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Theorem evlslem6OLD 18377
Description: Lemma for evlseu 18380. Finiteness and consistency of the top-level sum. (Contributed by Stefan O'Rear, 9-Mar-2015.) Obsolete version of evlslem6 18376 as of 26-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
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 )
evlslem6.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
evlslem6OLD  |-  ( ph  ->  ( ( b  e.  D  |->  ( ( F `
 ( Y `  b ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) : D --> C  /\  ( `' ( b  e.  D  |->  ( ( F `  ( Y `  b )
)  .x.  ( T  gsumg  ( b  oF  .^  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 evlslem6OLD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 evlslem1.s . . . . . 6  |-  ( ph  ->  S  e.  CRing )
2 crngring 17404 . . . . . 6  |-  ( S  e.  CRing  ->  S  e.  Ring )
31, 2syl 16 . . . . 5  |-  ( ph  ->  S  e.  Ring )
43adantr 463 . . . 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 17570 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  F : K
--> C )
95, 8syl 16 . . . . . 6  |-  ( ph  ->  F : K --> C )
109adantr 463 . . . . 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 18287 . . . . . 6  |-  ( ph  ->  Y : D --> K )
1615ffvelrnda 6007 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  ( Y `  b )  e.  K )
1710, 16ffvelrnd 6008 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( F `  ( Y `  b ) )  e.  C )
18 evlslem1.t . . . . . 6  |-  T  =  (mulGrp `  S )
1918, 7mgpbas 17342 . . . . 5  |-  C  =  ( Base `  T
)
20 evlslem1.x . . . . 5  |-  .^  =  (.g
`  T )
21 eqid 2454 . . . . 5  |-  ( 0g
`  T )  =  ( 0g `  T
)
2218crngmgp 17401 . . . . . . 7  |-  ( S  e.  CRing  ->  T  e. CMnd )
231, 22syl 16 . . . . . 6  |-  ( ph  ->  T  e. CMnd )
2423adantr 463 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  T  e. CMnd )
25 simpr 459 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  b  e.  D )
26 evlslem1.g . . . . . 6  |-  ( ph  ->  G : I --> C )
2726adantr 463 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  G : I --> C )
28 evlslem1.i . . . . . 6  |-  ( ph  ->  I  e.  _V )
2928adantr 463 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  I  e.  _V )
3013, 19, 20, 21, 24, 25, 27, 29psrbagev2 18374 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( T  gsumg  ( b  oF 
.^  G ) )  e.  C )
31 evlslem1.m . . . . 5  |-  .x.  =  ( .r `  S )
327, 31ringcl 17407 . . . 4  |-  ( ( S  e.  Ring  /\  ( F `  ( Y `  b ) )  e.  C  /\  ( T 
gsumg  ( b  oF 
.^  G ) )  e.  C )  -> 
( ( F `  ( Y `  b ) )  .x.  ( T 
gsumg  ( b  oF 
.^  G ) ) )  e.  C )
334, 17, 30, 32syl3anc 1226 . . 3  |-  ( (
ph  /\  b  e.  D )  ->  (
( F `  ( Y `  b )
)  .x.  ( T  gsumg  ( b  oF  .^  G ) ) )  e.  C )
34 eqid 2454 . . 3  |-  ( b  e.  D  |->  ( ( F `  ( Y `
 b ) ) 
.x.  ( T  gsumg  ( b  oF  .^  G
) ) ) )  =  ( b  e.  D  |->  ( ( F `
 ( Y `  b ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) )
3533, 34fmptd 6031 . 2  |-  ( ph  ->  ( b  e.  D  |->  ( ( F `  ( Y `  b ) )  .x.  ( T 
gsumg  ( b  oF 
.^  G ) ) ) ) : D --> C )
36 eqid 2454 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
37 evlslem1.r . . . 4  |-  ( ph  ->  R  e.  CRing )
3811, 12, 36, 14, 37mplelsfiOLD 18351 . . 3  |-  ( ph  ->  ( `' Y "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin )
3915feqmptd 5901 . . . . . . . 8  |-  ( ph  ->  Y  =  ( b  e.  D  |->  ( Y `
 b ) ) )
4039cnveqd 5167 . . . . . . 7  |-  ( ph  ->  `' Y  =  `' ( b  e.  D  |->  ( Y `  b
) ) )
4140imaeq1d 5324 . . . . . 6  |-  ( ph  ->  ( `' Y "
( _V  \  {
( 0g `  R
) } ) )  =  ( `' ( b  e.  D  |->  ( Y `  b ) ) " ( _V 
\  { ( 0g
`  R ) } ) ) )
42 eqimss2 3542 . . . . . 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 ) } ) ) )
4341, 42syl 16 . . . . 5  |-  ( ph  ->  ( `' ( b  e.  D  |->  ( Y `
 b ) )
" ( _V  \  { ( 0g `  R ) } ) )  C_  ( `' Y " ( _V  \  { ( 0g `  R ) } ) ) )
44 rhmghm 17569 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
45 eqid 2454 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
4636, 45ghmid 16472 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  ( F `  ( 0g `  R
) )  =  ( 0g `  S ) )
475, 44, 463syl 20 . . . . 5  |-  ( ph  ->  ( F `  ( 0g `  R ) )  =  ( 0g `  S ) )
48 fvex 5858 . . . . . 6  |-  ( Y `
 b )  e. 
