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Theorem fsumrev 13261
Description: Reversal of a finite sum. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
Hypotheses
Ref Expression
fsumrev.1  |-  ( ph  ->  K  e.  ZZ )
fsumrev.2  |-  ( ph  ->  M  e.  ZZ )
fsumrev.3  |-  ( ph  ->  N  e.  ZZ )
fsumrev.4  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )
fsumrev.5  |-  ( j  =  ( K  -  k )  ->  A  =  B )
Assertion
Ref Expression
fsumrev  |-  ( ph  -> 
sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( ( K  -  N ) ... ( K  -  M )
) B )
Distinct variable groups:    A, k    B, j    j, k, K   
j, M, k    j, N, k    ph, j, k
Allowed substitution hints:    A( j)    B( k)

Proof of Theorem fsumrev
StepHypRef Expression
1 fsumrev.5 . 2  |-  ( j  =  ( K  -  k )  ->  A  =  B )
2 fzfid 11810 . 2  |-  ( ph  ->  ( ( K  -  N ) ... ( K  -  M )
)  e.  Fin )
3 ovex 6131 . . . . 5  |-  ( K  -  j )  e. 
_V
4 eqid 2443 . . . . 5  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) )  =  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  |->  ( K  -  j ) )
53, 4fnmpti 5554 . . . 4  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) )  Fn  ( ( K  -  N ) ... ( K  -  M
) )
65a1i 11 . . 3  |-  ( ph  ->  ( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  |->  ( K  -  j ) )  Fn  ( ( K  -  N ) ... ( K  -  M ) ) )
7 ovex 6131 . . . . 5  |-  ( K  -  k )  e. 
_V
8 eqid 2443 . . . . 5  |-  ( k  e.  ( M ... N )  |->  ( K  -  k ) )  =  ( k  e.  ( M ... N
)  |->  ( K  -  k ) )
97, 8fnmpti 5554 . . . 4  |-  ( k  e.  ( M ... N )  |->  ( K  -  k ) )  Fn  ( M ... N )
10 simprr 756 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
k  =  ( K  -  j ) )
11 simprl 755 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
12 fsumrev.2 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  ZZ )
1312adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  M  e.  ZZ )
14 fsumrev.3 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  ZZ )
1514adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  N  e.  ZZ )
16 fsumrev.1 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  ZZ )
1716adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  K  e.  ZZ )
18 elfzelz 11468 . . . . . . . . . . . . 13  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  ->  j  e.  ZZ )
1911, 18syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  e.  ZZ )
20 fzrev 11534 . . . . . . . . . . . 12  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  j  e.  ZZ ) )  -> 
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  <->  ( K  -  j )  e.  ( M ... N
) ) )
2113, 15, 17, 19, 20syl22anc 1219 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  <->  ( K  -  j )  e.  ( M ... N
) ) )
2211, 21mpbid 210 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  j
)  e.  ( M ... N ) )
2310, 22eqeltrd 2517 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
k  e.  ( M ... N ) )
2410oveq2d 6122 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  k
)  =  ( K  -  ( K  -  j ) ) )
25 zcn 10666 . . . . . . . . . . . 12  |-  ( K  e.  ZZ  ->  K  e.  CC )
26 zcn 10666 . . . . . . . . . . . 12  |-  ( j  e.  ZZ  ->  j  e.  CC )
27 nncan 9653 . . . . . . . . . . . 12  |-  ( ( K  e.  CC  /\  j  e.  CC )  ->  ( K  -  ( K  -  j )
)  =  j )
2825, 26, 27syl2an 477 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  j  e.  ZZ )  ->  ( K  -  ( K  -  j )
)  =  j )
2917, 19, 28syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  ( K  -  j )
)  =  j )
3024, 29eqtr2d 2476 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  =  ( K  -  k ) )
3123, 30jca 532 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( k  e.  ( M ... N )  /\  j  =  ( K  -  k ) ) )
32 simprr 756 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
j  =  ( K  -  k ) )
33 simprl 755 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  e.  ( M ... N ) )
3412adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  M  e.  ZZ )
3514adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  N  e.  ZZ )
3616adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  K  e.  ZZ )
37 elfzelz 11468 . . . . . . . . . . . . 13  |-  ( k  e.  ( M ... N )  ->  k  e.  ZZ )
3833, 37syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  e.  ZZ )
39 fzrev2 11535 . . . . . . . . . . . 12  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  k  e.  ZZ ) )  -> 
( k  e.  ( M ... N )  <-> 
( K  -  k
)  e.  ( ( K  -  N ) ... ( K  -  M ) ) ) )
4034, 35, 36, 38, 39syl22anc 1219 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( k  e.  ( M ... N )  <-> 
( K  -  k
)  e.  ( ( K  -  N ) ... ( K  -  M ) ) ) )
4133, 40mpbid 210 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  k
)  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
