MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsummgp0 Structured version   Unicode version

Theorem gsummgp0 17033
Description: If one factor in a finite group sum of the multiplicative group of a commutative ring is 0, the whole "sum" (i.e. product) is 0. (Contributed by AV, 3-Jan-2019.)
Hypotheses
Ref Expression
gsummgp0.g  |-  G  =  (mulGrp `  R )
gsummgp0.0  |-  .0.  =  ( 0g `  R )
gsummgp0.r  |-  ( ph  ->  R  e.  CRing )
gsummgp0.n  |-  ( ph  ->  N  e.  Fin )
gsummgp0.a  |-  ( (
ph  /\  n  e.  N )  ->  A  e.  ( Base `  R
) )
gsummgp0.e  |-  ( (
ph  /\  n  =  i )  ->  A  =  B )
gsummgp0.b  |-  ( ph  ->  E. i  e.  N  B  =  .0.  )
Assertion
Ref Expression
gsummgp0  |-  ( ph  ->  ( G  gsumg  ( n  e.  N  |->  A ) )  =  .0.  )
Distinct variable groups:    A, i    B, n    i, n, G   
i, N, n    R, n    ph, i, n    .0. , i, n
Allowed substitution hints:    A( n)    B( i)    R( i)

Proof of Theorem gsummgp0
StepHypRef Expression
1 gsummgp0.b . 2  |-  ( ph  ->  E. i  e.  N  B  =  .0.  )
2 difsnid 4166 . . . . . . 7  |-  ( i  e.  N  ->  (
( N  \  {
i } )  u. 
{ i } )  =  N )
32eqcomd 2468 . . . . . 6  |-  ( i  e.  N  ->  N  =  ( ( N 
\  { i } )  u.  { i } ) )
43ad2antrl 727 . . . . 5  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  ->  N  =  ( ( N  \  { i } )  u.  { i } ) )
54mpteq1d 4521 . . . 4  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( n  e.  N  |->  A )  =  ( n  e.  ( ( N  \  { i } )  u.  {
i } )  |->  A ) )
65oveq2d 6291 . . 3  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( G  gsumg  ( n  e.  N  |->  A ) )  =  ( G  gsumg  ( n  e.  ( ( N  \  {
i } )  u. 
{ i } ) 
|->  A ) ) )
7 gsummgp0.g . . . . 5  |-  G  =  (mulGrp `  R )
8 eqid 2460 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
97, 8mgpbas 16930 . . . 4  |-  ( Base `  R )  =  (
Base `  G )
10 eqid 2460 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
117, 10mgpplusg 16928 . . . 4  |-  ( .r
`  R )  =  ( +g  `  G
)
12 gsummgp0.r . . . . . 6  |-  ( ph  ->  R  e.  CRing )
137crngmgp 16987 . . . . . 6  |-  ( R  e.  CRing  ->  G  e. CMnd )
1412, 13syl 16 . . . . 5  |-  ( ph  ->  G  e. CMnd )
1514adantr 465 . . . 4  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  ->  G  e. CMnd )
16 gsummgp0.n . . . . . 6  |-  ( ph  ->  N  e.  Fin )
17 diffi 7740 . . . . . 6  |-  ( N  e.  Fin  ->  ( N  \  { i } )  e.  Fin )
1816, 17syl 16 . . . . 5  |-  ( ph  ->  ( N  \  {
i } )  e. 
Fin )
1918adantr 465 . . . 4  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( N  \  {
i } )  e. 
Fin )
20 simpl 457 . . . . 5  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  ->  ph )
21 eldifi 3619 . . . . 5  |-  ( n  e.  ( N  \  { i } )  ->  n  e.  N
)
22 gsummgp0.a . . . . 5  |-  ( (
ph  /\  n  e.  N )  ->  A  e.  ( Base `  R
) )
2320, 21, 22syl2an 477 . . . 4  |-  ( ( ( ph  /\  (
i  e.  N  /\  B  =  .0.  )
)  /\  n  e.  ( N  \  { i } ) )  ->  A  e.  ( Base `  R ) )
24 simprl 755 . . . 4  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
i  e.  N )
25 neldifsnd 4148 . . . 4  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  ->  -.  i  e.  ( N  \  { i } ) )
26 crngrng 16989 . . . . . . . 8  |-  ( R  e.  CRing  ->  R  e.  Ring )
2712, 26syl 16 . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
28 rngmnd 16988 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
29 gsummgp0.0 . . . . . . . 8  |-  .0.  =  ( 0g `  R )
308, 29mndidcl 15745 . . . . . . 7  |-  ( R  e.  Mnd  ->  .0.  e.  ( Base `  R
) )
3127, 28, 303syl 20 . . . . . 6  |-  ( ph  ->  .0.  e.  ( Base `  R ) )
3231adantr 465 . . . . 5  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  ->  .0.  e.  ( Base `  R
) )
33 eleq1 2532 . . . . . 6  |-  ( B  =  .0.  ->  ( B  e.  ( Base `  R )  <->  .0.  e.  ( Base `  R )
) )
3433ad2antll 728 . . . . 5  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( B  e.  (
Base `  R )  <->  .0. 
