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Theorem gsumpr 30758
Description: Group sum of a pair. (Contributed by AV, 6-Dec-2018.) (Proof shortened by AV, 28-Jul-2019.)
Hypotheses
Ref Expression
gsumpr.b  |-  B  =  ( Base `  G
)
gsumpr.p  |-  .+  =  ( +g  `  G )
gsumpr.s  |-  ( k  =  M  ->  A  =  C )
gsumpr.t  |-  ( k  =  N  ->  A  =  D )
Assertion
Ref Expression
gsumpr  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  ( G  gsumg  ( k  e.  { M ,  N }  |->  A ) )  =  ( C  .+  D
) )
Distinct variable groups:    B, k    C, k    D, k    k, G   
k, M    k, N    k, V    k, W
Allowed substitution hints:    A( k)    .+ ( k)

Proof of Theorem gsumpr
StepHypRef Expression
1 gsumpr.b . . 3  |-  B  =  ( Base `  G
)
2 gsumpr.p . . 3  |-  .+  =  ( +g  `  G )
3 simp1 988 . . 3  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  G  e. CMnd )
4 prfi 7586 . . . 4  |-  { M ,  N }  e.  Fin
54a1i 11 . . 3  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  { M ,  N }  e.  Fin )
6 vex 2975 . . . . . 6  |-  k  e. 
_V
76elpr 3895 . . . . 5  |-  ( k  e.  { M ,  N }  <->  ( k  =  M  \/  k  =  N ) )
8 gsumpr.s . . . . . . 7  |-  ( k  =  M  ->  A  =  C )
9 eleq1a 2512 . . . . . . . . 9  |-  ( C  e.  B  ->  ( A  =  C  ->  A  e.  B ) )
109adantr 465 . . . . . . . 8  |-  ( ( C  e.  B  /\  D  e.  B )  ->  ( A  =  C  ->  A  e.  B
) )
11103ad2ant3 1011 . . . . . . 7  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  ( A  =  C  ->  A  e.  B ) )
128, 11syl5com 30 . . . . . 6  |-  ( k  =  M  ->  (
( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N
)  /\  ( C  e.  B  /\  D  e.  B ) )  ->  A  e.  B )
)
13 gsumpr.t . . . . . . 7  |-  ( k  =  N  ->  A  =  D )
14 eleq1a 2512 . . . . . . . . 9  |-  ( D  e.  B  ->  ( A  =  D  ->  A  e.  B ) )
1514adantl 466 . . . . . . . 8  |-  ( ( C  e.  B  /\  D  e.  B )  ->  ( A  =  D  ->  A  e.  B
) )
16153ad2ant3 1011 . . . . . . 7  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  ( A  =  D  ->  A  e.  B ) )
1713, 16syl5com 30 . . . . . 6  |-  ( k  =  N  ->  (
( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N
)  /\  ( C  e.  B  /\  D  e.  B ) )  ->  A  e.  B )
)
1812, 17jaoi 379 . . . . 5  |-  ( ( k  =  M  \/  k  =  N )  ->  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N
)  /\  ( C  e.  B  /\  D  e.  B ) )  ->  A  e.  B )
)
197, 18sylbi 195 . . . 4  |-  ( k  e.  { M ,  N }  ->  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  A  e.  B ) )
2019impcom 430 . . 3  |-  ( ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N
)  /\  ( C  e.  B  /\  D  e.  B ) )  /\  k  e.  { M ,  N } )  ->  A  e.  B )
21 disjsn2 3937 . . . . 5  |-  ( M  =/=  N  ->  ( { M }  i^i  { N } )  =  (/) )
22213ad2ant3 1011 . . . 4  |-  ( ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  -> 
( { M }  i^i  { N } )  =  (/) )
23223ad2ant2 1010 . . 3  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  ( { M }  i^i  { N } )  =  (/) )
24 df-pr 3880 . . . 4  |-  { M ,  N }  =  ( { M }  u.  { N } )
2524a1i 11 . . 3  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  { M ,  N }  =  ( { M }  u.  { N } ) )
26 eqid 2443 . . 3  |-  ( k  e.  { M ,  N }  |->  A )  =  ( k  e. 
{ M ,  N }  |->  A )
271, 2, 3, 5, 20, 23, 25, 26gsummptfidmsplitres 16425 . 2  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  ( G  gsumg  ( k  e.  { M ,  N }  |->  A ) )  =  ( ( G  gsumg  ( ( k  e.  { M ,  N }  |->  A )  |`  { M } ) )  .+  ( G 
gsumg  ( ( k  e. 
