MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumprval Structured version   Unicode version

Theorem gsumprval 15507
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 10871 . . . . 5  |-  ( M  e.  ZZ  ->  M  e.  ( ZZ>= `  M )
)
64, 5syl 16 . . . 4  |-  ( ph  ->  M  e.  ( ZZ>= `  M ) )
7 peano2uz 10904 . . . 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 11507 . . . . . . 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 2446 . . . . . . 7  |-  ( ph  ->  ( M  +  1 )  =  N )
1413preq2d 3958 . . . . . 6  |-  ( ph  ->  { M ,  ( M  +  1 ) }  =  { M ,  N } )
1511, 14eqtrd 2473 . . . . 5  |-  ( ph  ->  ( M ... ( M  +  1 ) )  =  { M ,  N } )
1615feq2d 5544 . . . 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 15506 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  (  seq M (  .+  ,  F ) `  ( M  +  1 ) ) )
19 seqp1 11817 . . 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 11815 . . . 4  |-  ( M  e.  ZZ  ->  (  seq M (  .+  ,  F ) `  M
)  =  ( F `
 M ) )
224, 21syl 16 . . 3  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 M )  =  ( F `  M
) )
2313fveq2d 5692 . . 3  |-  ( ph  ->  ( F `  ( M  +  1 ) )  =  ( F `
 N ) )
2422, 23oveq12d 6108 . 2  |-  ( ph  ->  ( (  seq M
(  .+  ,  F
) `  M )  .+  ( F `  ( M  +  1 ) ) )  =  ( ( F `  M
)  .+  ( F `  N ) ) )
2518, 20, 243eqtrd 2477 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( ( F `  M
)  .+  ( F `  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761   {cpr 3876   -->wf 5411   ` cfv 5415  (class class class)co 6090   1c1 9279    + caddc 9281   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433    seqcseq 11802   Basecbs 14170   +g cplusg 14234    gsumg cgsu 14375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-seq 11803  df-0g 14376  df-gsum 14377
This theorem is referenced by:  gsumpr12val  15508
  Copyright terms: Public domain W3C validator