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Theorem evlslem3 18737
Description: Lemma for evlseu 18739. Polynomial evaluation of a scaled monomial. (Contributed by Stefan O'Rear, 8-Mar-2015.)
Hypotheses
Ref Expression
evlslem1.p  |-  P  =  ( I mPoly  R )
evlslem1.b  |-  B  =  ( Base `  P
)
evlslem1.c  |-  C  =  ( Base `  S
)
evlslem1.k  |-  K  =  ( Base `  R
)
evlslem1.d  |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
evlslem1.t  |-  T  =  (mulGrp `  S )
evlslem1.x  |-  .^  =  (.g
`  T )
evlslem1.m  |-  .x.  =  ( .r `  S )
evlslem1.v  |-  V  =  ( I mVar  R )
evlslem1.e  |-  E  =  ( p  e.  B  |->  ( S  gsumg  ( b  e.  D  |->  ( ( F `  ( p `  b
) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) ) )
evlslem1.i  |-  ( ph  ->  I  e.  _V )
evlslem1.r  |-  ( ph  ->  R  e.  CRing )
evlslem1.s  |-  ( ph  ->  S  e.  CRing )
evlslem1.f  |-  ( ph  ->  F  e.  ( R RingHom  S ) )
evlslem1.g  |-  ( ph  ->  G : I --> C )
evlslem3.z  |-  .0.  =  ( 0g `  R )
evlslem3.k  |-  ( ph  ->  A  e.  D )
evlslem3.q  |-  ( ph  ->  H  e.  K )
Assertion
Ref Expression
evlslem3  |-  ( ph  ->  ( E `  (
x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) )  =  ( ( F `  H )  .x.  ( T  gsumg  ( A  oF 
.^  G ) ) ) )
Distinct variable groups:    p, b, x,  .0.    B, p    C, b    D, b, p, x    F, b, p    .^ , b, p   
h, b, A, p, x    h, I    x, K    ph, b, x    G, b, p    H, b, p, x    S, b, p    T, b, p    .x. , b, p   
x, R
Allowed substitution hints:    ph( h, p)    B( x, h, b)    C( x, h, p)    D( h)    P( x, h, p, b)    R( h, p, b)    S( x, h)    T( x, h)    .x. ( x, h)    E( x, h, p, b)    .^ ( x, h)    F( x, h)    G( x, h)    H( h)    I( x, p, b)    K( h, p, b)    V( x, h, p, b)    .0. ( h)

Proof of Theorem evlslem3
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlslem1.p . . . 4  |-  P  =  ( I mPoly  R )
2 evlslem1.d . . . 4  |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
3 evlslem3.z . . . 4  |-  .0.  =  ( 0g `  R )
4 evlslem1.k . . . 4  |-  K  =  ( Base `  R
)
5 evlslem1.i . . . 4  |-  ( ph  ->  I  e.  _V )
6 evlslem1.r . . . . 5  |-  ( ph  ->  R  e.  CRing )
7 crngring 17791 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
86, 7syl 17 . . . 4  |-  ( ph  ->  R  e.  Ring )
9 evlslem1.b . . . 4  |-  B  =  ( Base `  P
)
10 evlslem3.q . . . 4  |-  ( ph  ->  H  e.  K )
11 evlslem3.k . . . 4  |-  ( ph  ->  A  e.  D )
121, 2, 3, 4, 5, 8, 9, 10, 11mplmon2cl 18723 . . 3  |-  ( ph  ->  ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) )  e.  B
)
13 fveq1 5864 . . . . . . . 8  |-  ( p  =  ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) )  -> 
( p `  b
)  =  ( ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) `  b
) )
1413fveq2d 5869 . . . . . . 7  |-  ( p  =  ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) )  -> 
( F `  (
p `  b )
)  =  ( F `
 ( ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  )
) `  b )
) )
1514oveq1d 6305 . . . . . 6  |-  ( p  =  ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) )  -> 
( ( F `  ( p `  b
) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) )  =  ( ( F `  ( ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) `  b
) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) )
1615mpteq2dv 4490 . . . . 5  |-  ( p  =  ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) )  -> 
( b  e.  D  |->  ( ( F `  ( p `  b
) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) )  =  ( b  e.  D  |->  ( ( F `  (
( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) `  b
) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) )
1716oveq2d 6306 . . . 4  |-  ( p  =  ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) )  -> 
( S  gsumg  ( b  e.  D  |->  ( ( F `  ( p `  b
) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) )  =  ( S  gsumg  ( b  e.  D  |->  ( ( F `  ( ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) `  b ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) ) )
18 evlslem1.e . . . 4  |-  E  =  ( p  e.  B  |->  ( S  gsumg  ( b  e.  D  |->  ( ( F `  ( p `  b
) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) ) )
19 ovex 6318 . . . 4  |-  ( S 
gsumg  ( b  e.  D  |->  ( ( F `  ( ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) `  b ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) )  e. 
