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Theorem evlseu 19400
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 2283 . . . 4  |-  ( Base `  P )  =  (
Base `  P )
3 evlseu.c . . . 4  |-  C  =  ( Base `  S
)
4 eqid 2283 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
5 eqid 2283 . . . 4  |-  { z  e.  ( NN0  ^m  I )  |  ( `' z " NN )  e.  Fin }  =  { z  e.  ( NN0  ^m  I )  |  ( `' z
" NN )  e. 
Fin }
6 eqid 2283 . . . 4  |-  (mulGrp `  S )  =  (mulGrp `  S )
7 eqid 2283 . . . 4  |-  (.g `  (mulGrp `  S ) )  =  (.g `  (mulGrp `  S
) )
8 eqid 2283 . . . 4  |-  ( .r
`  S )  =  ( .r `  S
)
9 evlseu.v . . . 4  |-  V  =  ( I mVar  R )
10 eqid 2283 . . . 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 19399 . . 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 4841 . . . . . . 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 2291 . . . . . 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 4841 . . . . . . 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 2291 . . . . . 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 691 . . . . 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 2884 . . . 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 1147 . . 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 15 . 2  |-  ( ph  ->  E. m  e.  ( P RingHom  S ) ( ( m  o.  A )  =  F  /\  (
m  o.  V )  =  G ) )
26 crngrng 15351 . . . . . . . . . . 11  |-  ( R  e.  CRing  ->  R  e.  Ring )
2712, 26syl 15 . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
28 eqid 2283 . . . . . . . . . . 11  |-  (Scalar `  P )  =  (Scalar `  P )
291mplrng 16196 . . . . . . . . . . 11  |-  ( ( I  e.  _V  /\  R  e.  Ring )  ->  P  e.  Ring )
301mpllmod 16195 . . . . . . . . . . 11  |-  ( ( I  e.  _V  /\  R  e.  Ring )  ->  P  e.  LMod )
31 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
3216, 28, 29, 30, 31, 2asclf 16077 . . . . . . . . . 10  |-  ( ( I  e.  _V  /\  R  e.  Ring )  ->  A : ( Base `  (Scalar `  P ) ) --> (
Base `  P )
)
3311, 27, 32syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  A : ( Base `  (Scalar `  P )
) --> ( Base `  P
) )
34 ffun 5391 . . . . . . . . 9  |-  ( A : ( Base `  (Scalar `  P ) ) --> (
Base `  P )  ->  Fun  A )
3533, 34syl 15 . . . . . . . 8  |-  ( ph  ->  Fun  A )
36 funcoeqres 5504 . . . . . . . 8  |-  ( ( Fun  A  /\  (
m  o.  A )  =  F )  -> 
( m  |`  ran  A
)  =  ( F  o.  `' A ) )
3735, 36sylan 457 . . . . . . 7  |-  ( (
ph  /\  ( m  o.  A )  =  F )  ->  ( m  |` 
ran  A )  =  ( F  o.  `' A ) )
381, 9, 2, 11, 27mvrf2 16233 . . . . . . . . 9  |-  ( ph  ->  V : I --> ( Base `  P ) )
39 ffun 5391 . . . . . . . . 9  |-  ( V : I --> ( Base `  P )  ->  Fun  V )
4038, 39syl 15 . . . . . . . 8  |-  ( ph  ->  Fun  V )
41 funcoeqres 5504 . . . . . . . 8  |-  ( ( Fun  V  /\  (
m  o.  V )  =  G )  -> 
( m  |`  ran  V
)  =  ( G  o.  `' V ) )
4240, 41sylan 457 . . . . . . 7  |-  ( (
ph  /\  ( m  o.  V )  =  G )  ->  ( m  |` 
ran  V )  =  ( G  o.  `' V ) )
4337, 42anim12dan 810 . . . . . 6  |-  ( (
ph  /\  ( (
m  o.  A )  =  F  /\  (
m  o.  V )  =  G ) )  ->  ( ( m  |`  ran  A )  =  ( F  o.  `' A )  /\  (
m  |`  ran  V )  =  ( G  o.  `' V ) ) )
4443ex 423 . . . . 5  |-  ( ph  ->  ( ( ( m  o.  A )  =  F  /\  ( m  o.  V )  =  G )  ->  (
( m  |`  ran  A
)  =  ( F  o.  `' A )  /\  ( m  |`  ran  V )  =  ( G  o.  `' V
) ) ) )
45 resundi 4969 . . . . . 6  |-  ( m  |`  ( ran  A  u.  ran  V ) )  =  ( ( m  |`  ran  A )  u.  (
