MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  evlslem6 Structured version   Unicode version

Theorem evlslem6 18294
Description: Lemma for evlseu 18298. Finiteness and consistency of the top-level sum. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 26-Jul-2019.)
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
evlslem6  |-  ( 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 ) ) ) ) finSupp  ( 0g
`  S ) ) )
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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 evlslem1.s . . . . . 6  |-  ( ph  ->  S  e.  CRing )
2 crngring 17322 . . . . . 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 17488 . . . . . . 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 18205 . . . . . 6  |-  ( ph  ->  Y : D --> K )
1615ffvelrnda 5933 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  ( Y `  b )  e.  K )
1710, 16ffvelrnd 5934 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( F `  ( Y `  b ) )  e.  C )
18 evlslem1.t . . . . . 6  |-  T  =  (mulGrp `  S )
1918, 7mgpbas 17260 . . . . 5  |-  C  =  ( Base `  T
)
20 evlslem1.x . . . . 5  |-  .^  =  (.g
`  T )
21 eqid 2382 . . . . 5  |-  ( 0g
`  T )  =  ( 0g `  T
)
2218crngmgp 17319 . . . . . . 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 18292 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( T  gsumg  ( b  oF 
.^  G ) )  e.  C )
31 evlslem1.m . . . . 5  |-  .x.  =  ( .r `  S )
327, 31ringcl 17325 . . . 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 2382 . . 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 5957 . 2  |-  ( ph  ->  ( b  e.  D  |->  ( ( F `  ( Y `  b ) )  .x.  ( T 
gsumg  ( b  oF 
.^  G ) ) ) ) : D --> C )
36 ovex 6224 . . . . . 6  |-  ( NN0 
^m  I )  e. 
_V
3736a1i 11 . . . . 5  |-  ( ph  ->  ( NN0  ^m  I
)  e.  _V )
3813, 37rabexd 4517 . . . 4  |-  ( ph  ->  D  e.  _V )
39 mptexg 6043 . . . 4  |-  ( D  e.  _V  ->  (
b  e.  D  |->  ( ( F `  ( Y `  b )
)  .x.  ( T  gsumg  ( b  oF  .^  G ) ) ) )  e.  _V )
4038, 39syl 16 . . 3  |-  ( ph  ->  ( b  e.  D  |->  ( ( F `  ( Y `  b ) )  .x.  ( T 
gsumg  ( b  oF 
.^  G ) ) ) )  e.  _V )
41 funmpt 5532 . . . 4  |-  Fun  (
b  e.  D  |->  ( ( F `  ( Y `  b )
)  .x.  ( T  gsumg  ( b  oF  .^  G ) ) ) )
4241a1i 11 . . 3  |-  ( ph  ->  Fun  ( b  e.  D  |->  ( ( F `
 ( Y `  b ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) )
43 fvex 5784 . . . 4  |-  ( 0g
`  S )  e. 
_V
4443a1i 11 . . 3  |-  ( ph  ->  ( 0g `  S
)  e.  _V )
45 eqid 2382 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
46 evlslem1.r . . . . 5  |-  ( ph  ->  R  e.  CRing )
4711, 12, 45, 14, 46mplelsfi 18268 . . . 4  |-  ( ph  ->  Y finSupp  ( 0g `  R ) )
4847fsuppimpd 7751 . . 3  |-  ( ph  ->  ( Y supp  ( 0g
`  R ) )  e.  Fin )
4915feqmptd 5827 . . . . . . 7  |-  ( ph  ->  Y  =  ( b  e.  D  |->  ( Y `
 b ) ) )
5049oveq1d 6211 . . . . . 6  |-  ( ph  ->  ( Y supp  ( 0g
`  R ) )  =  ( ( b  e.  D  |->  ( Y `
 b ) ) supp  ( 0g `  R
) ) )
51 eqimss2 3470 . . . . . 6  |-  ( ( Y supp  ( 0g `  R ) )  =  ( ( b  e.  D  |->  ( Y `  b ) ) supp  ( 0g `  R ) )  ->  ( ( b  e.  D  |->  ( Y `
 b ) ) supp  ( 0g `  R
) )  C_  ( Y supp  ( 0g `  R
) ) )
5250, 51syl 16 . . . . 5  |-  ( ph  ->  ( ( b  e.  D  |->  ( Y `  b ) ) supp  ( 0g `  R ) ) 
C_  ( Y supp  ( 0g `  R ) ) )
53 rhmghm 17487 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
54 eqid 2382 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
5545, 54ghmid 16390 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  ( F `  ( 0g `  R
) )  =  ( 0g `  S ) )
565, 53, 553syl 20 . . . . 5  |-  ( ph  ->  ( F `  ( 0g `  R ) )  =  ( 0g `  S ) )
57 fvex 5784 . . . . . 6  |-  ( Y `
 b )  e. 