_V
2017, 18, 19fvmpt 5948 . . 3  |-  ( ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) )  e.  B  ->  ( E `  (
x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) )  =  ( S  gsumg  ( b  e.  D  |->  ( ( F `  ( ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) `  b ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) ) )
2112, 20syl 17 . 2  |-  ( ph  ->  ( E `  (
x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) )  =  ( S  gsumg  ( b  e.  D  |->  ( ( F `  ( ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) `  b ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) ) )
22 simpr 463 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  b  e.  D )
23 fvex 5875 . . . . . . . . . . . 12  |-  ( 0g
`  R )  e. 
_V
243, 23eqeltri 2525 . . . . . . . . . . 11  |-  .0.  e.  _V
2524a1i 11 . . . . . . . . . 10  |-  ( ph  ->  .0.  e.  _V )
26 ifexg 3950 . . . . . . . . . 10  |-  ( ( H  e.  K  /\  .0.  e.  _V )  ->  if ( b  =  A ,  H ,  .0.  )  e.  _V )
2710, 25, 26syl2anc 667 . . . . . . . . 9  |-  ( ph  ->  if ( b  =  A ,  H ,  .0.  )  e.  _V )
2827adantr 467 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  if ( b  =  A ,  H ,  .0.  )  e.  _V )
29 eqeq1 2455 . . . . . . . . . 10  |-  ( x  =  b  ->  (
x  =  A  <->  b  =  A ) )
3029ifbid 3903 . . . . . . . . 9  |-  ( x  =  b  ->  if ( x  =  A ,  H ,  .0.  )  =  if ( b  =  A ,  H ,  .0.  ) )
31 eqid 2451 . . . . . . . . 9  |-  ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  )
)  =  ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  )
)
3230, 31fvmptg 5946 . . . . . . . 8  |-  ( ( b  e.  D  /\  if ( b  =  A ,  H ,  .0.  )  e.  _V )  ->  ( ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) `  b )  =  if ( b  =  A ,  H ,  .0.  ) )
3322, 28, 32syl2anc 667 . . . . . . 7  |-  ( (
ph  /\  b  e.  D )  ->  (
( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) `  b
)  =  if ( b  =  A ,  H ,  .0.  )
)
3433fveq2d 5869 . . . . . 6  |-  ( (
ph  /\  b  e.  D )  ->  ( F `  ( (
x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) `  b
) )  =  ( F `  if ( b  =  A ,  H ,  .0.  )
) )
3534oveq1d 6305 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  (
( F `  (
( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) `  b
) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) )  =  ( ( F `  if ( b  =  A ,  H ,  .0.  )
)  .x.  ( T  gsumg  ( b  oF  .^  G ) ) ) )
3635mpteq2dva 4489 . . . 4  |-  ( ph  ->  ( b  e.  D  |->  ( ( F `  ( ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) `  b ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) )  =  ( b  e.  D  |->  ( ( F `  if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) )
3736oveq2d 6306 . . 3  |-  ( ph  ->  ( S  gsumg  ( b  e.  D  |->  ( ( F `  ( ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) `  b ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) )  =  ( S  gsumg  ( b  e.  D  |->  ( ( F `  if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) ) )
38 evlslem1.c . . . 4  |-  C  =  ( Base `  S
)
39 eqid 2451 . . . 4  |-  ( 0g
`  S )  =  ( 0g `  S
)
40 evlslem1.s . . . . . 6  |-  ( ph  ->  S  e.  CRing )
41 crngring 17791 . . . . . 6  |-  ( S  e.  CRing  ->  S  e.  Ring )
4240, 41syl 17 . . . . 5  |-  ( ph  ->  S  e.  Ring )
43 ringmnd 17789 . . . . 5  |-  ( S  e.  Ring  ->  S  e. 
