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Theorem evlslem3 17716
Description: Lemma for evlseu 17718. 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 crngrng 16770 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
86, 7syl 16 . . . 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 17698 . . 3  |-  ( ph  ->  ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) )  e.  B
)
13 fveq1 5791 . . . . . . . 8  |-  ( p  =  ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) )  -> 
( p `  b
)  =  ( ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) `  b
) )
1413fveq2d 5796 . . . . . . 7  |-  ( p  =  ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) )  -> 
( F `  (
p `  b )
)  =  ( F `
 ( ( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  )
) `  b )
) )
1514oveq1d 6208 . . . . . 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 4480 . . . . 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 6209 . . . 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 6218 . . . 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 5876 . . 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 16 . 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 461 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  b  e.  D )
23 fvex 5802 . . . . . . . . . . . 12  |-  ( 0g
`  R )  e. 
_V
243, 23eqeltri 2535 . . . . . . . . . . 11  |-  .0.  e.  _V
2524a1i 11 . . . . . . . . . 10  |-  ( ph  ->  .0.  e.  _V )
26 ifexg 3960 . . . . . . . . . 10  |-  ( ( H  e.  K  /\  .0.  e.  _V )  ->  if ( b  =  A ,  H ,  .0.  )  e.  _V )
2710, 25, 26syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  if ( b  =  A ,  H ,  .0.  )  e.  _V )
2827adantr 465 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  if ( b  =  A ,  H ,  .0.  )  e.  _V )
29 eqeq1 2455 . . . . . . . . . 10  |-  ( x  =  b  ->  (
x  =  A  <->  b  =  A ) )
3029ifbid 3912 . . . . . . . . 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 5874 . . . . . . . 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 661 . . . . . . 7  |-  ( (
ph  /\  b  e.  D )  ->  (
( x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) `  b
)  =  if ( b  =  A ,  H ,  .0.  )
)
3433fveq2d 5796 . . . . . 6  |-  ( (
ph  /\  b  e.  D )  ->  ( F `  ( (
x  e.  D  |->  if ( x  =  A ,  H ,  .0.  ) ) `  b
) )  =  ( F `  if ( b  =  A ,  H ,  .0.  )
) )
3534oveq1d 6208 . . . . 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 4479 . . . 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 6209 . . 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 crngrng 16770 . . . . . 6  |-  ( S  e.  CRing  ->  S  e.  Ring )
4240, 41syl 16 . . . . 5  |-  ( ph  ->  S  e.  Ring )
43 rngmnd 16769 . . . . 5  |-  ( S  e.  Ring  ->  S  e. 
Mnd )
4442, 43syl 16 . . . 4  |-  ( ph  ->  S  e.  Mnd )
45 ovex 6218 . . . . . 6  |-  ( NN0 
^m  I )  e. 
_V
462, 45rabex2 4546 . . . . 5  |-  D  e. 