_V
5857a1i 11 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  ( Y `  b )  e.  _V )
59 fvex 5784 . . . . . 6  |-  ( 0g
`  R )  e. 
_V
6059a1i 11 . . . . 5  |-  ( ph  ->  ( 0g `  R
)  e.  _V )
6152, 56, 58, 60suppssfv 6854 . . . 4  |-  ( ph  ->  ( ( b  e.  D  |->  ( F `  ( Y `  b ) ) ) supp  ( 0g
`  S ) ) 
C_  ( Y supp  ( 0g `  R ) ) )
627, 31, 54ringlz 17348 . . . . 5  |-  ( ( S  e.  Ring  /\  x  e.  C )  ->  (
( 0g `  S
)  .x.  x )  =  ( 0g `  S ) )
633, 62sylan 469 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  (
( 0g `  S
)  .x.  x )  =  ( 0g `  S ) )
64 fvex 5784 . . . . 5  |-  ( F `
 ( Y `  b ) )  e. 
_V
6564a1i 11 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( F `  ( Y `  b ) )  e. 
_V )
6661, 63, 65, 30, 44suppssov1 6850 . . 3  |-  ( ph  ->  ( ( b  e.  D  |->  ( ( F `
 ( Y `  b ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) supp  ( 0g
`  S ) ) 
C_  ( Y supp  ( 0g `  R ) ) )
67 suppssfifsupp 7759 . . 3  |-  ( ( ( ( b  e.  D  |->  ( ( F `
 ( Y `  b ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) )  e.  _V  /\ 
Fun  ( b  e.  D  |->  ( ( F `
 ( Y `  b ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) )  /\  ( 0g `  S )  e. 
_V )  /\  (
( Y supp  ( 0g `  R ) )  e. 
Fin  /\  ( (
b  e.  D  |->  ( ( F `  ( Y `  b )
)  .x.  ( T  gsumg  ( b  oF  .^  G ) ) ) ) supp  ( 0g `  S ) )  C_  ( Y supp  ( 0g `  R ) ) ) )  ->  ( b  e.  D  |->  ( ( F `  ( Y `
 b ) ) 
.x.  ( T  gsumg  ( b  oF  .^  G
) ) ) ) finSupp 
( 0g `  S
) )
6840, 42, 44, 48, 66, 67syl32anc 1234 . 2  |-  ( ph  ->  ( b  e.  D  |->  ( ( F `  ( Y `  b ) )  .x.  ( T 
gsumg  ( b  oF 
.^  G ) ) ) ) finSupp  ( 0g
`  S ) )
6935, 68jca 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 ) ) ) ) finSupp  ( 0g
`  S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   {crab 2736   _Vcvv 3034    C_ wss 3389   class class class wbr 4367    |-> cmpt 4425   `'ccnv 4912   "cima 4916   Fun wfun 5490   -->wf 5492   ` cfv 5496  (class class class)co 6196    oFcof 6437   supp csupp 6817    ^m cmap 7338   Fincfn 7435   finSupp cfsupp 7744   NNcn 10452   NN0cn0 10712   Basecbs 14634   .rcmulr 14703   0gc0g 14847    gsumg cgsu 14848  .gcmg 16173    GrpHom cghm 16381  CMndccmn 16915  mulGrpcmgp 17254   Ringcrg 17311   CRingccrg 17312   RingHom crh 17474   mVar cmvr 18114   mPoly cmpl 18115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-oi 7850  df-card 8233  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-seq 12011  df-hash 12308  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-sca 14718  df-vsca 14719  df-tset 14721  df-0g 14849  df-gsum 14850  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-mhm 16083  df-grp 16174  df-minusg 16175  df-mulg 16177  df-ghm 16382  df-cntz 16472  df-cmn 16917  df-mgp 17255  df-ur 17267  df-ring 17313  df-cring 17314  df-rnghom 17477  df-psr 18118  df-mpl 18120
This theorem is referenced by:  evlslem1  18297
  Copyright terms: Public domain W3C validator