Mnd )
4442, 43syl 17 . . . 4  |-  ( ph  ->  S  e.  Mnd )
45 ovex 6318 . . . . . 6  |-  ( NN0 
^m  I )  e. 
_V
462, 45rabex2 4556 . . . . 5  |-  D  e. 
_V
4746a1i 11 . . . 4  |-  ( ph  ->  D  e.  _V )
4842adantr 467 . . . . . 6  |-  ( (
ph  /\  b  e.  D )  ->  S  e.  Ring )
49 evlslem1.f . . . . . . . . 9  |-  ( ph  ->  F  e.  ( R RingHom  S ) )
504, 38rhmf 17954 . . . . . . . . 9  |-  ( F  e.  ( R RingHom  S
)  ->  F : K
--> C )
5149, 50syl 17 . . . . . . . 8  |-  ( ph  ->  F : K --> C )
524, 3ring0cl 17802 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  .0.  e.  K )
538, 52syl 17 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  K )
5410, 53ifcld 3924 . . . . . . . 8  |-  ( ph  ->  if ( b  =  A ,  H ,  .0.  )  e.  K
)
5551, 54ffvelrnd 6023 . . . . . . 7  |-  ( ph  ->  ( F `  if ( b  =  A ,  H ,  .0.  ) )  e.  C
)
5655adantr 467 . . . . . 6  |-  ( (
ph  /\  b  e.  D )  ->  ( F `  if (
b  =  A ,  H ,  .0.  )
)  e.  C )
57 evlslem1.t . . . . . . . 8  |-  T  =  (mulGrp `  S )
5857, 38mgpbas 17729 . . . . . . 7  |-  C  =  ( Base `  T
)
59 eqid 2451 . . . . . . 7  |-  ( 0g
`  T )  =  ( 0g `  T
)
6057crngmgp 17788 . . . . . . . . 9  |-  ( S  e.  CRing  ->  T  e. CMnd )
6140, 60syl 17 . . . . . . . 8  |-  ( ph  ->  T  e. CMnd )
6261adantr 467 . . . . . . 7  |-  ( (
ph  /\  b  e.  D )  ->  T  e. CMnd )
635adantr 467 . . . . . . 7  |-  ( (
ph  /\  b  e.  D )  ->  I  e.  _V )
64 cmnmnd 17445 . . . . . . . . . . 11  |-  ( T  e. CMnd  ->  T  e.  Mnd )
6561, 64syl 17 . . . . . . . . . 10  |-  ( ph  ->  T  e.  Mnd )
6665ad2antrr 732 . . . . . . . . 9  |-  ( ( ( ph  /\  b  e.  D )  /\  (
y  e.  NN0  /\  z  e.  C )
)  ->  T  e.  Mnd )
67 simprl 764 . . . . . . . . 9  |-  ( ( ( ph  /\  b  e.  D )  /\  (
y  e.  NN0  /\  z  e.  C )
)  ->  y  e.  NN0 )
68 simprr 766 . . . . . . . . 9  |-  ( ( ( ph  /\  b  e.  D )  /\  (
y  e.  NN0  /\  z  e.  C )
)  ->  z  e.  C )
69 evlslem1.x . . . . . . . . . 10  |-  .^  =  (.g
`  T )
7058, 69mulgnn0cl 16774 . . . . . . . . 9  |-  ( ( T  e.  Mnd  /\  y  e.  NN0  /\  z  e.  C )  ->  (
y  .^  z )  e.  C )
7166, 67, 68, 70syl3anc 1268 . . . . . . . 8  |-  ( ( ( ph  /\  b  e.  D )  /\  (
y  e.  NN0  /\  z  e.  C )
)  ->  ( y  .^  z )  e.  C
)
722psrbagf 18589 . . . . . . . . 9  |-  ( ( I  e.  _V  /\  b  e.  D )  ->  b : I --> NN0 )
735, 72sylan 474 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  b : I --> NN0 )
74 evlslem1.g . . . . . . . . 9  |-  ( ph  ->  G : I --> C )
7574adantr 467 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  G : I --> C )
76 inidm 3641 . . . . . . . 8  |-  ( I  i^i  I )  =  I
