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Theorem gsumpr 33223
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 994 . . 3  |-  ( ( G  e. CMnd  /\  ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  /\  ( C  e.  B  /\  D  e.  B
) )  ->  G  e. CMnd )
4 prfi 7787 . . . 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 3109 . . . . . 6  |-  k  e. 
_V
76elpr 4034 . . . . 5  |-  ( k  e.  { M ,  N }  <->  ( k  =  M  \/  k  =  N ) )
8 gsumpr.s . . . . . . 7  |-  ( k  =  M  ->  A  =  C )
9 eleq1a 2537 . . . . . . . . 9  |-  ( C  e.  B  ->  ( A  =  C  ->  A  e.  B ) )
109adantr 463 . . . . . . . 8  |-  ( ( C  e.  B  /\  D  e.  B )  ->  ( A  =  C  ->  A  e.  B
) )
11103ad2ant3 1017 . . . . . . 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 2537 . . . . . . . . 9  |-  ( D  e.  B  ->  ( A  =  D  ->  A  e.  B ) )
1514adantl 464 . . . . . . . 8  |-  ( ( C  e.  B  /\  D  e.  B )  ->  ( A  =  D  ->  A  e.  B
) )
16153ad2ant3 1017 . . . . . . 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 377 . . . . 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 428 . . 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 4077 . . . . 5  |-  ( M  =/=  N  ->  ( { M }  i^i  { N } )  =  (/) )
22213ad2ant3 1017 . . . 4  |-  ( ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  -> 
( { M }  i^i  { N } )  =  (/) )
23223ad2ant2 1016 . . 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 4019 . . . 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 2454 . . 3  |-  ( k  e.  { M ,  N }  |->  A )  =  ( k  e. 
{ M ,  N }  |->  A )
271, 2, 3, 5, 20, 23, 25, 26gsummptfidmsplitres 17152 . 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 4165 . . . . . 6  |-  { M }  C_  { M ,  N }
29 resmpt 5311 . . . . . 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 6286 . . . 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 17015 . . . . 5  |-  ( G  e. CMnd  ->  G  e.  Mnd )
33 simp1 994 . . . . 5  |-  ( ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  ->  M  e.  V )
34 simpl 455 . . . . 5  |-  ( ( C  e.  B  /\  D  e.  B )  ->  C  e.  B )
351, 8gsumsn 17180 . . . . 5  |-  ( ( G  e.  Mnd  /\  M  e.  V  /\  C  e.  B )  ->  ( G  gsumg  ( k  e.  { M }  |->  A ) )  =  C )
3632, 33, 34, 35syl3an 1268 . . . 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 2495 . . 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 4166 . . . . . 6  |-  { N }  C_  { M ,  N }
39 resmpt 5311 . . . . . 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 6286 . . . 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 995 . . . . 5  |-  ( ( M  e.  V  /\  N  e.  W  /\  M  =/=  N )  ->  N  e.  W )
43 simpr 459 . . . . 5  |-  ( ( C  e.  B  /\  D  e.  B )  ->  D  e.  B )
441, 13gsumsn 17180 . . . . 5  |-  ( ( G  e.  Mnd  /\  N  e.  W  /\  D  e.  B )  ->  ( G  gsumg  ( k  e.  { N }  |->  A ) )  =  D )
4532, 42, 43, 44syl3an 1268 . . . 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 2495 . . 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 6288 . 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 2495 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 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3783   {csn 4016   {cpr 4018    |-> cmpt 4497    |` cres 4990   ` cfv 5570  (class class class)co 6270   Fincfn 7509   Basecbs 14719   +g cplusg 14787    gsumg cgsu 14933   Mndcmnd 16121  CMndccmn 17000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  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 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12093  df-hash 12391  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-0g 14934  df-gsum 14935  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-mulg 16262  df-cntz 16557  df-cmn 17002
This theorem is referenced by:  lincvalpr  33292  zlmodzxzldeplem3  33376
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