4232, 41eqeltrd 2517 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
j  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
4332oveq2d 6122 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  j
)  =  ( K  -  ( K  -  k ) ) )
44 zcn 10666 . . . . . . . . . . . 12  |-  ( k  e.  ZZ  ->  k  e.  CC )
45 nncan 9653 . . . . . . . . . . . 12  |-  ( ( K  e.  CC  /\  k  e.  CC )  ->  ( K  -  ( K  -  k )
)  =  k )
4625, 44, 45syl2an 477 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  k  e.  ZZ )  ->  ( K  -  ( K  -  k )
)  =  k )
4736, 38, 46syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  ( K  -  k )
)  =  k )
4843, 47eqtr2d 2476 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  =  ( K  -  j ) )
4942, 48jca 532 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  /\  k  =  ( K  -  j ) ) )
5031, 49impbida 828 . . . . . . 7  |-  ( ph  ->  ( ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) )  <->  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) ) )
5150opabbidv 4370 . . . . . 6  |-  ( ph  ->  { <. k ,  j
>.  |  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) }  =  { <. k ,  j
>.  |  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) } )
52 df-mpt 4367 . . . . . . . 8  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) )  =  { <. j ,  k >.  |  ( j  e.  ( ( K  -  N ) ... ( K  -  M ) )  /\  k  =  ( K  -  j ) ) }
5352cnveqi 5029 . . . . . . 7  |-  `' ( j  e.  ( ( K  -  N ) ... ( K  -  M ) )  |->  ( K  -  j ) )  =  `' { <. j ,  k >.  |  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) }
54 cnvopab 5253 . . . . . . 7  |-  `' { <. j ,  k >.  |  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) }  =  { <. k ,  j
>.  |  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) }
5553, 54eqtri 2463 . . . . . 6  |-  `' ( j  e.  ( ( K  -  N ) ... ( K  -  M ) )  |->  ( K  -  j ) )  =  { <. k ,  j >.  |  ( j  e.  ( ( K  -  N ) ... ( K  -  M ) )  /\  k  =  ( K  -  j ) ) }
56 df-mpt 4367 . . . . . 6  |-  ( k  e.  ( M ... N )  |->  ( K  -  k ) )  =  { <. k ,  j >.  |  ( k  e.  ( M ... N )  /\  j  =  ( K  -  k ) ) }
5751, 55, 563eqtr4g 2500 . . . . 5  |-  ( ph  ->  `' ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  |->  ( K  -  j ) )  =  ( k  e.  ( M ... N ) 
|->  ( K  -  k
) ) )
5857fneq1d 5516 . . . 4  |-  ( ph  ->  ( `' ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) )  Fn  ( M ... N )  <->  ( k  e.  ( M ... N
)  |->  ( K  -  k ) )  Fn  ( M ... N
) ) )
599, 58mpbiri 233 . . 3  |-  ( ph  ->  `' ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  |->  ( K  -  j ) )  Fn  ( M ... N
) )
60 dff1o4 5664 . . 3  |-  ( ( j  e.  ( ( K  -  N ) ... ( K  -  M ) )  |->  ( K  -  j ) ) : ( ( K  -  N ) ... ( K  -  M ) ) -1-1-onto-> ( M ... N )  <->  ( (
j  e.  ( ( K  -  N ) ... ( K  -  M ) )  |->  ( K  -  j ) )  Fn  ( ( K  -  N ) ... ( K  -  M ) )  /\  `' ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  |->  ( K  -  j ) )  Fn  ( M ... N
) ) )
616, 59, 60sylanbrc 664 . 2  |-  ( ph  ->  ( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  |->  ( K  -  j ) ) : ( ( K  -  N ) ... ( K  -  M ) ) -1-1-onto-> ( M ... N ) )
62 oveq2 6114 . . . 4  |-  ( j  =  k  ->  ( K  -  j )  =  ( K  -  k ) )
6362, 4, 7fvmpt 5789 . . 3  |-  ( k  e.  ( ( K  -  N ) ... ( K  -  M
) )  ->  (
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  |->  ( K  -  j ) ) `  k )  =  ( K  -  k ) )
6463adantl 466 . 2  |-  ( (
ph  /\  k  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  (
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  |->  ( K  -  j ) ) `  k )  =  ( K  -  k ) )
65 fsumrev.4 . 2  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )
661, 2, 61, 64, 65fsumf1o 13215 1  |-  ( ph  -> 
sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( ( K  -  N ) ... ( K  -  M )
) B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {copab 4364    e. cmpt 4365   `'ccnv 4854    Fn wfn 5428   -1-1-onto->wf1o 5432   ` cfv 5433  (class class class)co 6106   CCcc 9295    - cmin 9610   ZZcz 10661   ...cfz 11452   sum_csu 13178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-inf2 7862  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-se 4695  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-isom 5442  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-sup 7706  df-oi 7739  df-card 8124  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-2 10395  df-3 10396  df-n0 10595  df-z 10662  df-uz 10877  df-rp 11007  df-fz 11453  df-fzo 11564  df-seq 11822  df-exp 11881  df-hash 12119  df-cj 12603  df-re 12604  df-im 12605  df-sqr 12739  df-abs 12740  df-clim 12981  df-sum 13179
This theorem is referenced by:  fsumrev2  13264  birthdaylem2  22361
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