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Theorem evlslem6 19345
Description: Lemma for evlseu 19348. Finiteness and consistency of the top level sum. (Contributed by Stefan O'Rear, 9-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  o F 
.^  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 )
evlslem6.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
evlslem6  |-  ( ph  ->  ( ( b  e.  D  |->  ( ( F `
 ( Y `  b ) )  .x.  ( T  gsumg  ( b  o F 
.^  G ) ) ) ) : D --> C  /\  ( `' ( b  e.  D  |->  ( ( F `  ( Y `  b )
)  .x.  ( T  gsumg  ( b  o F  .^  G ) ) ) ) " ( _V 
\  { ( 0g
`  S ) } ) )  e.  Fin ) )
Distinct variable groups:    ph, b    C, b    D, b    h, I    R, b    S, b    Y, b    h, b
Allowed substitution hints:    ph( h, p)    B( h, p, b)    C( h, p)    D( h, p)    P( h, p, b)    R( h, p)    S( h, p)    T( h, p, b)    .x. ( h, p, b)    E( h, p, b)    .^ ( h, p, b)    F( h, p, b)    G( h, p, b)    I( p, b)    K( h, p, b)    V( h, p, b)    Y( h, p)

Proof of Theorem evlslem6
StepHypRef Expression
1 evlslem1.s . . . . . 6  |-  ( ph  ->  S  e.  CRing )
2 crngrng 15299 . . . . . 6  |-  ( S  e.  CRing  ->  S  e.  Ring )
31, 2syl 17 . . . . 5  |-  ( ph  ->  S  e.  Ring )
43adantr 453 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  S  e.  Ring )
5 evlslem1.f . . . . . . 7  |-  ( ph  ->  F  e.  ( R RingHom  S ) )
6 evlslem1.k . . . . . . . 8  |-  K  =  ( Base `  R
)
7 evlslem1.c . . . . . . . 8  |-  C  =  ( Base `  S
)
86, 7rhmf 15452 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  F : K
--> C )
95, 8syl 17 . . . . . 6  |-  ( ph  ->  F : K --> C )
109adantr 453 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  F : K --> C )
11 evlslem1.p . . . . . . 7  |-  P  =  ( I mPoly  R )
12 evlslem1.b . . . . . . 7  |-  B  =  ( Base `  P
)
13 evlslem1.d . . . . . . 7  |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
14 evlslem6.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
1511, 6, 12, 13, 14mplelf 16126 . . . . . 6  |-  ( ph  ->  Y : D --> K )
16 ffvelrn 5583 . . . . . 6  |-  ( ( Y : D --> K  /\  b  e.  D )  ->  ( Y `  b
)  e.  K )
1715, 16sylan 459 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  ( Y `  b )  e.  K )
18 ffvelrn 5583 . . . . 5  |-  ( ( F : K --> C  /\  ( Y `  b )  e.  K )  -> 
( F `  ( Y `  b )
)  e.  C )
1910, 17, 18syl2anc 645 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( F `  ( Y `  b ) )  e.  C )
20 evlslem1.t . . . . . 6  |-  T  =  (mulGrp `  S )
2120, 7mgpbas 15279 . . . . 5  |-  C  =  ( Base `  T
)
22 evlslem1.x . . . . 5  |-  .^  =  (.g
`  T )
23 eqid 2256 . . . . 5  |-  ( 0g
`  T )  =  ( 0g `  T
)
2420crngmgp 15297 . . . . . . 7  |-  ( S  e.  CRing  ->  T  e. CMnd )
251, 24syl 17 . . . . . 6  |-  ( ph  ->  T  e. CMnd )
2625adantr 453 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  T  e. CMnd )
27 simpr 449 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  b  e.  D )
28 evlslem1.g . . . . . 6  |-  ( ph  ->  G : I --> C )