_V
4746a1i 11 . . . 4  |-  ( ph  ->  D  e.  _V )
4842adantr 465 . . . . . 6  |-  ( (
ph  /\  b  e.  D )  ->  S  e.  Ring )
49 evlslem1.f . . . . . . . . 9  |-  ( ph  ->  F  e.  ( R RingHom  S ) )
504, 38rhmf 16931 . . . . . . . . 9  |-  ( F  e.  ( R RingHom  S
)  ->  F : K
--> C )
5149, 50syl 16 . . . . . . . 8  |-  ( ph  ->  F : K --> C )
524, 3rng0cl 16781 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  .0.  e.  K )
538, 52syl 16 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  K )
5410, 53ifcld 3933 . . . . . . . 8  |-  ( ph  ->  if ( b  =  A ,  H ,  .0.  )  e.  K
)
5551, 54ffvelrnd 5946 . . . . . . 7  |-  ( ph  ->  ( F `  if ( b  =  A ,  H ,  .0.  ) )  e.  C
)
5655adantr 465 . . . . . 6  |-  ( (
ph  /\  b  e.  D )  ->  ( F `  if (
b  =  A ,  H ,  .0.  )
)  e.  C )
57 evlslem1.t . . . . . . . 8  |-  T  =  (mulGrp `  S )
5857, 38mgpbas 16711 . . . . . . 7  |-  C  =  ( Base `  T
)
59 eqid 2451 . . . . . . 7  |-  ( 0g
`  T )  =  ( 0g `  T
)
6057crngmgp 16768 . . . . . . . . 9  |-  ( S  e.  CRing  ->  T  e. CMnd )
6140, 60syl 16 . . . . . . . 8  |-  ( ph  ->  T  e. CMnd )
6261adantr 465 . . . . . . 7  |-  ( (
ph  /\  b  e.  D )  ->  T  e. CMnd )
635adantr 465 . . . . . . 7  |-  ( (
ph  /\  b  e.  D )  ->  I  e.  _V )
64 cmnmnd 16405 . . . . . . . . . . 11  |-  ( T  e. CMnd  ->  T  e.  Mnd )
6561, 64syl 16 . . . . . . . . . 10  |-  ( ph  ->  T  e.  Mnd )
6665ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  b  e.  D )  /\  (
y  e.  NN0  /\  z  e.  C )
)  ->  T  e.  Mnd )
67 simprl 755 . . . . . . . . 9  |-  ( ( ( ph  /\  b  e.  D )  /\  (
y  e.  NN0  /\  z  e.  C )
)  ->  y  e.  NN0 )
68 simprr 756 . . . . . . . . 9  |-  ( ( ( ph  /\  b  e.  D )  /\  (
y  e.  NN0  /\  z  e.  C )
)  ->  z  e.  C )
69 evlslem1.x . . . . . . . . . 10  |-  .^  =  (.g
`  T )
7058, 69mulgnn0cl 15754 . . . . . . . . 9  |-  ( ( T  e.  Mnd  /\  y  e.  NN0  /\  z  e.  C )  ->  (
y  .^  z )  e.  C )
7166, 67, 68, 70syl3anc 1219 . . . . . . . 8  |-  ( ( ( ph  /\  b  e.  D )  /\  (
y  e.  NN0  /\  z  e.  C )
)  ->  ( y  .^  z )  e.  C
)
722psrbagf 17547 . . . . . . . . 9  |-  ( ( I  e.  _V  /\  b  e.  D )  ->  b : I --> NN0 )
735, 72sylan 471 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  b : I --> NN0 )
74 evlslem1.g . . . . . . . . 9  |-  ( ph  ->  G : I --> C )
7574adantr 465 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  G : I --> C )
76 inidm 3660 . . . . . . . 8  |-  ( I  i^i  I )  =  I
7771, 73, 75, 63, 63, 76off 6437 . . . . . . 7  |-  ( (
ph  /\  b  e.  D )  ->  (
b  oF  .^  G ) : I --> C )
78 ovex 6218 . . . . . . . . 9  |-  ( b  oF  .^  G
)  e.  _V
7978a1i 11 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  (
b  oF  .^  G )  e.  _V )
80 ffun 5662 . . . . . . . . 9  |-  ( ( b  oF  .^  G ) : I --> C  ->  Fun  ( b  oF  .^  G
) )
8177, 80syl 16 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  Fun  ( b  oF 
.^  G ) )
82 fvex 5802 . . . . . . . . 9  |-  ( 0g
`  T )  e. 
_V
8382a1i 11 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  ( 0g `  T )  e. 