m  |`  ran  V ) )
46 uneq12 3324 . . . . . 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 2327 . . . . 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 29 . . . 4  |-  ( ph  ->  ( ( ( m  o.  A )  =  F  /\  ( m  o.  V )  =  G )  ->  (
m  |`  ( ran  A  u.  ran  V ) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V ) ) ) )
4948ralrimivw 2627 . . 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 2302 . . . . . 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 2283 . . . . . . . . . . . . 13  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
5251, 11, 12psrassa 16158 . . . . . . . . . . . 12  |-  ( ph  ->  ( I mPwSer  R )  e. AssAlg )
53 eqid 2283 . . . . . . . . . . . . . 14  |-  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  R ) )
5451, 9, 53, 11, 27mvrf 16169 . . . . . . . . . . . . 13  |-  ( ph  ->  V : I --> ( Base `  ( I mPwSer  R ) ) )
55 frn 5395 . . . . . . . . . . . . 13  |-  ( V : I --> ( Base `  ( I mPwSer  R ) )  ->  ran  V  C_  ( Base `  ( I mPwSer  R ) ) )
5654, 55syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ran  V  C_  ( Base `  ( I mPwSer  R
) ) )
57 eqid 2283 . . . . . . . . . . . . 13  |-  (AlgSpan `  (
I mPwSer  R ) )  =  (AlgSpan `  ( I mPwSer  R ) )
58 eqid 2283 . . . . . . . . . . . . 13  |-  (algSc `  ( I mPwSer  R ) )  =  (algSc `  (
I mPwSer  R ) )
59 eqid 2283 . . . . . . . . . . . . 13  |-  (mrCls `  (SubRing `  ( I mPwSer  R
) ) )  =  (mrCls `  (SubRing `  (
I mPwSer  R ) ) )
6057, 58, 59, 53aspval2 16086 . . . . . . . . . . . 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 642 . . . . . . . . . . 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 16212 . . . . . . . . . . 11  |-  ( ph  ->  ( (AlgSpan `  (
I mPwSer  R ) ) `  ran  V )  =  (
Base `  P )
)
6351, 1, 2, 11, 27mplsubrg 16184 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( Base `  P
)  e.  (SubRing `  (
I mPwSer  R ) ) )
641, 51, 2mplval2 16176 . . . . . . . . . . . . . . . 16  |-  P  =  ( ( I mPwSer  R
)s  ( Base `  P
) )
6564subsubrg2 15572 . . . . . . . . . . . . . . 15  |-  ( (
Base `  P )  e.  (SubRing `  ( I mPwSer  R ) )  ->  (SubRing `  P )  =  ( (SubRing `  ( I mPwSer  R ) )  i^i  ~P ( Base `  P )
) )
6663, 65syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  (SubRing `  P )  =  ( (SubRing `  (
I mPwSer  R ) )  i^i 
~P ( Base `  P
) ) )
6766fveq2d 5529 . . . . . . . . . . . . 13  |-  ( ph  ->  (mrCls `  (SubRing `  P
) )  =  (mrCls `  ( (SubRing `  (
I mPwSer  R ) )  i^i 
~P ( Base `  P
) ) ) )
6858, 64ressascl 16083 . . . . . . . . . . . . . . . . 17  |-  ( (
Base `  P )  e.  (SubRing `  ( I mPwSer  R ) )  ->  (algSc `  ( I mPwSer  R ) )  =  (algSc `  P ) )
6963, 68syl 15 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  (algSc `  ( I mPwSer  R ) )  =  (algSc `  P ) )
7069, 16syl6reqr 2334 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  =  (algSc `  ( I mPwSer  R ) ) )
7170rneqd 4906 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  A  =  ran  (algSc `  ( I mPwSer  R
) ) )
7271uneq1d 3328 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  A  u.  ran  V )  =  ( ran  (algSc `  (
I mPwSer  R ) )  u. 
ran  V ) )
7367, 72fveq12d 5531 . . . . . . . . . . . 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 16061 . . . . . . . . . . . . . 14  |-  ( ( I mPwSer  R )  e. AssAlg  ->  ( I mPwSer  R )  e.  Ring )
7553subrgmre 15569 . . . . . . . . . . . . . 14  |-  ( ( I mPwSer  R )  e. 
