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Theorem evlseu 18312
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 2457 . . . 4  |-  ( Base `  P )  =  (
Base `  P )
3 evlseu.c . . . 4  |-  C  =  ( Base `  S
)
4 eqid 2457 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
5 eqid 2457 . . . 4  |-  { z  e.  ( NN0  ^m  I )  |  ( `' z " NN )  e.  Fin }  =  { z  e.  ( NN0  ^m  I )  |  ( `' z
" NN )  e. 
Fin }
6 eqid 2457 . . . 4  |-  (mulGrp `  S )  =  (mulGrp `  S )
7 eqid 2457 . . . 4  |-  (.g `  (mulGrp `  S ) )  =  (.g `  (mulGrp `  S
) )
8 eqid 2457 . . . 4  |-  ( .r
`  S )  =  ( .r `  S
)
9 evlseu.v . . . 4  |-  V  =  ( I mVar  R )
10 eqid 2457 . . . 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 18311 . . 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 5170 . . . . . . 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 2459 . . . . . 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 5170 . . . . . . 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 2459 . . . . . 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 3210 . . . 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 crngring 17336 . . . . . . . . . . 11  |-  ( R  e.  CRing  ->  R  e.  Ring )
2712, 26syl 16 . . . . . . . . . 10  |-  ( ph  ->  R  e.  Ring )
28 eqid 2457 . . . . . . . . . . 11  |-  (Scalar `  P )  =  (Scalar `  P )
291mplring 18241 . . . . . . . . . . 11  |-  ( ( I  e.  _V  /\  R  e.  Ring )  ->  P  e.  Ring )
301mpllmod 18240 . . . . . . . . . . 11  |-  ( ( I  e.  _V  /\  R  e.  Ring )  ->  P  e.  LMod )
31 eqid 2457 . . . . . . . . . . 11  |-  ( Base `  (Scalar `  P )
)  =  ( Base `  (Scalar `  P )
)
3216, 28, 29, 30, 31, 2asclf 18113 . . . . . . . . . 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 5739 . . . . . . . . 9  |-  ( A : ( Base `  (Scalar `  P ) ) --> (
Base `  P )  ->  Fun  A )
3533, 34syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  A )
36 funcoeqres 5852 . . . . . . . 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 18284 . . . . . . . . 9  |-  ( ph  ->  V : I --> ( Base `  P ) )
39 ffun 5739 . . . . . . . . 9  |-  ( V : I --> ( Base `  P )  ->  Fun  V )
4038, 39syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  V )
41 funcoeqres 5852 . . . . . . . 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 837 . . . . . 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 5297 . . . . . 6  |-  ( m  |`  ( ran  A  u.  ran  V ) )  =  ( ( m  |`  ran  A )  u.  (
m  |`  ran  V ) )
46 uneq12 3649 . . . . . 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 2510 . . . . 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 2872 . . 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 2485 . . . . . 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 2457 . . . . . . . . . . . . 13  |-  ( I mPwSer  R )  =  ( I mPwSer  R )
5251, 11, 12psrassa 18196 . . . . . . . . . . . 12  |-  ( ph  ->  ( I mPwSer  R )  e. AssAlg )
53 eqid 2457 . . . . . . . . . . . . . 14  |-  ( Base `  ( I mPwSer  R ) )  =  ( Base `  ( I mPwSer  R ) )
5451, 9, 53, 11, 27mvrf 18207 . . . . . . . . . . . . 13  |-  ( ph  ->  V : I --> ( Base `  ( I mPwSer  R ) ) )
55 frn 5743 . . . . . . . . . . . . 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 2457 . . . . . . . . . . . . 13  |-  (AlgSpan `  (
I mPwSer  R ) )  =  (AlgSpan `  ( I mPwSer  R ) )
58 eqid 2457 . . . . . . . . . . . . 13  |-  (algSc `  ( I mPwSer  R ) )  =  (algSc `  (
I mPwSer  R ) )
59 eqid 2457 . . . . . . . . . . . . 13  |-  (mrCls `  (SubRing `  ( I mPwSer  R
) ) )  =  (mrCls `  (SubRing `  (
I mPwSer  R ) ) )
6057, 58, 59, 53aspval2 18123 . . . . . . . . . . . 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 18261 . . . . . . . . . . 11  |-  ( ph  ->  ( (AlgSpan `  (
I mPwSer  R ) ) `  ran  V )  =  (
Base `  P )
)
6351, 1, 2, 11, 27mplsubrg 18229 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( Base `  P
)  e.  (SubRing `  (
I mPwSer  R ) ) )
641, 51, 2mplval2 18217 . . . . . . . . . . . . . . . 16  |-  P  =  ( ( I mPwSer  R
)s  ( Base `  P
) )
6564subsubrg2 17583 . . . . . . . . . . . . . . 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 5876 . . . . . . . . . . . . 13  |-  ( ph  ->  (mrCls `  (SubRing `  P
) )  =  (mrCls `  ( (SubRing `  (
I mPwSer  R ) )  i^i 
~P ( Base `  P
) ) ) )
6858, 64ressascl 18120 . . . . . . . . . . . . . . . . 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 2517 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  =  (algSc `  ( I mPwSer  R ) ) )
7170rneqd 5240 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  A  =  ran  (algSc `  ( I mPwSer  R
) ) )
7271uneq1d 3653 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  A  u.  ran  V )  =  ( ran  (algSc `  (
I mPwSer  R ) )  u. 