7771, 73, 75, 63, 63, 76off 6546 . . . . . . 7  |-  ( (
ph  /\  b  e.  D )  ->  (
b  oF  .^  G ) : I --> C )
78 ovex 6318 . . . . . . . . 9  |-  ( b  oF  .^  G
)  e.  _V
7978a1i 11 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  (
b  oF  .^  G )  e.  _V )
80 ffun 5731 . . . . . . . . 9  |-  ( ( b  oF  .^  G ) : I --> C  ->  Fun  ( b  oF  .^  G
) )
8177, 80syl 17 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  Fun  ( b  oF 
.^  G ) )
82 fvex 5875 . . . . . . . . 9  |-  ( 0g
`  T )  e. 
_V
8382a1i 11 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  ( 0g `  T )  e. 
_V )
842psrbag 18588 . . . . . . . . . 10  |-  ( I  e.  _V  ->  (
b  e.  D  <->  ( b : I --> NN0  /\  ( `' b " NN )  e.  Fin )
) )
855, 84syl 17 . . . . . . . . 9  |-  ( ph  ->  ( b  e.  D  <->  ( b : I --> NN0  /\  ( `' b " NN )  e.  Fin )
) )
8685simplbda 630 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  ( `' b " NN )  e.  Fin )
87 ffn 5728 . . . . . . . . . . . . 13  |-  ( b : I --> NN0  ->  b  Fn  I )
8873, 87syl 17 . . . . . . . . . . . 12  |-  ( (
ph  /\  b  e.  D )  ->  b  Fn  I )
8988adantr 467 . . . . . . . . . . 11  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
b  Fn  I )
90 ffn 5728 . . . . . . . . . . . . 13  |-  ( G : I --> C  ->  G  Fn  I )
9174, 90syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  G  Fn  I )
9291ad2antrr 732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  ->  G  Fn  I )
935ad2antrr 732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  ->  I  e.  _V )
94 eldifi 3555 . . . . . . . . . . . 12  |-  ( y  e.  ( I  \ 
( `' b " NN ) )  ->  y  e.  I )
9594adantl 468 . . . . . . . . . . 11  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
y  e.  I )
96 fnfvof 6545 . . . . . . . . . . 11  |-  ( ( ( b  Fn  I  /\  G  Fn  I
)  /\  ( I  e.  _V  /\  y  e.  I ) )  -> 
( ( b  oF  .^  G ) `  y )  =  ( ( b `  y
)  .^  ( G `  y ) ) )
9789, 92, 93, 95, 96syl22anc 1269 . . . . . . . . . 10  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
( ( b  oF  .^  G ) `  y )  =  ( ( b `  y
)  .^  ( G `  y ) ) )
98 eldifn 3556 . . . . . . . . . . . . . 14  |-  ( y  e.  ( I  \ 
( `' b " NN ) )  ->  -.  y  e.  ( `' b " NN ) )
9998adantl 468 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  ->  -.  y  e.  ( `' b " NN ) )
10094ad2antlr 733 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  b  e.  D )  /\  y  e.  (
I  \  ( `' b " NN ) ) )  /\  ( b `
 y )  e.  NN )  ->  y  e.  I )
101 simpr 463 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  b  e.  D )  /\  y  e.  (
I  \  ( `' b " NN ) ) )  /\  ( b `
 y )  e.  NN )  ->  (