2928adantr 453 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  G : I --> C )
30 evlslem1.i . . . . . 6  |-  ( ph  ->  I  e.  _V )
3130adantr 453 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  I  e.  _V )
3213, 21, 22, 23, 26, 27, 29, 31psrbagev2 16196 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( T  gsumg  ( b  o F 
.^  G ) )  e.  C )
33 evlslem1.m . . . . 5  |-  .x.  =  ( .r `  S )
347, 33rngcl 15302 . . . 4  |-  ( ( S  e.  Ring  /\  ( F `  ( Y `  b ) )  e.  C  /\  ( T 
gsumg  ( b  o F 
.^  G ) )  e.  C )  -> 
( ( F `  ( Y `  b ) )  .x.  ( T 
gsumg  ( b  o F 
.^  G ) ) )  e.  C )
354, 19, 32, 34syl3anc 1187 . . 3  |-  ( (
ph  /\  b  e.  D )  ->  (
( F `  ( Y `  b )
)  .x.  ( T  gsumg  ( b  o F  .^  G ) ) )  e.  C )
36 eqid 2256 . . 3  |-  ( b  e.  D  |->  ( ( F `  ( Y `
 b ) ) 
.x.  ( T  gsumg  ( b  o F  .^  G
) ) ) )  =  ( b  e.  D  |->  ( ( F `
 ( Y `  b ) )  .x.  ( T  gsumg  ( b  o F 
.^  G ) ) ) )
3735, 36fmptd 5604 . 2  |-  ( ph  ->  ( b  e.  D  |->  ( ( F `  ( Y `  b ) )  .x.  ( T 
gsumg  ( b  o F 
.^  G ) ) ) ) : D --> C )
38 eqid 2256 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
39 evlslem1.r . . . 4  |-  ( ph  ->  R  e.  CRing )
4011, 12, 38, 14, 39mplelsfi 16180 . . 3  |-  ( ph  ->  ( `' Y "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin )
4115feqmptd 5495 . . . . . . . 8  |-  ( ph  ->  Y  =  ( b  e.  D  |->  ( Y `
 b ) ) )
4241cnveqd 4831 . . . . . . 7  |-  ( ph  ->  `' Y  =  `' ( b  e.  D  |->  ( Y `  b
) ) )
4342imaeq1d 4985 . . . . . 6  |-  ( ph  ->  ( `' Y "
( _V  \  {
( 0g `  R
) } ) )  =  ( `' ( b  e.  D  |->  ( Y `  b ) ) " ( _V 
\  { ( 0g
`  R ) } ) ) )
44 eqimss2 3192 . . . . . 6  |-  ( ( `' Y " ( _V 
\  { ( 0g
`  R ) } ) )  =  ( `' ( b  e.  D  |->  ( Y `  b ) ) "
( _V  \  {
( 0g `  R
) } ) )  ->  ( `' ( b  e.  D  |->  ( Y `  b ) ) " ( _V 
\  { ( 0g
`  R ) } ) )  C_  ( `' Y " ( _V 
\  { ( 0g
`  R ) } ) ) )
4543, 44syl 17 . . . . 5  |-  ( ph  ->  ( `' ( b  e.  D  |->  ( Y `
 b ) )
" ( _V  \  { ( 0g `  R ) } ) )  C_  ( `' Y " ( _V  \  { ( 0g `  R ) } ) ) )
46 rhmghm 15451 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
47 eqid 2256 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
4838, 47ghmid 14637 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  ( F `  ( 0g `  R
) )  =  ( 0g `  S ) )
495, 46, 483syl 20 . . . . 5  |-  ( ph  ->  ( F `  ( 0g `  R ) )  =  ( 0g `  S ) )
50 fvex 5458 . . . . . 6  |-  ( Y `
 b )  e. 
_V
5150a1i 12 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  ( Y `  b )  e.  _V )
5245, 49, 51suppssfv 5994 . . . 4  |-  ( ph  ->  ( `' ( b  e.  D  |->  ( F `
 ( Y `  b ) ) )
" ( _V  \  { ( 0g `  S ) } ) )  C_  ( `' Y " ( _V  \  { ( 0g `  R ) } ) ) )
537, 33, 47rnglz 15325 . . . . 5  |-  ( ( S  e.  Ring  /\  x  e.  C )  ->  (
( 0g `  S
)  .x.  x )  =  ( 0g `  S ) )
543, 53sylan 459 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  (
( 0g `  S
)  .x.  x )  =  ( 0g `  S ) )
55 fvex 5458 . . . . 5  |-  ( F `
 ( Y `  b ) )  e. 
