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Theorem fsumprd 25329
Description: Relation between  sum_ and  prod_. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)
Assertion
Ref Expression
fsumprd  |-  ( A. k  e.  ( M ... N ) A  e.  CC  ->  sum_ k  e.  ( M ... N
) A  =  prod_ k  e.  ( M ... N )  +  A
)
Distinct variable groups:    k, M    k, N
Allowed substitution hint:    A( k)

Proof of Theorem fsumprd
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 cnid 21018 . . . . 5  |-  0  =  (GId `  +  )
2 sum0 12194 . . . . 5  |-  sum_ k  e.  (/)  A  =  0
3 prod0 25305 . . . . 5  |-  prod_ k  e.  (/)  +  A  =  (GId `  +  )
41, 2, 33eqtr4i 2313 . . . 4  |-  sum_ k  e.  (/)  A  =  prod_ k  e.  (/)  +  A
5 sumeq1 12162 . . . 4  |-  ( ( M ... N )  =  (/)  ->  sum_ k  e.  ( M ... N
) A  =  sum_ k  e.  (/)  A )
6 prodeq1 25306 . . . 4  |-  ( ( M ... N )  =  (/)  ->  prod_ k  e.  ( M ... N
)  +  A  = 
prod_ k  e.  (/)  +  A
)
74, 5, 63eqtr4a 2341 . . 3  |-  ( ( M ... N )  =  (/)  ->  sum_ k  e.  ( M ... N
) A  =  prod_ k  e.  ( M ... N )  +  A
)
87a1d 22 . 2  |-  ( ( M ... N )  =  (/)  ->  ( A. k  e.  ( M ... N ) A  e.  CC  ->  sum_ k  e.  ( M ... N
) A  =  prod_ k  e.  ( M ... N )  +  A
) )
9 fzn0 10809 . . 3  |-  ( ( M ... N )  =/=  (/)  <->  N  e.  ( ZZ>=
`  M ) )
10 eqidd 2284 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  M )  /\  A. k  e.  ( M ... N ) A  e.  CC )  /\  m  e.  ( M ... N ) )  -> 
( ( k  e.  ( M ... N
)  |->  A ) `  m )  =  ( ( k  e.  ( M ... N ) 
|->  A ) `  m
) )
11 simpl 443 . . . . . . 7  |-  ( ( N  e.  ( ZZ>= `  M )  /\  A. k  e.  ( M ... N ) A  e.  CC )  ->  N  e.  ( ZZ>= `  M )
)
12 simpr 447 . . . . . . . . 9  |-  ( ( N  e.  ( ZZ>= `  M )  /\  A. k  e.  ( M ... N ) A  e.  CC )  ->  A. k  e.  ( M ... N
) A  e.  CC )
13 eqid 2283 . . . . . . . . . 10  |-  ( k  e.  ( M ... N )  |->  A )  =  ( k  e.  ( M ... N
)  |->  A )
1413fmpt 5681 . . . . . . . . 9  |-  ( A. k  e.  ( M ... N ) A  e.  CC  <->  ( k  e.  ( M ... N
)  |->  A ) : ( M ... N
) --> CC )
1512, 14sylib 188 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= `  M )  /\  A. k  e.  ( M ... N ) A  e.  CC )  ->  (
k  e.  ( M ... N )  |->  A ) : ( M ... N ) --> CC )
16 ffvelrn 5663 . . . . . . . 8  |-  ( ( ( k  e.  ( M ... N ) 
|->  A ) : ( M ... N ) --> CC  /\  m  e.  ( M ... N
) )  ->  (
( k  e.  ( M ... N ) 
|->  A ) `  m
)  e.  CC )
1715, 16sylan 457 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  M )  /\  A. k  e.  ( M ... N ) A  e.  CC )  /\  m  e.  ( M ... N ) )  -> 
( ( k  e.  ( M ... N
)  |->  A ) `  m )  e.  CC )
1810, 11, 17fsumser 12203 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  A. k  e.  ( M ... N ) A  e.  CC )  ->  sum_ m  e.  ( M ... N
) ( ( k  e.  ( M ... N )  |->  A ) `
 m )  =  (  seq  M (  +  ,  ( k  e.  ( M ... N )  |->  A ) ) `  N ) )
19 fprodser 25320 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  prod_ m  e.  ( M ... N
)  +  ( ( k  e.  ( M ... N )  |->  A ) `  m )  =  (  seq  M
(  +  ,  ( k  e.  ( M ... N )  |->  A ) ) `  N
) )
2019adantr 451 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  A. k  e.  ( M ... N ) A  e.  CC )  ->  prod_ m  e.  ( M ... N )  +  ( ( k  e.  ( M ... N ) 
|->  A ) `  m
)  =  (  seq 
M (  +  , 
( k  e.  ( M ... N ) 
|->  A ) ) `  N ) )
2118, 20eqtr4d 2318 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  A. k  e.  ( M ... N ) A  e.  CC )  ->  sum_ m  e.  ( M ... N
) ( ( k  e.  ( M ... N )  |->  A ) `
 m )  = 
prod_ m  e.  ( M ... N )  +  ( ( k  e.  ( M ... N
)  |->  A ) `  m ) )
22 sumfc 12182 . . . . 5  |-  sum_ m  e.  ( M ... N
) ( ( k  e.  ( M ... N )  |->  A ) `
 m )  = 
sum_ k  e.  ( M ... N ) A
23 ssid 3197 . . . . . 6  |-  ( M ... N )  C_  ( M ... N )
2413prodeqfv 25318 . . . . . 6  |-  ( ( M ... N ) 
C_  ( M ... N )  ->  prod_ m  e.  ( M ... N )  +  ( ( k  e.  ( M ... N ) 
|->  A ) `  m
)  =  prod_ k  e.  ( M ... N
)  +  A )
2523, 24ax-mp 8 . . . . 5  |-  prod_ m  e.  ( M ... N
)  +  ( ( k  e.  ( M ... N )  |->  A ) `  m )  =  prod_ k  e.  ( M ... N )  +  A
2621, 22, 253eqtr3g 2338 . . . 4  |-  ( ( N  e.  ( ZZ>= `  M )  /\  A. k  e.  ( M ... N ) A  e.  CC )  ->  sum_ k  e.  ( M ... N
) A  =  prod_ k  e.  ( M ... N )  +  A
)
2726ex 423 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( A. k  e.  ( M ... N ) A  e.  CC  ->  sum_ k  e.  ( M ... N
) A  =  prod_ k  e.  ( M ... N )  +  A
) )
289, 27sylbi 187 . 2  |-  ( ( M ... N )  =/=  (/)  ->  ( A. k  e.  ( M ... N ) A  e.  CC  ->  sum_ k  e.  ( M ... N
) A  =  prod_ k  e.  ( M ... N )  +  A
) )
298, 28pm2.61ine 2522 1  |-  ( A. k  e.  ( M ... N ) A  e.  CC  ->  sum_ k  e.  ( M ... N
) A  =  prod_ k  e.  ( M ... N )  +  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   (/)c0 3455    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737    + caddc 8740   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046   sum_csu 12158  GIdcgi 20854   prod_cprd 25298
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-grpo 20858  df-gid 20859  df-ablo 20949  df-prod 25299
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