_V )
842psrbag 17546 . . . . . . . . . 10  |-  ( I  e.  _V  ->  (
b  e.  D  <->  ( b : I --> NN0  /\  ( `' b " NN )  e.  Fin )
) )
855, 84syl 16 . . . . . . . . 9  |-  ( ph  ->  ( b  e.  D  <->  ( b : I --> NN0  /\  ( `' b " NN )  e.  Fin )
) )
8685simplbda 624 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  ( `' b " NN )  e.  Fin )
87 ffn 5660 . . . . . . . . . . . . 13  |-  ( b : I --> NN0  ->  b  Fn  I )
8873, 87syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  b  e.  D )  ->  b  Fn  I )
8988adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
b  Fn  I )
90 ffn 5660 . . . . . . . . . . . . 13  |-  ( G : I --> C  ->  G  Fn  I )
9174, 90syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  G  Fn  I )
9291ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  ->  G  Fn  I )
935ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  ->  I  e.  _V )
94 eldifi 3579 . . . . . . . . . . . 12  |-  ( y  e.  ( I  \ 
( `' b " NN ) )  ->  y  e.  I )
9594adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
y  e.  I )
96 fnfvof 6436 . . . . . . . . . . 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 1220 . . . . . . . . . 10  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
( ( b  oF  .^  G ) `  y )  =  ( ( b `  y
)  .^  ( G `  y ) ) )
98 eldifn 3580 . . . . . . . . . . . . . 14  |-  ( y  e.  ( I  \ 
( `' b " NN ) )  ->  -.  y  e.  ( `' b " NN ) )
9998adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  ->  -.  y  e.  ( `' b " NN ) )
10094ad2antlr 726 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  b  e.  D )  /\  y  e.  (
I  \  ( `' b " NN ) ) )  /\  ( b `
 y )  e.  NN )  ->  y  e.  I )
101 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  b  e.  D )  /\  y  e.  (
I  \  ( `' b " NN ) ) )  /\  ( b `
 y )  e.  NN )  ->  (
b `  y )  e.  NN )
10288ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  b  e.  D )  /\  y  e.  (
I  \  ( `' b " NN ) ) )  /\  ( b `
 y )  e.  NN )  ->  b  Fn  I )
103 elpreima 5925 . . . . . . . . . . . . . . 15  |-  ( b  Fn  I  ->  (
y  e.  ( `' b " NN )  <-> 
( y  e.  I  /\  ( b `  y
)  e.  NN ) ) )
104102, 103syl 16 . . . . . . . . . . . . . 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 913 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  b  e.  D )  /\  y  e.  (
I  \  ( `' b " NN ) ) )  /\  ( b `
 y )  e.  NN )  ->  y  e.  ( `' b " NN ) )
10699, 105mtand 659 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  ->  -.  ( b `  y
)  e.  NN )
107 ffvelrn 5943 . . . . . . . . . . . . . 14  |-  ( ( b : I --> NN0  /\  y  e.  I )  ->  ( b `  y
)  e.  NN0 )
10873, 94, 107syl2an 477 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
( b `  y
)  e.  NN0 )
109 elnn0 10685 . . . . . . . . . . . . 13  |-  ( ( b `  y )  e.  NN0  <->  ( ( b `
 y )  e.  NN  \/  ( b `
 y )  =  0 ) )
110108, 109sylib 196 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
( ( b `  y )  e.  NN  \/  ( b `  y
)  =  0 ) )
111 orel1 382 . . . . . . . . . . . 12  |-  ( -.  ( b `  y
)  e.  NN  ->  ( ( ( b `  y )  e.  NN  \/  ( b `  y
)  =  0 )  ->  ( b `  y )  =  0 ) )
112106, 110, 111sylc 60 . . . . . . . . . . 11  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
( b `  y