_V
5655a1i 12 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( F `  ( Y `  b ) )  e. 
_V )
5752, 54, 56, 32suppssov1 5995 . . 3  |-  ( ph  ->  ( `' ( b  e.  D  |->  ( ( F `  ( Y `
 b ) ) 
.x.  ( T  gsumg  ( b  o F  .^  G
) ) ) )
" ( _V  \  { ( 0g `  S ) } ) )  C_  ( `' Y " ( _V  \  { ( 0g `  R ) } ) ) )
58 ssfi 7037 . . 3  |-  ( ( ( `' Y "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin  /\  ( `' ( b  e.  D  |->  ( ( F `
 ( Y `  b ) )  .x.  ( T  gsumg  ( b  o F 
.^  G ) ) ) ) " ( _V  \  { ( 0g
`  S ) } ) )  C_  ( `' Y " ( _V 
\  { ( 0g
`  R ) } ) ) )  -> 
( `' ( b  e.  D  |->  ( ( F `  ( Y `
 b ) ) 
.x.  ( T  gsumg  ( b  o F  .^  G
) ) ) )
" ( _V  \  { ( 0g `  S ) } ) )  e.  Fin )
5940, 57, 58syl2anc 645 . 2  |-  ( ph  ->  ( `' ( b  e.  D  |->  ( ( F `  ( Y `
 b ) ) 
.x.  ( T  gsumg  ( b  o F  .^  G
) ) ) )
" ( _V  \  { ( 0g `  S ) } ) )  e.  Fin )
6037, 59jca 520 1  |-  ( ph  ->  ( ( b  e.  D  |->  ( ( F `
 ( Y `  b ) )  .x.  ( T  gsumg  ( b  o F 
.^  G ) ) ) ) : D --> C  /\  ( `' ( b  e.  D  |->  ( ( F `  ( Y `  b )
)  .x.  ( T  gsumg  ( b  o F  .^  G ) ) ) ) " ( _V 
\  { ( 0g
`  S ) } ) )  e.  Fin ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   {crab 2520   _Vcvv 2757    \ cdif 3110    C_ wss 3113   {csn 3600    e. cmpt 4037   `'ccnv 4646   "cima 4650   -->wf 4655   ` cfv 4659  (class class class)co 5778    o Fcof 5996    ^m cmap 6726   Fincfn 6817   NNcn 9700   NN0cn0 9918   Basecbs 13096   .rcmulr 13157   0gc0g 13348    gsumg cgsu 13349  .gcmg 14314    GrpHom cghm 14628  CMndccmn 15037  mulGrpcmgp 15273   Ringcrg 15285   CRingccrg 15286   RingHom crh 15442   mVar cmvr 16036   mPoly cmpl 16037
This theorem is referenced by:  evlslem1  19347
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-of 5998  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-oadd 6437  df-er 6614  df-map 6728  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-oi 7179  df-card 7526  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-5 9761  df-6 9762  df-7 9763  df-8 9764  df-9 9765  df-n0 9919  df-z 9978  df-uz 10184  df-fz 10735  df-fzo 10823  df-seq 10999  df-hash 11290  df-struct 13098  df-ndx 13099  df-slot 13100  df-base 13101  df-sets 13102  df-ress 13103  df-plusg 13169  df-mulr 13170  df-sca 13172  df-vsca 13173  df-tset 13175  df-0g 13352  df-gsum 13353  df-mnd 14315  df-mhm 14363  df-grp 14437  df-minusg 14438  df-mulg 14440  df-ghm 14629  df-cntz 14741  df-cmn 15039  df-mgp 15274  df-ring 15288  df-cring 15289  df-ur 15290  df-rnghom 15444  df-psr 16046  df-mpl 16048
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