . . . . . 7  |-  ( (
ph  /\  ( m  o.  A )  =  F )  ->  ( m  |` 
ran  A )  =  ( F  o.  `' A ) )
381, 9, 2, 11, 27mvrf2 16557 . . . . . . . . 9  |-  ( ph  ->  V : I --> ( Base `  P ) )
39 ffun 5596 . . . . . . . . 9  |-  ( V : I --> ( Base `  P )  ->  Fun  V )
4038, 39syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  V )
41 funcoeqres 5709 . . . . . . . 8  |-  ( ( Fun  V  /\  (
m  o.  V )  =  G )  -> 
( m  |`  ran  V
)  =  ( G  o.  `' V ) )
4240, 41sylan 459 . . . . . . 7  |-  ( (
ph  /\  ( m  o.  V )  =  G )  ->  ( m  |` 
ran  V )  =  ( G  o.  `' V ) )
4337, 42anim12dan 812 . . . . . 6  |-  ( (
ph  /\  ( (
m  o.  A )  =  F  /\  (
m  o.  V )  =  G ) )  ->  ( ( m  |`  ran  A )  =  ( F  o.  `' A )  /\  (
m  |`  ran  V )  =  ( G  o.  `' V ) ) )
4443ex 425 . . . . 5  |-  ( ph  ->  ( ( ( m  o.  A )  =  F  /\  ( m  o.  V )  =  G )  ->  (
( m  |`  ran  A
)  =  ( F  o.  `' A )  /\  ( m  |`  ran  V )  =  ( G  o.  `' V
) ) ) )
45 resundi 5163 . . . . . 6  |-  ( m  |`  ( ran  A  u.  ran  V ) )  =  ( ( m  |`  ran  A )  u.  (
m  |`  ran  V ) )
46 uneq12 3498 . . . . . 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 2482 . . . . 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 32 . . . 4  |-  ( ph  ->  ( ( ( m  o.  A )  =  F  /\  ( m  o.  V )  =  G )  ->  (
m  |`  ( ran  A  u.  ran  V ) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V ) ) ) )
4948ralrimivw 2792 . . 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 2457 . . . . . 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 2438 . . . . . . . . . . . . 13  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
5251, 11, 12psrassa 16482 . . . . . . . . . . . 12  |-  ( ph  ->  ( I mPwSer  R )  e. AssAlg )
53 eqid 2438 . . . . . . . . . . . . . 14  |-  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  R ) )
5451, 9, 53, 11, 27mvrf 16493 . . . . . . . . . . . . 13  |-  ( ph  ->  V : I --> ( Base `  ( I mPwSer  R ) ) )
55 frn 5600 . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . 13  |-  (AlgSpan `  (
I mPwSer  R ) )  =  (AlgSpan `  ( I mPwSer  R ) )
58 eqid 2438 . . . . . . . . . . . . 13  |-  (algSc `  ( I mPwSer  R ) )  =  (algSc `  (
I mPwSer  R ) )
59 eqid 2438 . . . . . . . . . . . . 13  |-  (mrCls `  (SubRing `  ( I mPwSer  R
) ) )  =  (mrCls `  (SubRing `  (
I mPwSer  R ) ) )
6057, 58, 59, 53aspval2 16410 . . . . . . . . . . . 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 644 . . . . . . . . . . 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 16536 . . . . . . . . . . 11  |-  ( ph  ->  ( (AlgSpan `  (
I mPwSer  R ) ) `  ran  V )  =  (
Base `  P )
)
6351, 1, 2, 11, 27mplsubrg 16508 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( Base `  P
)  e.  (SubRing `  (
I mPwSer  R ) ) )
641, 51, 2mplval2 16500 . . . . . . . . . . . . . . . 16  |-  P  =  ( ( I mPwSer  R
)s  ( Base `  P
) )
6564subsubrg2 15900 . . . . . . . . . . . . . . 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 5735 . . . . . . . . . . . . 13  |-  ( ph  ->  (mrCls `  (SubRing `  P
) )  =  (mrCls `  ( (SubRing `  (
I mPwSer  R ) )  i^i 
~P ( Base `  P
) ) ) )
6858, 64ressascl 16407 . . . . . . . . . . . . . . . . 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 2489 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  =  (algSc `  ( I mPwSer  R ) ) )
7170rneqd 5100 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  A  =  ran  (algSc `  ( I mPwSer  R
) ) )
7271uneq1d 3502 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  A  u.  ran  V )  =  ( ran  (algSc `  (
I mPwSer  R ) )  u. 