b `  y )  e.  NN )
10288ad2antrr 732 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  b  e.  D )  /\  y  e.  (
I  \  ( `' b " NN ) ) )  /\  ( b `
 y )  e.  NN )  ->  b  Fn  I )
103 elpreima 6002 . . . . . . . . . . . . . . 15  |-  ( b  Fn  I  ->  (
y  e.  ( `' b " NN )  <-> 
( y  e.  I  /\  ( b `  y
)  e.  NN ) ) )
104102, 103syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  b  e.  D )  /\  y  e.  (
I  \  ( `' b " NN ) ) )  /\  ( b `
 y )  e.  NN )  ->  (
y  e.  ( `' b " NN )  <-> 
( y  e.  I  /\  ( b `  y
)  e.  NN ) ) )
105100, 101, 104mpbir2and 933 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  b  e.  D )  /\  y  e.  (
I  \  ( `' b " NN ) ) )  /\  ( b `
 y )  e.  NN )  ->  y  e.  ( `' b " NN ) )
10699, 105mtand 665 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  ->  -.  ( b `  y
)  e.  NN )
107 ffvelrn 6020 . . . . . . . . . . . . . 14  |-  ( ( b : I --> NN0  /\  y  e.  I )  ->  ( b `  y
)  e.  NN0 )
10873, 94, 107syl2an 480 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
( b `  y
)  e.  NN0 )
109 elnn0 10871 . . . . . . . . . . . . 13  |-  ( ( b `  y )  e.  NN0  <->  ( ( b `
 y )  e.  NN  \/  ( b `
 y )  =  0 ) )
110108, 109sylib 200 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
( ( b `  y )  e.  NN  \/  ( b `  y
)  =  0 ) )
111 orel1 384 . . . . . . . . . . . 12  |-  ( -.  ( b `  y
)  e.  NN  ->  ( ( ( b `  y )  e.  NN  \/  ( b `  y
)  =  0 )  ->  ( b `  y )  =  0 ) )
112106, 110, 111sylc 62 . . . . . . . . . . 11  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
( b `  y
)  =  0 )
113112oveq1d 6305 . . . . . . . . . 10  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
( ( b `  y )  .^  ( G `  y )
)  =  ( 0 
.^  ( G `  y ) ) )
114 ffvelrn 6020 . . . . . . . . . . . 12  |-  ( ( G : I --> C  /\  y  e.  I )  ->  ( G `  y
)  e.  C )
11575, 94, 114syl2an 480 . . . . . . . . . . 11  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
( G `  y
)  e.  C )
11658, 59, 69mulg0 16763 . . . . . . . . . . 11  |-  ( ( G `  y )  e.  C  ->  (
0  .^  ( G `  y ) )  =  ( 0g `  T
) )
117115, 116syl 17 . . . . . . . . . 10  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
( 0  .^  ( G `  y )
)  =  ( 0g
`  T ) )
11897, 113, 1173eqtrd 2489 . . . . . . . . 9  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
( ( b  oF  .^  G ) `  y )  =  ( 0g `  T ) )
11977, 118suppss 6945 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  (
( b  oF 
.^  G ) supp  ( 0g `  T ) ) 
C_  ( `' b
" NN ) )
120 suppssfifsupp 7898 . . . . . . . 8  |-  ( ( ( ( b  oF  .^  G )  e.  _V  /\  Fun  (
b  oF  .^  G )  /\  ( 0g `  T )  e. 