Ring  ->  (SubRing `  ( I mPwSer  R ) )  e.  (Moore `  ( Base `  (
I mPwSer  R ) ) ) )
7652, 74, 753syl 18 . . . . . . . . . . . . 13  |-  ( ph  ->  (SubRing `  ( I mPwSer  R ) )  e.  (Moore `  ( Base `  (
I mPwSer  R ) ) ) )
77 frn 5395 . . . . . . . . . . . . . . . 16  |-  ( A : ( Base `  (Scalar `  P ) ) --> (
Base `  P )  ->  ran  A  C_  ( Base `  P ) )
7833, 77syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  A  C_  ( Base `  P ) )
7971, 78eqsstr3d 3213 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  (algSc `  (
I mPwSer  R ) )  C_  ( Base `  P )
)
80 frn 5395 . . . . . . . . . . . . . . 15  |-  ( V : I --> ( Base `  P )  ->  ran  V 
C_  ( Base `  P
) )
8138, 80syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  V  C_  ( Base `  P ) )
8279, 81unssd 3351 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  (algSc `  ( I mPwSer  R ) )  u.  ran  V ) 
C_  ( Base `  P
) )
83 eqid 2283 . . . . . . . . . . . . . 14  |-  (mrCls `  ( (SubRing `  ( I mPwSer  R ) )  i^i  ~P ( Base `  P )
) )  =  (mrCls `  ( (SubRing `  (
I mPwSer  R ) )  i^i 
~P ( Base `  P
) ) )
8459, 83submrc 13530 . . . . . . . . . . . . 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 1182 . . . . . . . . . . . 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 2316 . . . . . . . . . . 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 2323 . . . . . . . . . 10  |-  ( ph  ->  ( Base `  P
)  =  ( (mrCls `  (SubRing `  P )
) `  ( ran  A  u.  ran  V ) ) )
8887ad2antrr 706 . . . . . . . . 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 642 . . . . . . . . . . . 12  |-  ( ph  ->  P  e.  Ring )
902subrgmre 15569 . . . . . . . . . . . 12  |-  ( P  e.  Ring  ->  (SubRing `  P
)  e.  (Moore `  ( Base `  P )
) )
9189, 90syl 15 . . . . . . . . . . 11  |-  ( ph  ->  (SubRing `  P )  e.  (Moore `  ( Base `  P ) ) )
9291ad2antrr 706 . . . . . . . . . 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 447 . . . . . . . . . 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 15575 . . . . . . . . . . 11  |-  ( ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S )
)  ->  dom  ( m  i^i  n )  e.  (SubRing `  P )
)
9594ad2antlr 707 . . . . . . . . . 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 2283 . . . . . . . . . . 11  |-  (mrCls `  (SubRing `  P ) )  =  (mrCls `  (SubRing `  P ) )
9796mrcsscl 13522 . . . . . . . . . 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 1182 . . . . . . . . 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 3212 . . . . . . . 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 423 . . . . . . 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 732 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  m  e.  ( P RingHom  S ) )
1022, 3rhmf 15504 . . . . . . . . 9  |-  ( m  e.  ( P RingHom  S
)  ->  m :
( Base `  P ) --> C )
103 ffn 5389 . . . . . . . . 9  |-  ( m : ( Base `  P
) --> C  ->  m  Fn  ( Base `  P
) )
104101, 102, 1033syl 18 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  m  Fn  ( Base `  P ) )
105 simprr 733 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  n  e.  ( P RingHom  S ) )
1062, 3rhmf 15504 . . . . . . . . 9  |-  ( n  e.  ( P RingHom  S
)  ->  n :
( Base `  P ) --> C )
107 ffn 5389 . . . . . . . . 9  |-  ( n : ( Base `  P
) --> C  ->  n  Fn  ( Base `  P
) )
108105, 106, 1073syl 18 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  n  Fn  ( Base `  P ) )
10978adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  ran  A  C_  ( Base `  P ) )
11081adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  ran  V  C_  ( Base `  P ) )
111109, 110unssd 3351 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  ( ran  A  u.  ran  V )  C_  ( Base `  P )
)
112 fnreseql 5635 . . . . . . . 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 1182 . . . . . . 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 5634 . . . . . . . 8  |-  ( ( m  Fn  ( Base `  P )  /\  n  Fn  ( Base `  P
) )  ->  (
m  =  n  <->  ( Base `  P )  C_  dom  ( m  i^i  n
) ) )
115104, 108, 114syl2anc 642 . . . . . . 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 259 . . . . . 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 28 . . . . 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 2635 . . . 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 4949 . . . . . 6  |-  ( m  =  n  ->  (
m  |`  ( ran  A  u.  ran  V ) )  =  ( n  |`  ( ran  A  u.  ran  V ) ) )
120119eqeq1d 2291 . . . . 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 2958 . . . 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 203 . . 3  |-  ( ph  ->  E* m  e.  ( P RingHom  S ) ( m  |`  ( ran  A  u.  ran  V ) )  =  ( ( F  o.  `' A )  u.  ( G  o.  `' V
) ) )
123 rmoim 2964 . . 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 56 . 2  |-  ( ph  ->  E* m  e.  ( P RingHom  S ) ( ( m  o.  A )  =  F  /\  (
m  o.  V )  =  G ) )
125 reu5 2753 . 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 645 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   E!wreu 2545   E*wrmo 2546   {crab 2547   _Vcvv 2788    u. cun 3150    i^i cin 3151    C_ wss 3152   ~Pcpw 3625    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692    o. ccom 4693   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076    ^m cmap 6772   Fincfn 6863   NNcn 9746   NN0cn0 9965   Basecbs 13148   .rcmulr 13209  Scalarcsca 13211    gsumg cgsu 13401  Moorecmre 13484  mrClscmrc 13485  .gcmg 14366  mulGrpcmgp 15325   Ringcrg 15337   CRingccrg 15338   RingHom crh 15494  SubRingcsubrg 15541  AssAlgcasa 16050  AlgSpancasp 16051  algSccascl 16052   mPwSer cmps 16087   mVar cmvr 16088   mPoly cmpl 16089
This theorem is referenced by:  evlsval2  19404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-rnghom 15496  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-assa 16053  df-asp 16054  df-ascl 16055  df-psr 16098  df-mvr 16099  df-mpl 16100
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