Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fsumprd Unicode version

Theorem fsumprd 25432
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 21034 . . . . 5  |-  0  =  (GId `  +  )
2 sum0 12210 . . . . 5  |-  sum_ k  e.  (/)  A  =  0
3 prod0 25408 . . . . 5  |-  prod_ k  e.  (/)  +  A  =  (GId `  +  )
41, 2, 33eqtr4i 2326 . . . 4  |-  sum_ k  e.  (/)  A  =  prod_ k  e.  (/)  +  A
5 sumeq1 12178 . . . 4  |-  ( ( M ... N )  =  (/)  ->  sum_ k  e.  ( M ... N
) A  =  sum_ k  e.  (/)  A )
6 prodeq1 25409 . . . 4  |-  ( ( M ... N )  =  (/)  ->  prod_ k  e.  ( M ... N
)  +  A  = 
prod_ k  e.  (/)  +  A
)
74, 5, 63eqtr4a 2354 . . 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 10825 . . 3  |-  ( ( M ... N )  =/=  (/)  <->  N  e.  ( ZZ>=
`  M ) )
10 eqidd 2297 . . . . . . 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 2296 . . . . . . . . . 10  |-  ( k  e.  ( M ... N )  |->  A )  =  ( k  e.  ( M ... N
)  |->  A )
1413fmpt 5697 . . . . . . . . 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 5679 . . . . . . . 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 12219 . . . . . 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 25423 . . . . . . 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 2331 . . . . 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 12198 . . . . 5  |-  sum_ m  e.  ( M ... N
) ( ( k  e.  ( M ... N )  |->  A ) `
 m )  = 
sum_ k  e.  ( M ... N ) A
23 ssid 3210 . . . . . 6  |-  ( M ... N )  C_  ( M ... N )
2413prodeqfv 25421 . . . . . 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 2351 . . . 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 2535 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   (/)c0 3468    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753    + caddc 8756   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062   sum_csu 12174  GIdcgi 20870   prod_cprd 25401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-grpo 20874  df-gid 20875  df-ablo 20965  df-prod 25402
  Copyright terms: Public domain W3C validator