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Theorem evlslem6 19229
Description: Lemma for evlseu 19232. 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 15186 . . . . . 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 15339 . . . . . . 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 16010 . . . . . 6  |-  ( ph  ->  Y : D --> K )
16 ffvelrn 5515 . . . . . 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 5515 . . . . 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 15166 . . . . 5  |-  C  =  ( Base `  T
)
22 evlslem1.x . . . . 5  |-  .^  =  (.g
`  T )
23 eqid 2253 . . . . 5  |-  ( 0g
`  T )  =  ( 0g `  T
)
2420crngmgp 15184 . . . . . . 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 16080 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( T  gsumg  ( b  o F 
.^  G ) )  e.  C )
33 evlslem1.m . . . . 5  |-  .x.  =  ( .r `  S )
347, 33rngcl 15189 . . . 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 2253 . . 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 5536 . 2  |-  ( ph  ->  ( b  e.  D  |->  ( ( F `  ( Y `  b ) )  .x.  ( T 
gsumg  ( b  o F 
.^  G ) ) ) ) : D --> C )
38 eqid 2253 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
39 evlslem1.r . . . 4  |-  ( ph  ->  R  e.  CRing )
4011, 12, 38, 14, 39mplelsfi 16064 . . 3  |-  ( ph  ->  ( `' Y "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin )
4115feqmptd 5427 . . . . . . . 8  |-  ( ph  ->  Y  =  ( b  e.  D  |->  ( Y `
 b ) ) )
4241cnveqd 4764 . . . . . . 7  |-  ( ph  ->  `' Y  =  `' ( b  e.  D  |->  ( Y `  b
) ) )
4342imaeq1d 4918 . . . . . 6  |-  ( ph  ->  ( `' Y "
( _V  \  {
( 0g `  R
) } ) )  =  ( `' ( b  e.  D  |->  ( Y `  b ) ) " ( _V 
\  { ( 0g
`  R ) } ) ) )
44 eqimss2 3152 . . . . . 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 15338 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
47 eqid 2253 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
4838, 47ghmid 14524 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  ( F `  ( 0g `  R
) )  =  ( 0g `  S ) )
495, 46, 483syl 20 . . . . 5  |-  ( ph  ->  ( F `  ( 0g `  R ) )  =  ( 0g `  S ) )
50 fvex 5391 . . . . . 6  |-  ( Y `
 b )  e. 
_V
5150a1i 12 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  ( Y `  b )  e.  _V )
5245, 49, 51suppssfv 5926 . . . 4  |-  ( ph  ->  ( `' ( b  e.  D  |->  ( F `
 ( Y `  b ) ) )
" ( _V  \  { ( 0g `  S ) } ) )  C_  ( `' Y " ( _V  \  { ( 0g `  R ) } ) ) )
537, 33, 47rnglz 15212 . . . . 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 5391 . . . . 5  |-  ( F `
 ( Y `  b ) )  e. 
_V
5655a1i 12 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( F `  ( Y `  b ) )  e. 
_V )
5752, 54, 56, 32suppssov1 5927 . . 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 6968 . . 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 2512   _Vcvv 2727    \ cdif 3075    C_ wss 3078   {csn 3544    e. cmpt 3974   `'ccnv 4579   "cima 4583   -->wf 4588   ` cfv 4592  (class class class)co 5710    o Fcof 5928    ^m cmap 6658   Fincfn 6749   NNcn 9626   NN0cn0 9844   Basecbs 13022   .rcmulr 13083   0gc0g 13274    gsumg cgsu 13275  .gcmg 14201    GrpHom cghm 14515  CMndccmn 14924  mulGrpcmgp 15160   Ringcrg 15172   CRingccrg 15173   RingHom crh 15329   mVar cmvr 15920   mPoly cmpl 15921
This theorem is referenced by:  evlslem1  19231
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-of 5930  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-oi 7109  df-card 7456  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-7 9689  df-8 9690  df-9 9691  df-n0 9845  df-z 9904  df-uz 10110  df-fz 10661  df-fzo 10749  df-seq 10925  df-hash 11216  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-sca 13098  df-vsca 13099  df-tset 13101  df-0g 13278  df-gsum 13279  df-mnd 14202  df-mhm 14250  df-grp 14324  df-minusg 14325  df-mulg 14327  df-ghm 14516  df-cntz 14628  df-cmn 14926  df-mgp 15161  df-ring 15175  df-cring 15176  df-ur 15177  df-rnghom 15331  df-psr 15930  df-mpl 15932
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