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Theorem gsumprval 15628
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 10981 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
64, 5syl 16 . . . 4  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
7 peano2uz 11014 . . . 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 11623 . . . . . . 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 2460 . . . . . . 7  |-  ( ph  ->  ( M  +  1 )  =  N )
1413preq2d 4064 . . . . . 6  |-  ( ph  ->  { M ,  ( M  +  1 ) }  =  { M ,  N } )
1511, 14eqtrd 2493 . . . . 5  |-  ( ph  ->  ( M ... ( M  +  1 ) )  =  { M ,  N } )
1615feq2d 5650 . . . 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 15627 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  (  seq M (  .+  ,  F ) `  ( M  +  1 ) ) )
19 seqp1 11933 . . 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 11931 . . . 4  |-  ( M  e.  ZZ  ->  (  seq M (  .+  ,  F ) `  M
)  =  ( F `
 M ) )
224, 21syl 16 . . 3  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 M )  =  ( F `  M
) )
2313fveq2d 5798 . . 3  |-  ( ph  ->  ( F `  ( M  +  1 ) )  =  ( F `
 N ) )
2422, 23oveq12d 6213 . 2  |-  ( ph  ->  ( (  seq M
(  .+  ,  F
) `  M )  .+  ( F `  ( M  +  1 ) ) )  =  ( ( F `  M
)  .+  ( F `  N ) ) )
2518, 20, 243eqtrd 2497 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( ( F `  M
)  .+  ( F `  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   {cpr 3982   -->wf 5517   ` cfv 5521  (class class class)co 6195   1c1 9389    + caddc 9391   ZZcz 10752   ZZ>=cuz 10967   ...cfz 11549    seqcseq 11918   Basecbs 14287   +g cplusg 14352    gsumg cgsu 14493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-n0 10686  df-z 10753  df-uz 10968  df-fz 11550  df-seq 11919  df-0g 14494  df-gsum 14495
This theorem is referenced by:  gsumpr12val  15629
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