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Theorem evlslem6 19324
Description: Lemma for evlseu 19327. 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 15278 . . . . . 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 15431 . . . . . . 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 16105 . . . . . 6  |-  ( ph  ->  Y : D --> K )
16 ffvelrn 5562 . . . . . 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 5562 . . . . 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 15258 . . . . 5  |-  C  =  ( Base `  T
)
22 evlslem1.x . . . . 5  |-  .^  =  (.g
`  T )
23 eqid 2256 . . . . 5  |-  ( 0g
`  T )  =  ( 0g `  T
)
2420crngmgp 15276 . . . . . . 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 16175 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( T  gsumg  ( b  o F 
.^  G ) )  e.  C )
33 evlslem1.m . . . . 5  |-  .x.  =  ( .r `  S )
347, 33rngcl 15281 . . . 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 5583 . 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 16159 . . 3  |-  ( ph  ->  ( `' Y "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin )
4115feqmptd 5474 . . . . . . . 8  |-  ( ph  ->  Y  =  ( b  e.  D  |->  ( Y `
 b ) ) )
4241cnveqd 4810 . . . . . . 7  |-  ( ph  ->  `' Y  =  `' ( b  e.  D  |->  ( Y `  b
) ) )
4342imaeq1d 4964 . . . . . 6  |-  ( ph  ->  ( `' Y "
( _V  \  {
( 0g `  R
) } ) )  =  ( `' ( b  e.  D  |->  ( Y `  b ) ) " ( _V 
\  { ( 0g
`  R ) } ) ) )
44 eqimss2 3173 . . . . . 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 15430 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
47 eqid 2256 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
4838, 47ghmid 14616 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  ( F `  ( 0g `  R
) )  =  ( 0g `  S ) )
495, 46, 483syl 20 . . . . 5  |-  ( ph  ->  ( F `  ( 0g `  R ) )  =  ( 0g `  S ) )
50 fvex 5437 . . . . . 6  |-  ( Y `
 b )  e. 
_V
5150a1i 12 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  ( Y `  b )  e.  _V )
5245, 49, 51suppssfv 5973 . . . 4  |-  ( ph  ->  ( `' ( b  e.  D  |->  ( F `
 ( Y `  b ) ) )
" ( _V  \  { ( 0g `  S ) } ) )  C_  ( `' Y " ( _V  \  { ( 0g `  R ) } ) ) )
537, 33, 47rnglz 15304 . . . . 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 5437 . . . . 5  |-  ( F `
 ( Y `  b ) )  e. 
_V
5655a1i 12 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( F `  ( Y `  b ) )  e. 
_V )
5752, 54, 56, 32suppssov1 5974 . . 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 7016 . . 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 2519   _Vcvv 2740    \ cdif 3091    C_ wss 3094   {csn 3581    e. cmpt 4017   `'ccnv 4625   "cima 4629   -->wf 4634   ` cfv 4638  (class class class)co 5757    o Fcof 5975    ^m cmap 6705   Fincfn 6796   NNcn 9679   NN0cn0 9897   Basecbs 13075   .rcmulr 13136   0gc0g 13327    gsumg cgsu 13328  .gcmg 14293    GrpHom cghm 14607  CMndccmn 15016  mulGrpcmgp 15252   Ringcrg 15264   CRingccrg 15265   RingHom crh 15421   mVar cmvr 16015   mPoly cmpl 16016
This theorem is referenced by:  evlslem1  19326
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 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747
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 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-oi 7158  df-card 7505  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-7 9742  df-8 9743  df-9 9744  df-n0 9898  df-z 9957  df-uz 10163  df-fz 10714  df-fzo 10802  df-seq 10978  df-hash 11269  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-sca 13151  df-vsca 13152  df-tset 13154  df-0g 13331  df-gsum 13332  df-mnd 14294  df-mhm 14342  df-grp 14416  df-minusg 14417  df-mulg 14419  df-ghm 14608  df-cntz 14720  df-cmn 15018  df-mgp 15253  df-ring 15267  df-cring 15268  df-ur 15269  df-rnghom 15423  df-psr 16025  df-mpl 16027
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