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Theorem gsummgp0 17576
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 4118 . . . . . . 7  |-  ( i  e.  N  ->  (
( N  \  {
i } )  u. 
{ i } )  =  N )
32eqcomd 2410 . . . . . 6  |-  ( i  e.  N  ->  N  =  ( ( N 
\  { i } )  u.  { i } ) )
43ad2antrl 726 . . . . 5  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  ->  N  =  ( ( N  \  { i } )  u.  { i } ) )
54mpteq1d 4476 . . . 4  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( n  e.  N  |->  A )  =  ( n  e.  ( ( N  \  { i } )  u.  {
i } )  |->  A ) )
65oveq2d 6294 . . 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 2402 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
97, 8mgpbas 17467 . . . 4  |-  ( Base `  R )  =  (
Base `  G )
10 eqid 2402 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
117, 10mgpplusg 17465 . . . 4  |-  ( .r
`  R )  =  ( +g  `  G
)
12 gsummgp0.r . . . . . 6  |-  ( ph  ->  R  e.  CRing )
137crngmgp 17526 . . . . . 6  |-  ( R  e.  CRing  ->  G  e. CMnd )
1412, 13syl 17 . . . . 5  |-  ( ph  ->  G  e. CMnd )
1514adantr 463 . . . 4  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  ->  G  e. CMnd )
16 gsummgp0.n . . . . . 6  |-  ( ph  ->  N  e.  Fin )
17 diffi 7786 . . . . . 6  |-  ( N  e.  Fin  ->  ( N  \  { i } )  e.  Fin )
1816, 17syl 17 . . . . 5  |-  ( ph  ->  ( N  \  {
i } )  e. 
Fin )
1918adantr 463 . . . 4  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( N  \  {
i } )  e. 
Fin )
20 simpl 455 . . . . 5  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  ->  ph )
21 eldifi 3565 . . . . 5  |-  ( n  e.  ( N  \  { i } )  ->  n  e.  N
)
22 gsummgp0.a . . . . 5  |-  ( (
ph  /\  n  e.  N )  ->  A  e.  ( Base `  R
) )
2320, 21, 22syl2an 475 . . . 4  |-  ( ( ( ph  /\  (
i  e.  N  /\  B  =  .0.  )
)  /\  n  e.  ( N  \  { i } ) )  ->  A  e.  ( Base `  R ) )
24 simprl 756 . . . 4  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
i  e.  N )
25 neldifsnd 4100 . . . 4  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  ->  -.  i  e.  ( N  \  { i } ) )
26 crngring 17529 . . . . . . . 8  |-  ( R  e.  CRing  ->  R  e.  Ring )
2712, 26syl 17 . . . . . . 7  |-  ( ph  ->  R  e.  Ring )
28 ringmnd 17527 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Mnd )
29 gsummgp0.0 . . . . . . . 8  |-  .0.  =  ( 0g `  R )
308, 29mndidcl 16262 . . . . . . 7  |-  ( R  e.  Mnd  ->  .0.  e.  ( Base `  R
) )
3127, 28, 303syl 18 . . . . . 6  |-  ( ph  ->  .0.  e.  ( Base `  R ) )
3231adantr 463 . . . . 5  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  ->  .0.  e.  ( Base `  R
) )
33 eleq1 2474 . . . . . 6  |-  ( B  =  .0.  ->  ( B  e.  ( Base `  R )  <->  .0.  e.  ( Base `  R )
) )
3433ad2antll 727 . . . . 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 713 . . . 4  |-  ( ( ( ph  /\  (
i  e.  N  /\  B  =  .0.  )
)  /\  n  =  i )  ->  A  =  B )
389, 11, 15, 19, 23, 24, 25, 35, 37gsumunsnd 17305 . . 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 6286 . . . . 5  |-  ( B  =  .0.  ->  (
( G  gsumg  ( n  e.  ( N  \  { i } )  |->  A ) ) ( .r `  R ) B )  =  ( ( G 
gsumg  ( n  e.  ( N  \  { i } )  |->  A ) ) ( .r `  R
)  .0.  ) )
4039ad2antll 727 . . . 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 463 . . . . 5  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  ->  R  e.  Ring )
4221, 22sylan2 472 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( N  \  { i } ) )  ->  A  e.  ( Base `  R ) )
4342ralrimiva 2818 . . . . . . 7  |-  ( ph  ->  A. n  e.  ( N  \  { i } ) A  e.  ( Base `  R
) )
449, 14, 18, 43gsummptcl 17315 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( n  e.  ( N  \  { i } )  |->  A ) )  e.  ( Base `  R ) )
4544adantr 463 . . . . 5  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( G  gsumg  ( n  e.  ( N  \  { i } )  |->  A ) )  e.  ( Base `  R ) )
468, 10, 29ringrz 17556 . . . . 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 659 . . . 4  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( ( G  gsumg  ( n  e.  ( N  \  { i } ) 
|->  A ) ) ( .r `  R )  .0.  )  =  .0.  )
4840, 47eqtrd 2443 . . 3  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( ( G  gsumg  ( n  e.  ( N  \  { i } ) 
|->  A ) ) ( .r `  R ) B )  =  .0.  )
496, 38, 483eqtrd 2447 . 2  |-  ( (
ph  /\  ( i  e.  N  /\  B  =  .0.  ) )  -> 
( G  gsumg  ( n  e.  N  |->  A ) )  =  .0.  )
501, 49rexlimddv 2900 1  |-  ( ph  ->  ( G  gsumg  ( n  e.  N  |->  A ) )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   E.wrex 2755    \ cdif 3411    u. cun 3412   {csn 3972    |-> cmpt 4453   ` cfv 5569  (class class class)co 6278   Fincfn 7554   Basecbs 14841   .rcmulr 14910   0gc0g 15054    gsumg cgsu 15055   Mndcmnd 16243  CMndccmn 17122  mulGrpcmgp 17461   Ringcrg 17518   CRingccrg 17519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-fzo 11855  df-seq 12152  df-hash 12453  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-0g 15056  df-gsum 15057  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-grp 16381  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-mgp 17462  df-ring 17520  df-cring 17521
This theorem is referenced by:  smadiadetlem0  19455
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