e.  ( Base `  R
) ) )
3532, 34mpbird 232 . . . 4  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  ->  B  e.  ( Base `  R ) )
36 gsummgp0.e . . . . 5  |-  ( (
ph  /\  n  =  i )  ->  A  =  B )
3736adantlr 714 . . . 4  |-  ( ( ( ph  /\  (
i  e.  N  /\  B  =  .0.  )
)  /\  n  =  i )  ->  A  =  B )
389, 11, 15, 19, 23, 24, 25, 35, 37gsumunsnd 16768 . . 3  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( G  gsumg  ( n  e.  ( ( N  \  {
i } )  u. 
{ i } ) 
|->  A ) )  =  ( ( G  gsumg  ( n  e.  ( N  \  { i } ) 
|->  A ) ) ( .r `  R ) B ) )
39 oveq2 6283 . . . . 5  |-  ( B  =  .0.  ->  (
( G  gsumg  ( n  e.  ( N  \  { i } )  |->  A ) ) ( .r `  R ) B )  =  ( ( G 
gsumg  ( n  e.  ( N  \  { i } )  |->  A ) ) ( .r `  R
)  .0.  ) )
4039ad2antll 728 . . . 4  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( ( G  gsumg  ( n  e.  ( N  \  { i } ) 
|->  A ) ) ( .r `  R ) B )  =  ( ( G  gsumg  ( n  e.  ( N  \  { i } )  |->  A ) ) ( .r `  R )  .0.  )
)
4127adantr 465 . . . . 5  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  ->  R  e.  Ring )
4221, 22sylan2 474 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( N  \  { i } ) )  ->  A  e.  ( Base `  R ) )
4342ralrimiva 2871 . . . . . . 7  |-  ( ph  ->  A. n  e.  ( N  \  { i } ) A  e.  ( Base `  R
) )
449, 14, 18, 43gsummptcl 16778 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( n  e.  ( N  \  { i } )  |->  A ) )  e.  ( Base `  R ) )
4544adantr 465 . . . . 5  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( G  gsumg  ( n  e.  ( N  \  { i } )  |->  A ) )  e.  ( Base `  R ) )
468, 10, 29rngrz 17016 . . . . 5  |-  ( ( R  e.  Ring  /\  ( G  gsumg  ( n  e.  ( N  \  { i } )  |->  A ) )  e.  ( Base `  R ) )  -> 
( ( G  gsumg  ( n  e.  ( N  \  { i } ) 
|->  A ) ) ( .r `  R )  .0.  )  =  .0.  )
4741, 45, 46syl2anc 661 . . . 4  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( ( G  gsumg  ( n  e.  ( N  \  { i } ) 
|->  A ) ) ( .r `  R )  .0.  )  =  .0.  )
4840, 47eqtrd 2501 . . 3  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( ( G  gsumg  ( n  e.  ( N  \  { i } ) 
|->  A ) ) ( .r `  R ) B )  =  .0.  )
496, 38, 483eqtrd 2505 . 2  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( G  gsumg  ( n  e.  N  |->  A ) )  =  .0.  )
501, 49rexlimddv 2952 1  |-  ( ph  ->  ( G  gsumg  ( n  e.  N  |->  A ) )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   E.wrex 2808    \ cdif 3466    u. cun 3467   {csn 4020    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275   Fincfn 7506   Basecbs 14479   .rcmulr 14545   0gc0g 14684    gsumg cgsu 14685   Mndcmnd 15715  CMndccmn 16587  mulGrpcmgp 16924   Ringcrg 16979   CRingccrg 16980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-seq 12064  df-hash 12361  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-0g 14686  df-gsum 14687  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-grp 15851  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-mgp 16925  df-rng 16981  df-cring 16982
This theorem is referenced by:  smadiadetlem0  18923
  Copyright terms: Public domain W3C validator