_V )  /\  (
( `' b " NN )  e.  Fin  /\  ( ( b  oF  .^  G ) supp  ( 0g `  T ) )  C_  ( `' b " NN ) ) )  ->  ( b  oF  .^  G ) finSupp 
( 0g `  T
) )
12179, 81, 83, 86, 119, 120syl32anc 1276 . . . . . . 7  |-  ( (
ph  /\  b  e.  D )  ->  (
b  oF  .^  G ) finSupp  ( 0g `  T ) )
12258, 59, 62, 63, 77, 121gsumcl 17549 . . . . . 6  |-  ( (
ph  /\  b  e.  D )  ->  ( T  gsumg  ( b  oF 
.^  G ) )  e.  C )
123 evlslem1.m . . . . . . 7  |-  .x.  =  ( .r `  S )
12438, 123ringcl 17794 . . . . . 6  |-  ( ( S  e.  Ring  /\  ( F `  if (
b  =  A ,  H ,  .0.  )
)  e.  C  /\  ( T  gsumg  ( b  oF 
.^  G ) )  e.  C )  -> 
( ( F `  if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) )  e.  C )
12548, 56, 122, 124syl3anc 1268 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  (
( F `  if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) )  e.  C )
126 eqid 2451 . . . . 5  |-  ( b  e.  D  |->  ( ( F `  if ( b  =  A ,  H ,  .0.  )
)  .x.  ( T  gsumg  ( b  oF  .^  G ) ) ) )  =  ( b  e.  D  |->  ( ( F `  if ( b  =  A ,  H ,  .0.  )
)  .x.  ( T  gsumg  ( b  oF  .^  G ) ) ) )
127125, 126fmptd 6046 . . . 4  |-  ( ph  ->  ( b  e.  D  |->  ( ( F `  if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) : D --> C )
128 eldifsni 4098 . . . . . . . . . . . 12  |-  ( b  e.  ( D  \  { A } )  -> 
b  =/=  A )
129128neneqd 2629 . . . . . . . . . . 11  |-  ( b  e.  ( D  \  { A } )  ->  -.  b  =  A
)
130129iffalsed 3892 . . . . . . . . . 10  |-  ( b  e.  ( D  \  { A } )  ->  if ( b  =  A ,  H ,  .0.  )  =  .0.  )
131130adantl 468 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  ( D  \  { A } ) )  ->  if ( b  =  A ,  H ,  .0.  )  =  .0.  )
132131fveq2d 5869 . . . . . . . 8  |-  ( (
ph  /\  b  e.  ( D  \  { A } ) )  -> 
( F `  if ( b  =  A ,  H ,  .0.  ) )  =  ( F `  .0.  )
)
133 rhmghm 17953 . . . . . . . . . . 11  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
13449, 133syl 17 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ( R 
GrpHom  S ) )
1353, 39ghmid 16889 . . . . . . . . . 10  |-  ( F  e.  ( R  GrpHom  S )  ->  ( F `  .0.  )  =  ( 0g `  S ) )
136134, 135syl 17 . . . . . . . . 9  |-  ( ph  ->  ( F `  .0.  )  =  ( 0g `  S ) )
137136adantr 467 . . . . . . . 8  |-  ( (
ph  /\  b  e.  ( D  \  { A } ) )  -> 
( F `  .0.  )  =  ( 0g `  S ) )
138132, 137eqtrd 2485 . . . . . . 7  |-  ( (
ph  /\  b  e.  ( D  \  { A } ) )  -> 
( F `  if ( b  =  A ,  H ,  .0.  ) )  =  ( 0g `  S ) )
139138oveq1d 6305 . . . . . 6  |-  ( (
ph  /\  b  e.  ( D  \  { A } ) )  -> 
( ( F `  if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) )  =  ( ( 0g `  S ) 
.x.  ( T  gsumg  ( b  oF  .^  G
) ) ) )
14042adantr 467 . . . . . . 7  |-  ( (
ph  /\  b  e.  ( D  \  { A } ) )  ->  S  e.  Ring )
141 eldifi 3555 . . . . . . . 8  |-  ( b  e.  ( D  \  { A } )  -> 
b  e.  D )
142141, 122sylan2 477 . . . . . . 7  |-  ( (
ph  /\  b  e.  ( D  \  { A } ) )  -> 
( T  gsumg  ( b  oF 
.^  G ) )  e.  C )
14338, 123, 39ringlz 17817 . . . . . . 7  |-  ( ( S  e.  Ring  /\  ( T  gsumg  ( b  oF 
.^  G ) )  e.  C )  -> 
( ( 0g `  S )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) )  =  ( 0g
`  S ) )
144140, 142, 143syl2anc 667 . . . . . 6  |-  ( (
ph  /\  b  e.  ( D  \  { A } ) )  -> 
( ( 0g `  S )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) )  =  ( 0g
`  S ) )
145139, 144eqtrd 2485 . . . . 5  |-  ( (