ran  V ) )
7367, 72fveq12d 5878 . . . . . . . . . . . 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 assaring 18096 . . . . . . . . . . . . . 14  |-  ( ( I mPwSer  R )  e. AssAlg  ->  ( I mPwSer  R )  e.  Ring )
7553subrgmre 17580 . . . . . . . . . . . . . 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 5743 . . . . . . . . . . . . . . . 16  |-  ( A : ( Base `  (Scalar `  P ) ) --> (
Base `  P )  ->  ran  A  C_  ( Base `  P ) )
7833, 77syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ran  A  C_  ( Base `  P ) )
7971, 78eqsstr3d 3534 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  (algSc `  (
I mPwSer  R ) )  C_  ( Base `  P )
)
80 frn 5743 . . . . . . . . . . . . . . 15  |-  ( V : I --> ( Base `  P )  ->  ran  V 
C_  ( Base `  P
) )
8138, 80syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  V  C_  ( Base `  P ) )
8279, 81unssd 3676 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ran  (algSc `  ( I mPwSer  R ) )  u.  ran  V ) 
C_  ( Base `  P
) )
83 eqid 2457 . . . . . . . . . . . . . 14  |-  (mrCls `  ( (SubRing `  ( I mPwSer  R ) )  i^i  ~P ( Base `  P )
) )  =  (mrCls `  ( (SubRing `  (
I mPwSer  R ) )  i^i 
~P ( Base `  P
) ) )
8459, 83submrc 15045 . . . . . . . . . . . . 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 2499 . . . . . . . . . . 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 2506 . . . . . . . . . 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 17580 . . . . . . . . . . . 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 17586 . . . . . . . . . . 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 2457 . . . . . . . . . . 11  |-  (mrCls `  (SubRing `  P ) )  =  (mrCls `  (SubRing `  P ) )
9796mrcsscl 15037 . . . . . . . . . 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 3533 . . . . . . . 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 756 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  m  e.  ( P RingHom  S ) )
1022, 3rhmf 17502 . . . . . . . . 9  |-  ( m  e.  ( P RingHom  S
)  ->  m :
( Base `  P ) --> C )
103 ffn 5737 . . . . . . . . 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 757 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  n  e.  ( P RingHom  S ) )
1062, 3rhmf 17502 . . . . . . . . 9  |-  ( n  e.  ( P RingHom  S
)  ->  n :
( Base `  P ) --> C )
107 ffn 5737 . . . . . . . . 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 3676 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  ( P RingHom  S )  /\  n  e.  ( P RingHom  S ) ) )  ->  ( ran  A  u.  ran  V )  C_  ( Base `  P )
)
112 fnreseql 5998 . . . . . . . 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 5997 . . . . . . . 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 2878 . . . 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 5277 . . . . . 6  |-  ( m  =  n  ->  (
m  |`  ( ran  A  u.  ran  V ) )  =  ( n  |`  ( ran  A  u.  ran  V ) ) )
120119eqeq1d 2459 . . . . 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 3292 . . . 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 3299 . . 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 3073 . 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 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   E!wreu 2809   E*wrmo 2810   {crab 2811   _Vcvv 3109    u. cun 3469    i^i cin 3470    C_ wss 3471   ~Pcpw 4015    |-> cmpt 4515   `'ccnv 5007   dom cdm 5008   ran crn 5009    |` cres 5010   "cima 5011    o. ccom 5012   Fun wfun 5588    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296    oFcof 6537    ^m cmap 7438   Fincfn 7535   NNcn 10556   NN0cn0 10816   Basecbs 14644   .rcmulr 14713  Scalarcsca 14715    gsumg cgsu 14858  Moorecmre 14999  mrClscmrc 15000  .gcmg 16183  mulGrpcmgp 17268   Ringcrg 17325   CRingccrg 17326   RingHom crh 17488  SubRingcsubrg 17552  AssAlgcasa 18085  AlgSpancasp 18086  algSccascl 18087   mPwSer cmps 18127   mVar cmvr 18128   mPoly cmpl 18129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-ofr 6540  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-sca 14728  df-vsca 14729  df-tset 14731  df-0g 14859  df-gsum 14860  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-submnd 16094  df-grp 16184  df-minusg 16185  df-sbg 16186  df-mulg 16187  df-subg 16325  df-ghm 16392  df-cntz 16482  df-cmn 16927  df-abl 16928  df-mgp 17269  df-ur 17281  df-srg 17285  df-ring 17327  df-cring 17328  df-rnghom 17491  df-subrg 17554  df-lmod 17641  df-lss 17706  df-lsp 17745  df-assa 18088  df-asp 18089  df-ascl 18090  df-psr 18132  df-mvr 18133  df-mpl 18134
This theorem is referenced by:  evlsval2  18316
  Copyright terms: Public domain W3C validator