_V
4948a1i 11 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  ( Y `  b )  e.  _V )
5043, 47, 49suppssfvOLD 6504 . . . 4  |-  ( ph  ->  ( `' ( b  e.  D  |->  ( F `
 ( Y `  b ) ) )
" ( _V  \  { ( 0g `  S ) } ) )  C_  ( `' Y " ( _V  \  { ( 0g `  R ) } ) ) )
517, 31, 45ringlz 17430 . . . . 5  |-  ( ( S  e.  Ring  /\  x  e.  C )  ->  (
( 0g `  S
)  .x.  x )  =  ( 0g `  S ) )
523, 51sylan 469 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  (
( 0g `  S
)  .x.  x )  =  ( 0g `  S ) )
53 fvex 5858 . . . . 5  |-  ( F `
 ( Y `  b ) )  e. 
_V
5453a1i 11 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( F `  ( Y `  b ) )  e. 
_V )
5550, 52, 54, 30suppssov1OLD 6505 . . 3  |-  ( ph  ->  ( `' ( b  e.  D  |->  ( ( F `  ( Y `
 b ) ) 
.x.  ( T  gsumg  ( b  oF  .^  G
) ) ) )
" ( _V  \  { ( 0g `  S ) } ) )  C_  ( `' Y " ( _V  \  { ( 0g `  R ) } ) ) )
56 ssfi 7733 . . 3  |-  ( ( ( `' Y "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin  /\  ( `' ( b  e.  D  |->  ( ( F `
 ( Y `  b ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) " ( _V  \  { ( 0g
`  S ) } ) )  C_  ( `' Y " ( _V 
\  { ( 0g
`  R ) } ) ) )  -> 
( `' ( b  e.  D  |->  ( ( F `  ( Y `
 b ) ) 
.x.  ( T  gsumg  ( b  oF  .^  G
) ) ) )
" ( _V  \  { ( 0g `  S ) } ) )  e.  Fin )
5738, 55, 56syl2anc 659 . 2  |-  ( ph  ->  ( `' ( b  e.  D  |->  ( ( F `  ( Y `
 b ) ) 
.x.  ( T  gsumg  ( b  oF  .^  G
) ) ) )
" ( _V  \  { ( 0g `  S ) } ) )  e.  Fin )
5835, 57jca 530 1  |-  ( ph  ->  ( ( b  e.  D  |->  ( ( F `
 ( Y `  b ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) : D --> C  /\  ( `' ( b  e.  D  |->  ( ( F `  ( Y `  b )
)  .x.  ( T  gsumg  ( b  oF  .^  G ) ) ) ) " ( _V 
\  { ( 0g
`  S ) } ) )  e.  Fin ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {crab 2808   _Vcvv 3106    \ cdif 3458    C_ wss 3461   {csn 4016    |-> cmpt 4497   `'ccnv 4987   "cima 4991   -->wf 5566   ` cfv 5570  (class class class)co 6270    oFcof 6511    ^m cmap 7412   Fincfn 7509   NNcn 10531   NN0cn0 10791   Basecbs 14716   .rcmulr 14785   0gc0g 14929    gsumg cgsu 14930  .gcmg 16255    GrpHom cghm 16463  CMndccmn 16997  mulGrpcmgp 17336   Ringcrg 17393   CRingccrg 17394   RingHom crh 17556   mVar cmvr 18196   mPoly cmpl 18197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-sca 14800  df-vsca 14801  df-tset 14803  df-0g 14931  df-gsum 14932  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165  df-grp 16256  df-minusg 16257  df-mulg 16259  df-ghm 16464  df-cntz 16554  df-cmn 16999  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-rnghom 17559  df-psr 18200  df-mpl 18202
This theorem is referenced by: (None)
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