ran  V ) )
7367, 72fveq12d 5737 . . . . . . . . . . . 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 16385 . . . . . . . . . . . . . 14  |-  ( ( I mPwSer  R )  e. AssAlg  ->  ( I mPwSer  R )  e.  Ring )
7553subrgmre 15897 . . . . . . . . . . . . . 14  |-  ( ( I mPwSer  R )  e. 
Ring  ->  (SubRing `  ( I mPwSer  R ) )  e.  (Moore `  ( Base `  (
I mPwSer  R ) ) ) )
7652, 74, 753syl 19 . . . . . . . . . . . . 13  |-  ( ph  ->  (SubRing `  ( I mPwSer  R ) )  e.  (Moore `  ( Base `  (
I mPwSer  R ) ) ) )
77 frn 5600 . . . . . . . . . . . . . . . 16  |-  ( A : ( Base `  (Scalar `  P ) ) --> (
Base `  P )  ->  ran  A  C_  ( Base `  P ) )
7833, 77syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  A  C_  ( Base `  P ) )
7971, 78eqsstr3d 3385 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  (algSc `  (
I mPwSer  R ) )  C_  ( Base `  P )
)
80 frn 5600 . . . . . . . . . . . . . . 15  |-  ( V : I --> ( Base `  P )  ->  ran  V 
C_  ( Base `  P
) )
8138, 80syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  V  C_  ( Base `  P ) )
8279, 81unssd 3525 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  (algSc `  ( I mPwSer  R ) )  u.  ran  V ) 
C_  ( Base `  P
) )
83 eqid 2438 . . . . . . . . . . . . . 14  |-  (mrCls `  ( (SubRing `  ( I mPwSer  R ) )  i^i  ~P ( Base `  P )
) )  =  (mrCls `  ( (SubRing `  (
I mPwSer  R ) )  i^i 
~P ( Base `  P
) ) )
8459, 83submrc 13858 . . . . . . . . . . . . 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 1185 . . . . . . . . . . . 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 2471 . . . . . . . . . . 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 2478 . . . . . . . . . 10  |-  ( ph  ->  ( Base `  P
)  =  ( (mrCls `  (SubRing `  P )
) `  ( ran  A  u.  ran  V ) ) )
8887ad2antrr 708 . . . . . . . . 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 644 . . . . . . . . . . . 12  |-  ( ph  ->  P  e.  Ring )
902subrgmre 15897 . . . . . . . . . . . 12  |-  ( P  e.  Ring  ->  (SubRing `  P
)  e.  (Moore `  ( Base `  P )
) )
9189, 90syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (SubRing `  P )  e.  (Moore `  ( Base `  P ) ) )
9291ad2antrr 708 . . . . . . . . . 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 449 . . . . . . . . . 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 15903 . . . . . . . . . . 11  |-  ( ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S )
)  ->  dom  ( m  i^i  n )  e.  (SubRing `  P )
)
9594ad2antlr 709 . . . . . . . . . 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 2438 . . . . . . . . . . 11  |-  (mrCls `  (SubRing `  P ) )  =  (mrCls `  (SubRing `  P ) )
9796mrcsscl 13850 . . . . . . . . . 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 1185 . . . . . . . . 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 3384 . . . . . . . 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 425 . . . . . . 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 734 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  m  e.  ( P RingHom  S ) )
1022, 3rhmf 15832 . . . . . . . . 9  |-  ( m  e.  ( P RingHom  S
)  ->  m :
( Base `  P ) --> C )
103 ffn 5594 . . . . . . . . 9  |-  ( m : ( Base `  P
) --> C  ->  m  Fn  ( Base `  P
) )
104101, 102, 1033syl 19 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  m  Fn  ( Base `  P ) )
105 simprr 735 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  n  e.  ( P RingHom  S ) )
1062, 3rhmf 15832 . . . . . . . . 9  |-  ( n  e.  ( P RingHom  S