)  =  0 )
113112oveq1d 6208 . . . . . . . . . 10  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
( ( b `  y )  .^  ( G `  y )
)  =  ( 0 
.^  ( G `  y ) ) )
114 ffvelrn 5943 . . . . . . . . . . . 12  |-  ( ( G : I --> C  /\  y  e.  I )  ->  ( G `  y
)  e.  C )
11575, 94, 114syl2an 477 . . . . . . . . . . 11  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
( G `  y
)  e.  C )
11658, 59, 69mulg0 15743 . . . . . . . . . . 11  |-  ( ( G `  y )  e.  C  ->  (
0  .^  ( G `  y ) )  =  ( 0g `  T
) )
117115, 116syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
( 0  .^  ( G `  y )
)  =  ( 0g
`  T ) )
11897, 113, 1173eqtrd 2496 . . . . . . . . 9  |-  ( ( ( ph  /\  b  e.  D )  /\  y  e.  ( I  \  ( `' b " NN ) ) )  -> 
( ( b  oF  .^  G ) `  y )  =  ( 0g `  T ) )
11977, 118suppss 6822 . . . . . . . 8  |-  ( (
ph  /\  b  e.  D )  ->  (
( b  oF 
.^  G ) supp  ( 0g `  T ) ) 
C_  ( `' b
" NN ) )
120 suppssfifsupp 7739 . . . . . . . 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 1227 . . . . . . 7  |-  ( (
ph  /\  b  e.  D )  ->  (
b  oF  .^  G ) finSupp  ( 0g `  T ) )
12258, 59, 62, 63, 77, 121gsumcl 16510 . . . . . 6  |-  ( (
ph  /\  b  e.  D )  ->  ( T  gsumg  ( b  oF 
.^  G ) )  e.  C )
123 evlslem1.m . . . . . . 7  |-  .x.  =  ( .r `  S )
12438, 123rngcl 16773 . . . . . 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 1219 . . . . 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 5969 . . . 4  |-  ( ph  ->  ( b  e.  D  |->  ( ( F `  if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) : D --> C )
128 eldifsni 4102 . . . . . . . . . . . 12  |-  ( b  e.  ( D  \  { A } )  -> 
b  =/=  A )
129128neneqd 2651 . . . . . . . . . . 11  |-  ( b  e.  ( D  \  { A } )  ->  -.  b  =  A
)
130 iffalse 3900 . . . . . . . . . . 11  |-  ( -.  b  =  A  ->  if ( b  =  A ,  H ,  .0.  )  =  .0.  )
131129, 130syl 16 . . . . . . . . . 10  |-  ( b  e.  ( D  \  { A } )  ->  if ( b  =  A ,  H ,  .0.  )  =  .0.  )
132131adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  b  e.  ( D  \  { A } ) )  ->  if ( b  =  A ,  H ,  .0.  )  =  .0.  )
133132fveq2d 5796 . . . . . . . 8  |-  ( (
ph  /\  b  e.  ( D  \  { A } ) )  -> 
( F `  if ( b  =  A ,  H ,  .0.  ) )  =  ( F `  .0.  )
)
134 rhmghm 16930 . . . . . . . . . . 11  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
13549, 134syl 16 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ( R 
GrpHom  S ) )
1363, 39ghmid 15864 . . . . . . . . . 10  |-  ( F  e.  ( R  GrpHom  S )  ->  ( F `  .0.  )  =  ( 0g `  S ) )
137135, 136syl 16 . . . . . . . . 9  |-  ( ph  ->  ( F `  .0.  )  =  ( 0g `  S ) )
138137adantr 465 . . . . . . . 8  |-  ( (
ph  /\  b  e.  ( D  \  { A } ) )  -> 
( F `  .0.  )  =  ( 0g `  S ) )
139133, 138eqtrd 2492 . . . . . . 7  |-  ( (
ph  /\  b  e.  ( D  \  { A } ) )  -> 
( F `  if ( b  =  A ,  H ,  .0.  ) )  =  ( 0g `  S ) )
140139oveq1d 6208 . . . . . 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
) ) ) )
14142adantr 465 . . . . . . 7  |-  ( (
ph  /\  b  e.  ( D  \  { A } ) )  ->  S  e.  Ring )
142 eldifi 3579 . . . . . . . 8  |-  ( b  e.  ( D  \  { A } )  -> 
b  e.  D )
143142, 122sylan2 474 . . . . . . 7  |-  ( (
ph  /\  b  e.  ( D  \  { A } ) )  -> 
