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Theorem gsummgp0 16721
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 4040 . . . . . . 7  |-  ( i  e.  N  ->  (
( N  \  {
i } )  u. 
{ i } )  =  N )
32eqcomd 2448 . . . . . 6  |-  ( i  e.  N  ->  N  =  ( ( N 
\  { i } )  u.  { i } ) )
43ad2antrl 727 . . . . 5  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  ->  N  =  ( ( N  \  { i } )  u.  { i } ) )
54mpteq1d 4394 . . . 4  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( n  e.  N  |->  A )  =  ( n  e.  ( ( N  \  { i } )  u.  {
i } )  |->  A ) )
65oveq2d 6128 . . 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 2443 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
97, 8mgpbas 16619 . . . 4  |-  ( Base `  R )  =  (
Base `  G )
10 eqid 2443 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
117, 10mgpplusg 16617 . . . 4  |-  ( .r
`  R )  =  ( +g  `  G
)
12 gsummgp0.r . . . . . 6  |-  ( ph  ->  R  e.  CRing )
137crngmgp 16675 . . . . . 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 7564 . . . . . 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 3499 . . . . 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 4024 . . . 4  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  ->  -.  i  e.  ( N  \  { i } ) )
26 crngrng 16677 . . . . . . . 8  |-  ( R  e.  CRing  ->  R  e.  Ring )
2712, 26syl 16 . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
28 rngmnd 16676 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
29 gsummgp0.0 . . . . . . . 8  |-  .0.  =  ( 0g `  R )
308, 29mndidcl 15460 . . . . . . 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 2503 . . . . . 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 16474 . . 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 6120 . . . . 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 2820 . . . . . . 7  |-  ( ph  ->  A. n  e.  ( N  \  { i } ) A  e.  ( Base `  R
) )
449, 14, 18, 43gsummptcl 16480 . . . . . 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 16704 . . . . 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 2475 . . 3  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( ( G  gsumg  ( n  e.  ( N  \  { i } ) 
|->  A ) ) ( .r `  R ) B )  =  .0.  )
496, 38, 483eqtrd 2479 . 2  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( G  gsumg  ( n  e.  N  |->  A ) )  =  .0.  )
501, 49rexlimddv 2866 1  |-  ( ph  ->  ( G  gsumg  ( n  e.  N  |->  A ) )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2737    \ cdif 3346    u. cun 3347   {csn 3898    e. cmpt 4371   ` cfv 5439  (class class class)co 6112   Fincfn 7331   Basecbs 14195   .rcmulr 14260   0gc0g 14399    gsumg cgsu 14400   Mndcmnd 15430  CMndccmn 16298  mulGrpcmgp 16613   Ringcrg 16667   CRingccrg 16668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-om 6498  df-1st 6598  df-2nd 6599  df-supp 6712  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fsupp 7642  df-oi 7745  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-fzo 11570  df-seq 11828  df-hash 12125  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-0g 14401  df-gsum 14402  df-mre 14545  df-mrc 14546  df-acs 14548  df-mnd 15436  df-submnd 15486  df-grp 15566  df-mulg 15569  df-cntz 15856  df-cmn 16300  df-mgp 16614  df-rng 16669  df-cring 16670
This theorem is referenced by:  smadiadetlem0  18489
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