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Theorem evlslem6 17732
Description: Lemma for evlseu 17736. 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 crngrng 16788 . . . . . 6  |-  ( S  e.  CRing  ->  S  e.  Ring )
31, 2syl 16 . . . . 5  |-  ( ph  ->  S  e.  Ring )
43adantr 465 . . . 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 16949 . . . . . . 7  |-  ( F  e.  ( R RingHom  S
)  ->  F : K
--> C )
95, 8syl 16 . . . . . 6  |-  ( ph  ->  F : K --> C )
109adantr 465 . . . . 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 17643 . . . . . 6  |-  ( ph  ->  Y : D --> K )
1615ffvelrnda 5955 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  ( Y `  b )  e.  K )
1710, 16ffvelrnd 5956 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( F `  ( Y `  b ) )  e.  C )
18 evlslem1.t . . . . . 6  |-  T  =  (mulGrp `  S )
1918, 7mgpbas 16729 . . . . 5  |-  C  =  ( Base `  T
)
20 evlslem1.x . . . . 5  |-  .^  =  (.g
`  T )
21 eqid 2454 . . . . 5  |-  ( 0g
`  T )  =  ( 0g `  T
)
2218crngmgp 16786 . . . . . . 7  |-  ( S  e.  CRing  ->  T  e. CMnd )
231, 22syl 16 . . . . . 6  |-  ( ph  ->  T  e. CMnd )
2423adantr 465 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  T  e. CMnd )
25 simpr 461 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  b  e.  D )
26 evlslem1.g . . . . . 6  |-  ( ph  ->  G : I --> C )
2726adantr 465 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  G : I --> C )
28 evlslem1.i . . . . . 6  |-  ( ph  ->  I  e.  _V )
2928adantr 465 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  I  e.  _V )
3013, 19, 20, 21, 24, 25, 27, 29psrbagev2 17730 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( T  gsumg  ( b  oF 
.^  G ) )  e.  C )
31 evlslem1.m . . . . 5  |-  .x.  =  ( .r `  S )
327, 31rngcl 16791 . . . 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 1219 . . 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 5979 . 2  |-  ( ph  ->  ( b  e.  D  |->  ( ( F `  ( Y `  b ) )  .x.  ( T 
gsumg  ( b  oF 
.^  G ) ) ) ) : D --> C )
36 ovex 6228 . . . . . 6  |-  ( NN0 
^m  I )  e. 
_V
3736a1i 11 . . . . 5  |-  ( ph  ->  ( NN0  ^m  I
)  e.  _V )
3813, 37rabexd 4555 . . . 4  |-  ( ph  ->  D  e.  _V )
39 mptexg 6059 . . . 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 5565 . . . 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 5812 . . . 4  |-  ( 0g
`  S )  e. 
_V
4443a1i 11 . . 3  |-  ( ph  ->  ( 0g `  S
)  e.  _V )
45 eqid 2454 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
46 evlslem1.r . . . . 5  |-  ( ph  ->  R  e.  CRing )
4711, 12, 45, 14, 46mplelsfi 17706 . . . 4  |-  ( ph  ->  Y finSupp  ( 0g `  R ) )
4847fsuppimpd 7741 . . 3  |-  ( ph  ->  ( Y supp  ( 0g
`  R ) )  e.  Fin )
4915feqmptd 5856 . . . . . . 7  |-  ( ph  ->  Y  =  ( b  e.  D  |->  ( Y `
 b ) ) )
5049oveq1d 6218 . . . . . 6  |-  ( ph  ->  ( Y supp  ( 0g
`  R ) )  =  ( ( b  e.  D  |->  ( Y `
 b ) ) supp  ( 0g `  R
) ) )
51 eqimss2 3520 . . . . . 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 16948 . . . . . 6  |-  ( F  e.  ( R RingHom  S
)  ->  F  e.  ( R  GrpHom  S ) )
54 eqid 2454 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
5545, 54ghmid 15876 . . . . . 6  |-  ( F  e.  ( R  GrpHom  S )  ->  ( F `  ( 0g `  R
) )  =  ( 0g `  S ) )
565, 53, 553syl 20 . . . . 5  |-  ( ph  ->  ( F `  ( 0g `  R ) )  =  ( 0g `  S ) )
57 fvex 5812 . . . . . 6  |-  ( Y `
 b )  e. 
_V
5857a1i 11 . . . . 5  |-  ( (
ph  /\  b  e.  D )  ->  ( Y `  b )  e.  _V )
59 fvex 5812 . . . . . 6  |-  ( 0g
`  R )  e. 
_V
6059a1i 11 . . . . 5  |-  ( ph  ->  ( 0g `  R
)  e.  _V )
6152, 56, 58, 60suppssfv 6838 . . . 4  |-  ( ph  ->  ( ( b  e.  D  |->  ( F `  ( Y `  b ) ) ) supp  ( 0g
`  S ) ) 
C_  ( Y supp  ( 0g `  R ) ) )
627, 31, 54rnglz 16814 . . . . 5  |-  ( ( S  e.  Ring  /\  x  e.  C )  ->  (
( 0g `  S
)  .x.  x )  =  ( 0g `  S ) )
633, 62sylan 471 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  (
( 0g `  S
)  .x.  x )  =  ( 0g `  S ) )
64 fvex 5812 . . . . 5  |-  ( F `
 ( Y `  b ) )  e. 
_V
6564a1i 11 . . . 4  |-  ( (
ph  /\  b  e.  D )  ->  ( F `  ( Y `  b ) )  e. 
_V )
6661, 63, 65, 30, 44suppssov1 6834 . . 3  |-  ( ph  ->  ( ( b  e.  D  |->  ( ( F `
 ( Y `  b ) )  .x.  ( T  gsumg  ( b  oF 
.^  G ) ) ) ) supp  ( 0g
`  S ) ) 
C_  ( Y supp  ( 0g `  R ) ) )
67 suppssfifsupp 7749 . . 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 1227 . 2  |-  ( ph  ->  ( b  e.  D  |->  ( ( F `  ( Y `  b ) )  .x.  ( T 
gsumg  ( b  oF 
.^  G ) ) ) ) finSupp  ( 0g
`  S ) )
6935, 68jca 532 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 369    = wceq 1370    e. wcel 1758   {crab 2803   _Vcvv 3078    C_ wss 3439   class class class wbr 4403    |-> cmpt 4461   `'ccnv 4950   "cima 4954   Fun wfun 5523   -->wf 5525   ` cfv 5529  (class class class)co 6203    oFcof 6431   supp csupp 6803    ^m cmap 7327   Fincfn 7423   finSupp cfsupp 7734   NNcn 10437   NN0cn0 10694   Basecbs 14296   .rcmulr 14362   0gc0g 14501    gsumg cgsu 14502  .gcmg 15537    GrpHom cghm 15867  CMndccmn 16402  mulGrpcmgp 16723   Ringcrg 16778   CRingccrg 16779   RingHom crh 16937   mVar cmvr 17552   mPoly cmpl 17553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-oi 7839  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-fzo 11670  df-seq 11928  df-hash 12225  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-ress 14303  df-plusg 14374  df-mulr 14375  df-sca 14377  df-vsca 14378  df-tset 14380  df-0g 14503  df-gsum 14504  df-mnd 15538  df-mhm 15587  df-grp 15668  df-minusg 15669  df-mulg 15671  df-ghm 15868  df-cntz 15958  df-cmn 16404  df-mgp 16724  df-ur 16736  df-rng 16780  df-cring 16781  df-rnghom 16939  df-psr 17556  df-mpl 17558
This theorem is referenced by:  evlslem1  17735
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