{ M ,  N }  |->  A )  |`  { N } ) ) ) )
28 snsspr1 4022 . . . . . 6  |-  { M }  C_  { M ,  N }
29 resmpt 5156 . . . . . 6  |-  ( { M }  C_  { M ,  N }  ->  (
( k  e.  { M ,  N }  |->  A )  |`  { M } )  =  ( k  e.  { M }  |->  A ) )
3028, 29mp1i 12 . . . . 5  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  (
( k  e.  { M ,  N }  |->  A )  |`  { M } )  =  ( k  e.  { M }  |->  A ) )
3130oveq2d 6107 . . . 4  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  ( G  gsumg  ( ( k  e. 
{ M ,  N }  |->  A )  |`  { M } ) )  =  ( G  gsumg  ( k  e.  { M }  |->  A ) ) )
32 cmnmnd 16292 . . . . 5  |-  ( G  e. CMnd  ->  G  e.  Mnd )
33 simp1 988 . . . . 5  |-  ( ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  ->  M  e.  V )
34 simpl 457 . . . . 5  |-  ( ( C  e.  B  /\  D  e.  B )  ->  C  e.  B )
351, 8gsumsn 16449 . . . . 5  |-  ( ( G  e.  Mnd  /\  M  e.  V  /\  C  e.  B )  ->  ( G  gsumg  ( k  e.  { M }  |->  A ) )  =  C )
3632, 33, 34, 35syl3an 1260 . . . 4  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  ( G  gsumg  ( k  e.  { M }  |->  A ) )  =  C )
3731, 36eqtrd 2475 . . 3  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  ( G  gsumg  ( ( k  e. 
{ M ,  N }  |->  A )  |`  { M } ) )  =  C )
38 snsspr2 4023 . . . . . 6  |-  { N }  C_  { M ,  N }
39 resmpt 5156 . . . . . 6  |-  ( { N }  C_  { M ,  N }  ->  (
( k  e.  { M ,  N }  |->  A )  |`  { N } )  =  ( k  e.  { N }  |->  A ) )
4038, 39mp1i 12 . . . . 5  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  (
( k  e.  { M ,  N }  |->  A )  |`  { N } )  =  ( k  e.  { N }  |->  A ) )
4140oveq2d 6107 . . . 4  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  ( G  gsumg  ( ( k  e. 
{ M ,  N }  |->  A )  |`  { N } ) )  =  ( G  gsumg  ( k  e.  { N }  |->  A ) ) )
42 simp2 989 . . . . 5  |-  ( ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  ->  N  e.  W )
43 simpr 461 . . . . 5  |-  ( ( C  e.  B  /\  D  e.  B )  ->  D  e.  B )
441, 13gsumsn 16449 . . . . 5  |-  ( ( G  e.  Mnd  /\  N  e.  W  /\  D  e.  B )  ->  ( G  gsumg  ( k  e.  { N }  |->  A ) )  =  D )
4532, 42, 43, 44syl3an 1260 . . . 4  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  ( G  gsumg  ( k  e.  { N }  |->  A ) )  =  D )
4641, 45eqtrd 2475 . . 3  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  ( G  gsumg  ( ( k  e. 
{ M ,  N }  |->  A )  |`  { N } ) )  =  D )
4737, 46oveq12d 6109 . 2  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  (
( G  gsumg  ( ( k  e. 
{ M ,  N }  |->  A )  |`  { M } ) ) 
.+  ( G  gsumg  ( ( k  e.  { M ,  N }  |->  A )  |`  { N } ) ) )  =  ( C  .+  D ) )
4827, 47eqtrd 2475 1  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  ( G  gsumg  ( k  e.  { M ,  N }  |->  A ) )  =  ( C  .+  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606    u. cun 3326    i^i cin 3327    C_ wss 3328   (/)c0 3637   {csn 3877   {cpr 3879    e. cmpt 4350    |` cres 4842   ` cfv 5418  (class class class)co 6091   Fincfn 7310   Basecbs 14174   +g cplusg 14238    gsumg cgsu 14379   Mndcmnd 15409  CMndccmn 16277
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-0g 14380  df-gsum 14381  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279
This theorem is referenced by:  lincvalpr  30952  zlmodzxzldeplem3  31044
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