ph  /\  b  e.  ( D  \  { A } ) )  -> 
( ( F `  if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) )  =  ( 0g
`  S ) )
146145, 47suppss2 6949 . . . 4  |-  ( ph  ->  ( ( b  e.  D  |->  ( ( F `
 if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) supp  ( 0g
`  S ) ) 
C_  { A }
)
14738, 39, 44, 47, 11, 127, 146gsumpt 17594 . . 3  |-  ( ph  ->  ( S  gsumg  ( b  e.  D  |->  ( ( F `  if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) )  =  ( ( b  e.  D  |->  ( ( F `
 if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) `  A
) )
14837, 147eqtrd 2485 . 2  |-  ( ph  ->  ( S  gsumg  ( b  e.  D  |->  ( ( F `  ( ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) `  b ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) )  =  ( ( b  e.  D  |->  ( ( F `
 if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) `  A
) )
149 iftrue 3887 . . . . . 6  |-  ( b  =  A  ->  if ( b  =  A ,  H ,  .0.  )  =  H )
150149fveq2d 5869 . . . . 5  |-  ( b  =  A  ->  ( F `  if (
b  =  A ,  H ,  .0.  )
)  =  ( F `
 H ) )
151 oveq1 6297 . . . . . 6  |-  ( b  =  A  ->  (
b  oF  .^  G )  =  ( A  oF  .^  G ) )
152151oveq2d 6306 . . . . 5  |-  ( b  =  A  ->  ( T  gsumg  ( b  oF 
.^  G ) )  =  ( T  gsumg  ( A  oF  .^  G
) ) )
153150, 152oveq12d 6308 . . . 4  |-  ( b  =  A  ->  (
( F `  if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) )  =  ( ( F `  H ) 
.x.  ( T  gsumg  ( A  oF  .^  G
) ) ) )
154 ovex 6318 . . . 4  |-  ( ( F `  H ) 
.x.  ( T  gsumg  ( A  oF  .^  G
) ) )  e. 
_V
155153, 126, 154fvmpt 5948 . . 3  |-  ( A  e.  D  ->  (
( b  e.  D  |->  ( ( F `  if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) `  A
)  =  ( ( F `  H ) 
.x.  ( T  gsumg  ( A  oF  .^  G
) ) ) )
15611, 155syl 17 . 2  |-  ( ph  ->  ( ( b  e.  D  |->  ( ( F `
 if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) `  A
)  =  ( ( F `  H ) 
.x.  ( T  gsumg  ( A  oF  .^  G
) ) ) )
15721, 148, 1563eqtrd 2489 1  |-  ( ph  ->  ( E `  (
x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) )  =  ( ( F `  H )  .x.  ( T  gsumg  ( A  oF 
.^  G ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1444    e. wcel 1887   {crab 2741   _Vcvv 3045    \ cdif 3401    C_ wss 3404   ifcif 3881   {csn 3968   class class class wbr 4402    |-> cmpt 4461   `'ccnv 4833   "cima 4837   Fun wfun 5576    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290    oFcof 6529   supp csupp 6914    ^m cmap 7472   Fincfn 7569   finSupp cfsupp 7883   0cc0 9539   NNcn 10609   NN0cn0 10869   Basecbs 15121   .rcmulr 15191   0gc0g 15338    gsumg cgsu 15339   Mndcmnd 16535  .gcmg 16672    GrpHom cghm 16880  CMndccmn 17430  mulGrpcmgp 17723   Ringcrg 17780   CRingccrg 17781   RingHom crh 17940   mVar cmvr 18576   mPoly cmpl 18577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-seq 12214  df-hash 12516  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-sca 15206  df-vsca 15207  df-tset 15209  df-0g 15340  df-gsum 15341  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-submnd 16583  df-grp 16673  df-minusg 16674  df-sbg 16675  df-mulg 16676  df-subg 16814  df-ghm 16881  df-cntz 16971  df-cmn 17432  df-mgp 17724  df-ur 17736  df-ring 17782  df-cring 17783  df-rnghom 17943  df-lmod 18093  df-lss 18156  df-psr 18580  df-mpl 18582
This theorem is referenced by:  evlslem1  18738
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