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Theorem evlslem6 19391
Description: Lemma for evlseu 19394. 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 15345 . . . . . 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 15498 . . . . . . 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 16172 . . . . . 6  |-  ( ph  ->  Y : D --> K )
16 ffvelrn 5624 . . . . . 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 5624 . . . . 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 15325 . . . . 5  |-  C  =  ( Base `  T
)
22 evlslem1.x . . . . 5  |-  .^  =  (.g
`  T )
23 eqid 2284 . . . . 5  |-  ( 0g
`  T )  =  ( 0g `  T
)
2420crngmgp 15343 . . . . . . 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 16242 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( T  gsumg  ( b  o F 
.^  G ) )  e.  C )
33 evlslem1.m . . . . 5  |-  .x.  =  ( .r `  S )
347, 33rngcl 15348 . . . 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 2284 . . 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 5645 . 2  |-  ( ph  ->  ( b  e.  D  |->  ( ( F `  ( Y `  b ) )  .x.  ( T 
gsumg  ( b  o F 
.^  G ) ) ) ) : D --> C )
38 eqid 2284 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
39 evlslem1.r . . . 4  |-  ( ph  ->  R  e.  CRing )
4011, 12, 38, 14, 39mplelsfi 16226 . . 3  |-  ( ph  ->  ( `' Y "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin )
4115feqmptd 5536 . . . . . . . 8  |-  ( ph  ->  Y  =  ( b  e.  D  |->  ( Y `
 b ) ) )
4241cnveqd 4856 . . . . . . 7  |-  ( ph  ->  `' Y  =  `' ( b  e.  D  |->  ( Y `  b
) ) )
4342imaeq1d 5010 . . . . . 6  |-  ( ph  ->  ( `' Y "
( _V  \  {
( 0g `  R
) } ) )  =  ( `' ( b  e.  D  |->  ( Y `  b ) ) " ( _V 
\  { ( 0g
`  R ) } ) ) )
44 eqimss2 3232 . . . . . 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 15497 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
47 eqid 2284 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
4838, 47ghmid 14683 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  ( F `  ( 0g `  R
) )  =  ( 0g `  S ) )
495, 46, 483syl 20 . . . . 5  |-  ( ph  ->  ( F `  ( 0g `  R ) )  =  ( 0g `  S ) )
50 fvex 5499 . . . . . 6  |-  ( Y `
 b )  e. 
_V
5150a1i 12 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  ( Y `  b )  e.  _V )
5245, 49, 51suppssfv 6035 . . . 4  |-  ( ph  ->  ( `' ( b  e.  D  |->  ( F `
 ( Y `  b ) ) )
" ( _V  \  { ( 0g `  S ) } ) )  C_  ( `' Y " ( _V  \  { ( 0g `  R ) } ) ) )
537, 33, 47rnglz 15371 . . . . 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 5499 . . . . 5  |-  ( F `
 ( Y `  b ) )  e. 
_V
5655a1i 12 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( F `  ( Y `  b ) )  e. 
_V )
5752, 54, 56, 32suppssov1 6036 . . 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 7078 . . 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 1628    e. wcel 1688   {crab 2548   _Vcvv 2789    \ cdif 3150    C_ wss 3153   {csn 3641    e. cmpt 4078   `'ccnv 4687   "cima 4691   -->wf 5217   ` cfv 5221  (class class class)co 5819    o Fcof 6037    ^m cmap 6767   Fincfn 6858   NNcn 9741   NN0cn0 9960   Basecbs 13142   .rcmulr 13203   0gc0g 13394    gsumg cgsu 13395  .gcmg 14360    GrpHom cghm 14674  CMndccmn 15083  mulGrpcmgp 15319   Ringcrg 15331   CRingccrg 15332   RingHom crh 15488   mVar cmvr 16082   mPoly cmpl 16083
This theorem is referenced by:  evlslem1  19393
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-oi 7220  df-card 7567  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-7 9804  df-8 9805  df-9 9806  df-n0 9961  df-z 10020  df-uz 10226  df-fz 10777  df-fzo 10865  df-seq 11041  df-hash 11332  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-sca 13218  df-vsca 13219  df-tset 13221  df-0g 13398  df-gsum 13399  df-mnd 14361  df-mhm 14409  df-grp 14483  df-minusg 14484  df-mulg 14486  df-ghm 14675  df-cntz 14787  df-cmn 15085  df-mgp 15320  df-rng 15334  df-cring 15335  df-ur 15336  df-rnghom 15490  df-psr 16092  df-mpl 16094
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