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Theorem gsumprval 15824
Description: Value of the group sum operation over a pair of sequential integers. (Contributed by AV, 14-Dec-2018.)
Hypotheses
Ref Expression
gsumprval.b  |-  B  =  ( Base `  G
)
gsumprval.p  |-  .+  =  ( +g  `  G )
gsumprval.g  |-  ( ph  ->  G  e.  V )
gsumprval.m  |-  ( ph  ->  M  e.  ZZ )
gsumprval.n  |-  ( ph  ->  N  =  ( M  +  1 ) )
gsumprval.f  |-  ( ph  ->  F : { M ,  N } --> B )
Assertion
Ref Expression
gsumprval  |-  ( ph  ->  ( G  gsumg  F )  =  ( ( F `  M
)  .+  ( F `  N ) ) )

Proof of Theorem gsumprval
StepHypRef Expression
1 gsumprval.b . . 3  |-  B  =  ( Base `  G
)
2 gsumprval.p . . 3  |-  .+  =  ( +g  `  G )
3 gsumprval.g . . 3  |-  ( ph  ->  G  e.  V )
4 gsumprval.m . . . . 5  |-  ( ph  ->  M  e.  ZZ )
5 uzid 11092 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
64, 5syl 16 . . . 4  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
7 peano2uz 11130 . . . 4  |-  ( M  e.  ( ZZ>= `  M
)  ->  ( M  +  1 )  e.  ( ZZ>= `  M )
)
86, 7syl 16 . . 3  |-  ( ph  ->  ( M  +  1 )  e.  ( ZZ>= `  M ) )
9 gsumprval.f . . . 4  |-  ( ph  ->  F : { M ,  N } --> B )
10 fzpr 11731 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M ... ( M  + 
1 ) )  =  { M ,  ( M  +  1 ) } )
114, 10syl 16 . . . . . 6  |-  ( ph  ->  ( M ... ( M  +  1 ) )  =  { M ,  ( M  + 
1 ) } )
12 gsumprval.n . . . . . . . 8  |-  ( ph  ->  N  =  ( M  +  1 ) )
1312eqcomd 2475 . . . . . . 7  |-  ( ph  ->  ( M  +  1 )  =  N )
1413preq2d 4113 . . . . . 6  |-  ( ph  ->  { M ,  ( M  +  1 ) }  =  { M ,  N } )
1511, 14eqtrd 2508 . . . . 5  |-  ( ph  ->  ( M ... ( M  +  1 ) )  =  { M ,  N } )
1615feq2d 5716 . . . 4  |-  ( ph  ->  ( F : ( M ... ( M  +  1 ) ) --> B  <->  F : { M ,  N } --> B ) )
179, 16mpbird 232 . . 3  |-  ( ph  ->  F : ( M ... ( M  + 
1 ) ) --> B )
181, 2, 3, 8, 17gsumval2 15823 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  (  seq M (  .+  ,  F ) `  ( M  +  1 ) ) )
19 seqp1 12085 . . 3  |-  ( M  e.  ( ZZ>= `  M
)  ->  (  seq M (  .+  ,  F ) `  ( M  +  1 ) )  =  ( (  seq M (  .+  ,  F ) `  M
)  .+  ( F `  ( M  +  1 ) ) ) )
206, 19syl 16 . 2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 ( M  + 
1 ) )  =  ( (  seq M
(  .+  ,  F
) `  M )  .+  ( F `  ( M  +  1 ) ) ) )
21 seq1 12083 . . . 4  |-  ( M  e.  ZZ  ->  (  seq M (  .+  ,  F ) `  M
)  =  ( F `
 M ) )
224, 21syl 16 . . 3  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 M )  =  ( F `  M
) )
2313fveq2d 5868 . . 3  |-  ( ph  ->  ( F `  ( M  +  1 ) )  =  ( F `
 N ) )
2422, 23oveq12d 6300 . 2  |-  ( ph  ->  ( (  seq M
(  .+  ,  F
) `  M )  .+  ( F `  ( M  +  1 ) ) )  =  ( ( F `  M
)  .+  ( F `  N ) ) )
2518, 20, 243eqtrd 2512 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( ( F `  M
)  .+  ( F `  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {cpr 4029   -->wf 5582   ` cfv 5586  (class class class)co 6282   1c1 9489    + caddc 9491   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668    seqcseq 12070   Basecbs 14483   +g cplusg 14548    gsumg cgsu 14689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-seq 12071  df-0g 14690  df-gsum 14691
This theorem is referenced by:  gsumpr12val  15825
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