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Theorem evlseu 17600
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 2441 . . . 4  |-  ( Base `  P )  =  (
Base `  P )
3 evlseu.c . . . 4  |-  C  =  ( Base `  S
)
4 eqid 2441 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
5 eqid 2441 . . . 4  |-  { z  e.  ( NN0  ^m  I )  |  ( `' z " NN )  e.  Fin }  =  { z  e.  ( NN0  ^m  I )  |  ( `' z
" NN )  e. 
Fin }
6 eqid 2441 . . . 4  |-  (mulGrp `  S )  =  (mulGrp `  S )
7 eqid 2441 . . . 4  |-  (.g `  (mulGrp `  S ) )  =  (.g `  (mulGrp `  S
) )
8 eqid 2441 . . . 4  |-  ( .r
`  S )  =  ( .r `  S
)
9 evlseu.v . . . 4  |-  V  =  ( I mVar  R )
10 eqid 2441 . . . 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 17599 . . 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 4995 . . . . . . 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 2449 . . . . . 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 4995 . . . . . . 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 2449 . . . . . 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 3071 . . . 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 1183 . . 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 16653 . . . . . . . . . . 11  |-  ( R  e.  CRing  ->  R  e.  Ring )
2712, 26syl 16 . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
28 eqid 2441 . . . . . . . . . . 11  |-  (Scalar `  P )  =  (Scalar `  P )
291mplrng 17529 . . . . . . . . . . 11  |-  ( ( I  e.  _V  /\  R  e.  Ring )  ->  P  e.  Ring )
301mpllmod 17528 . . . . . . . . . . 11  |-  ( ( I  e.  _V  /\  R  e.  Ring )  ->  P  e.  LMod )
31 eqid 2441 . . . . . . . . . . 11  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
3216, 28, 29, 30, 31, 2asclf 17406 . . . . . . . . . 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 5559 . . . . . . . . 9  |-  ( A : ( Base `  (Scalar `  P ) ) --> (
Base `  P )  ->  Fun  A )
3533, 34syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  A )
36 funcoeqres 5669 . . . . . . . 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 17572 . . . . . . . . 9  |-  ( ph  ->  V : I --> ( Base `  P ) )
39 ffun 5559 . . . . . . . . 9  |-  ( V : I --> ( Base `  P )  ->  Fun  V )
4038, 39syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  V )
41 funcoeqres 5669 . . . . . . . 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 833 . . . . . 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 5122 . . . . . 6  |-  ( m  |`  ( ran  A  u.  ran  V ) )  =  ( ( m  |`  ran  A )  u.  (
m  |`  ran  V ) )
46 uneq12 3503 . . . . . 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 2485 . . . . 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 2798 . . 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 2460 . . . . . 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 2441 . . . . . . . . . . . . 13  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
5251, 11, 12psrassa 17484 . . . . . . . . . . . 12  |-  ( ph  ->  ( I mPwSer  R )  e. AssAlg )
53 eqid 2441 . . . . . . . . . . . . . 14  |-  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  R ) )
5451, 9, 53, 11, 27mvrf 17495 . . . . . . . . . . . . 13  |-  ( ph  ->  V : I --> ( Base `  ( I mPwSer  R ) ) )
55 frn 5563 . . . . . . . . . . . . 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 2441 . . . . . . . . . . . . 13  |-  (AlgSpan `  (
I mPwSer  R ) )  =  (AlgSpan `  ( I mPwSer  R ) )
58 eqid 2441 . . . . . . . . . . . . 13  |-  (algSc `  ( I mPwSer  R ) )  =  (algSc `  (
I mPwSer  R ) )
59 eqid 2441 . . . . . . . . . . . . 13  |-  (mrCls `  (SubRing `  ( I mPwSer  R
) ) )  =  (mrCls `  (SubRing `  (
I mPwSer  R ) ) )
6057, 58, 59, 53aspval2 17415 . . . . . . . . . . . 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 17549 . . . . . . . . . . 11  |-  ( ph  ->  ( (AlgSpan `  (
I mPwSer  R ) ) `  ran  V )  =  (
Base `  P )
)
6351, 1, 2, 11, 27mplsubrg 17517 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( Base `  P
)  e.  (SubRing `  (
I mPwSer  R ) ) )
641, 51, 2mplval2 17505 . . . . . . . . . . . . . . . 16  |-  P  =  ( ( I mPwSer  R
)s  ( Base `  P
) )
6564subsubrg2 16890 . . . . . . . . . . . . . . 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 5693 . . . . . . . . . . . . 13  |-  ( ph  ->  (mrCls `  (SubRing `  P
) )  =  (mrCls `  ( (SubRing `  (
I mPwSer  R ) )  i^i 
~P ( Base `  P
) ) ) )
6858, 64ressascl 17412 . . . . . . . . . . . . . . . . 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 2492 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  =  (algSc `  ( I mPwSer  R ) ) )
7170rneqd 5065 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  A  =  ran  (algSc `  ( I mPwSer  R
) ) )
7271uneq1d 3507 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  A  u.  ran  V )  =  ( ran  (algSc `  (
I mPwSer  R ) )  u. 