( T  gsumg  ( b  oF 
.^  G ) )  e.  C )
14438, 123, 39rnglz 16796 . . . . . . 7  |-  ( ( S  e.  Ring  /\  ( T  gsumg  ( b  oF 
.^  G ) )  e.  C )  -> 
( ( 0g `  S )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) )  =  ( 0g
`  S ) )
145141, 143, 144syl2anc 661 . . . . . 6  |-  ( (
ph  /\  b  e.  ( D  \  { A } ) )  -> 
( ( 0g `  S )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) )  =  ( 0g
`  S ) )
146140, 145eqtrd 2492 . . . . 5  |-  ( (
ph  /\  b  e.  ( D  \  { A } ) )  -> 
( ( F `  if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) )  =  ( 0g
`  S ) )
147146, 47suppss2 6826 . . . 4  |-  ( ph  ->  ( ( b  e.  D  |->  ( ( F `
 if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) supp  ( 0g
`  S ) ) 
C_  { A }
)
14838, 39, 44, 47, 11, 127, 147gsumpt 16568 . . 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
) )
14937, 148eqtrd 2492 . 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
) )
150 iftrue 3898 . . . . . 6  |-  ( b  =  A  ->  if ( b  =  A ,  H ,  .0.  )  =  H )
151150fveq2d 5796 . . . . 5  |-  ( b  =  A  ->  ( F `  if (
b  =  A ,  H ,  .0.  )
)  =  ( F `
 H ) )
152 oveq1 6200 . . . . . 6  |-  ( b  =  A  ->  (
b  oF  .^  G )  =  ( A  oF  .^  G ) )
153152oveq2d 6209 . . . . 5  |-  ( b  =  A  ->  ( T  gsumg  ( b  oF 
.^  G ) )  =  ( T  gsumg  ( A  oF  .^  G
) ) )
154151, 153oveq12d 6211 . . . 4  |-  ( b  =  A  ->  (
( F `  if ( b  =  A ,  H ,  .0.  ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) )  =  ( ( F `  H ) 
.x.  ( T  gsumg  ( A  oF  .^  G
) ) ) )
155 ovex 6218 . . . 4  |-  ( ( F `  H ) 
.x.  ( T  gsumg  ( A  oF  .^  G
) ) )  e. 
_V
156154, 126, 155fvmpt 5876 . . 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
) ) ) )
15711, 156syl 16 . 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
) ) ) )
15821, 149, 1573eqtrd 2496 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 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   {crab 2799   _Vcvv 3071    \ cdif 3426    C_ wss 3429   ifcif 3892   {csn 3978   class class class wbr 4393    |-> cmpt 4451   `'ccnv 4940   "cima 4944   Fun wfun 5513    Fn wfn 5514   -->wf 5515   ` cfv 5519  (class class class)co 6193    oFcof 6421   supp csupp 6793    ^m cmap 7317   Fincfn 7413   finSupp cfsupp 7724   0cc0 9386   NNcn 10426   NN0cn0 10683   Basecbs 14285   .rcmulr 14350   0gc0g 14489    gsumg cgsu 14490   Mndcmnd 15520  .gcmg 15525    GrpHom cghm 15855  CMndccmn 16390  mulGrpcmgp 16705   Ringcrg 16760   CRingccrg 16761   RingHom crh 16919   mVar cmvr 17534   mPoly cmpl 17535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-fzo 11659  df-seq 11917  df-hash 12214  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-sca 14365  df-vsca 14366  df-tset 14368  df-0g 14491  df-gsum 14492  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-mhm 15575  df-submnd 15576  df-grp 15656  df-minusg 15657  df-sbg 15658  df-mulg 15659  df-subg 15789  df-ghm 15856  df-cntz 15946  df-cmn 16392  df-mgp 16706  df-ur 16718  df-rng 16762  df-cring 16763  df-rnghom 16921  df-lmod 17065  df-lss 17129  df-psr 17538  df-mpl 17540
This theorem is referenced by:  evlslem1  17717
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