)  ->  n :
( Base `  P ) --> C )
107 ffn 5594 . . . . . . . . 9  |-  ( n : ( Base `  P
) --> C  ->  n  Fn  ( Base `  P
) )
108105, 106, 1073syl 19 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  n  Fn  ( Base `  P ) )
10978adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  ran  A  C_  ( Base `  P ) )
11081adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  ran  V  C_  ( Base `  P ) )
111109, 110unssd 3525 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  ( ran  A  u.  ran  V )  C_  ( Base `  P )
)
112 fnreseql 5843 . . . . . . . 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 1185 . . . . . . 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 5842 . . . . . . . 8  |-  ( ( m  Fn  ( Base `  P )  /\  n  Fn  ( Base `  P
) )  ->  (
m  =  n  <->  ( Base `  P )  C_  dom  ( m  i^i  n
) ) )
115104, 108, 114syl2anc 644 . . . . . . 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 261 . . . . . 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 31 . . . . 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 2800 . . . 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 5143 . . . . . 6  |-  ( m  =  n  ->  (
m  |`  ( ran  A  u.  ran  V ) )  =  ( n  |`  ( ran  A  u.  ran  V ) ) )
120119eqeq1d 2446 . . . . 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 3129 . . . 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 205 . . 3  |-  ( ph  ->  E* m  e.  ( P RingHom  S ) ( m  |`  ( ran  A  u.  ran  V ) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V
) ) )
123 rmoim 3135 . . 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 59 . 2  |-  ( ph  ->  E* m  e.  ( P RingHom  S ) ( ( m  o.  A )  =  F  /\  (
m  o.  V )  =  G ) )
125 reu5 2923 . 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 647 1  |-  ( ph  ->  E! m  e.  ( P RingHom  S ) ( ( m  o.  A )  =  F  /\  (
m  o.  V )  =  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   E!wreu 2709   E*wrmo 2710   {crab 2711   _Vcvv 2958    u. cun 3320    i^i cin 3321    C_ wss 3322   ~Pcpw 3801    e. cmpt 4269   `'ccnv 4880   dom cdm 4881   ran crn 4882    |` cres 4883   "cima 4884    o. ccom 4885   Fun wfun 5451    Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084    o Fcof 6306    ^m cmap 7021   Fincfn 7112   NNcn 10005   NN0cn0 10226   Basecbs 13474   .rcmulr 13535  Scalarcsca 13537    gsumg cgsu 13729  Moorecmre 13812  mrClscmrc 13813  .gcmg 14694  mulGrpcmgp 15653   Ringcrg 15665   CRingccrg 15666   RingHom crh 15822  SubRingcsubrg 15869  AssAlgcasa 16374  AlgSpancasp 16375  algSccascl 16376   mPwSer cmps 16411   mVar cmvr 16412   mPoly cmpl 16413
This theorem is referenced by:  evlsval2  19946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-ofr 6309  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-fzo 11141  df-seq 11329  df-hash 11624  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-sca 13550  df-vsca 13551  df-tset 13553  df-0g 13732  df-gsum 13733  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-mhm 14743  df-submnd 14744  df-grp 14817  df-minusg 14818  df-sbg 14819  df-mulg 14820  df-subg 14946  df-ghm 15009  df-cntz 15121  df-cmn 15419  df-abl 15420  df-mgp 15654  df-rng 15668  df-cring 15669  df-ur 15670  df-rnghom 15824  df-subrg 15871  df-lmod 15957  df-lss 16014  df-lsp 16053  df-assa 16377  df-asp 16378  df-ascl 16379  df-psr 16422  df-mvr 16423  df-mpl 16424
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