ran  V ) )
7367, 72fveq12d 5695 . . . . . . . . . . . 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 17390 . . . . . . . . . . . . . 14  |-  ( ( I mPwSer  R )  e. AssAlg  ->  ( I mPwSer  R )  e.  Ring )
7553subrgmre 16887 . . . . . . . . . . . . . 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 5563 . . . . . . . . . . . . . . . 16  |-  ( A : ( Base `  (Scalar `  P ) ) --> (
Base `  P )  ->  ran  A  C_  ( Base `  P ) )
7833, 77syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  A  C_  ( Base `  P ) )
7971, 78eqsstr3d 3389 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  (algSc `  (
I mPwSer  R ) )  C_  ( Base `  P )
)
80 frn 5563 . . . . . . . . . . . . . . 15  |-  ( V : I --> ( Base `  P )  ->  ran  V 
C_  ( Base `  P
) )
8138, 80syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  V  C_  ( Base `  P ) )
8279, 81unssd 3530 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  (algSc `  ( I mPwSer  R ) )  u.  ran  V ) 
C_  ( Base `  P
) )
83 eqid 2441 . . . . . . . . . . . . . 14  |-  (mrCls `  ( (SubRing `  ( I mPwSer  R ) )  i^i  ~P ( Base `  P )
) )  =  (mrCls `  ( (SubRing `  (
I mPwSer  R ) )  i^i 
~P ( Base `  P
) ) )
8459, 83submrc 14564 . . . . . . . . . . . . 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 1218 . . . . . . . . . . . 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 2474 . . . . . . . . . . 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 2481 . . . . . . . . . 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 16887 . . . . . . . . . . . 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 16893 . . . . . . . . . . 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 2441 . . . . . . . . . . 11  |-  (mrCls `  (SubRing `  P ) )  =  (mrCls `  (SubRing `  P ) )
9796mrcsscl 14556 . . . . . . . . . 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 1218 . . . . . . . . 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 3388 . . . . . . . 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 16814 . . . . . . . . 9  |-  ( m  e.  ( P RingHom  S
)  ->  m :
( Base `  P ) --> C )
103 ffn 5557 . . . . . . . . 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 16814 . . . . . . . . 9  |-  ( n  e.  ( P RingHom  S
)  ->  n :
( Base `  P ) --> C )
107 ffn 5557 . . . . . . . . 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 3530 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  ( ran  A  u.  ran  V )  C_  ( Base `  P )
)
112 fnreseql 5811 . . . . . . . 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 1218 . . . . . . 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 5810 . . . . . . . 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 2806 . . . 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 5102 . . . . . 6  |-  ( m  =  n  ->  (
m  |`  ( ran  A  u.  ran  V ) )  =  ( n  |`  ( ran  A  u.  ran  V ) ) )
120119eqeq1d 2449 . . . . 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 3150 . . . 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 3156 . . 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 2934 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2713   E.wrex 2714   E!wreu 2715   E*wrmo 2716   {crab 2717   _Vcvv 2970    u. cun 3324    i^i cin 3325    C_ wss 3326   ~Pcpw 3858    e. cmpt 4348   `'ccnv 4837   dom cdm 4838   ran crn 4839    |` cres 4840   "cima 4841    o. ccom 4842   Fun wfun 5410    Fn wfn 5411   -->wf 5412   ` cfv 5416  (class class class)co 6089    oFcof 6316    ^m cmap 7212   Fincfn 7308   NNcn 10320   NN0cn0 10577   Basecbs 14172   .rcmulr 14237  Scalarcsca 14239    gsumg cgsu 14377  Moorecmre 14518  mrClscmrc 14519  .gcmg 15412  mulGrpcmgp 16589   Ringcrg 16643   CRingccrg 16644   RingHom crh 16802  SubRingcsubrg 16859  AssAlgcasa 17379  AlgSpancasp 17380  algSccascl 17381   mPwSer cmps 17416   mVar cmvr 17417   mPoly cmpl 17418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-iin 4172  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-ofr 6319  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-er 7099  df-map 7214  df-pm 7215  df-ixp 7262  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fsupp 7619  df-oi 7722  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-seq 11805  df-hash 12102  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-sca 14252  df-vsca 14253  df-tset 14255  df-0g 14378  df-gsum 14379  df-mre 14522  df-mrc 14523  df-acs 14525  df-mnd 15413  df-mhm 15462  df-submnd 15463  df-grp 15543  df-minusg 15544  df-sbg 15545  df-mulg 15546  df-subg 15676  df-ghm 15743  df-cntz 15833  df-cmn 16277  df-abl 16278  df-mgp 16590  df-ur 16602  df-srg 16606  df-rng 16645  df-cring 16646  df-rnghom 16804  df-subrg 16861  df-lmod 16948  df-lss 17012  df-lsp 17051  df-assa 17382  df-asp 17383  df-ascl 17384  df-psr 17421  df-mvr 17422  df-mpl 17423
This theorem is referenced by:  evlsval2  17604
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