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Theorem fsumrev 12257
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 11051 . 2  |-  ( ph  ->  ( ( K  -  N ) ... ( K  -  M )
)  e.  Fin )
3 ovex 5899 . . . . 5  |-  ( K  -  j )  e. 
_V
4 eqid 2296 . . . . 5  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) )  =  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  |->  ( K  -  j ) )
53, 4fnmpti 5388 . . . 4  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) )  Fn  ( ( K  -  N ) ... ( K  -  M
) )
65a1i 10 . . 3  |-  ( ph  ->  ( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  |->  ( K  -  j ) )  Fn  ( ( K  -  N ) ... ( K  -  M ) ) )
7 ovex 5899 . . . . 5  |-  ( K  -  k )  e. 
_V
8 eqid 2296 . . . . 5  |-  ( k  e.  ( M ... N )  |->  ( K  -  k ) )  =  ( k  e.  ( M ... N
)  |->  ( K  -  k ) )
97, 8fnmpti 5388 . . . 4  |-  ( k  e.  ( M ... N )  |->  ( K  -  k ) )  Fn  ( M ... N )
10 simprr 733 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
k  =  ( K  -  j ) )
11 simprl 732 . . . . . . . . . . 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 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  M  e.  ZZ )
14 fsumrev.3 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  ZZ )
1514adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  N  e.  ZZ )
16 fsumrev.1 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  ZZ )
1716adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  K  e.  ZZ )
18 elfzelz 10814 . . . . . . . . . . . . 13  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  ->  j  e.  ZZ )
1911, 18syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  e.  ZZ )
20 fzrev 10862 . . . . . . . . . . . 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 1183 . . . . . . . . . . 11  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  <->  ( K  -  j )  e.  ( M ... N
) ) )
2211, 21mpbid 201 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  j
)  e.  ( M ... N ) )
2310, 22eqeltrd 2370 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
k  e.  ( M ... N ) )
2410oveq2d 5890 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  k
)  =  ( K  -  ( K  -  j ) ) )
25 zcn 10045 . . . . . . . . . . . 12  |-  ( K  e.  ZZ  ->  K  e.  CC )
26 zcn 10045 . . . . . . . . . . . 12  |-  ( j  e.  ZZ  ->  j  e.  CC )
27 nncan 9092 . . . . . . . . . . . 12  |-  ( ( K  e.  CC  /\  j  e.  CC )  ->  ( K  -  ( K  -  j )
)  =  j )
2825, 26, 27syl2an 463 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  j  e.  ZZ )  ->  ( K  -  ( K  -  j )
)  =  j )
2917, 19, 28syl2anc 642 . . . . . . . . . 10  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  ( K  -  j )
)  =  j )
3024, 29eqtr2d 2329 . . . . . . . . 9  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  =  ( K  -  k ) )
3123, 30jca 518 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( k  e.  ( M ... N )  /\  j  =  ( K  -  k ) ) )
32 simprr 733 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
j  =  ( K  -  k ) )
33 simprl 732 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  e.  ( M ... N ) )
3412adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  M  e.  ZZ )
3514adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  N  e.  ZZ )
3616adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  K  e.  ZZ )
37 elfzelz 10814 . . . . . . . . . . . . 13  |-  ( k  e.  ( M ... N )  ->  k  e.  ZZ )
3833, 37syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  e.  ZZ )
39 fzrev2 10863 . . . . . . . . . . . 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 1183 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( k  e.  ( M ... N )  <-> 
( K  -  k
)  e.  ( ( K  -  N ) ... ( K  -  M ) ) ) )
4133, 40mpbid 201 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  k
)  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
4232, 41eqeltrd 2370 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
j  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
4332oveq2d 5890 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  j
)  =  ( K  -  ( K  -  k ) ) )
44 zcn 10045 . . . . . . . . . . . 12  |-  ( k  e.  ZZ  ->  k  e.  CC )
45 nncan 9092 . . . . . . . . . . . 12  |-  ( ( K  e.  CC  /\  k  e.  CC )  ->  ( K  -  ( K  -  k )
)  =  k )
4625, 44, 45syl2an 463 . . . . . . . . . . 11  |-  ( ( K  e.  ZZ  /\  k  e.  ZZ )  ->  ( K  -  ( K  -  k )
)  =  k )
4736, 38, 46syl2anc 642 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  ( K  -  k )
)  =  k )
4843, 47eqtr2d 2329 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  =  ( K  -  j ) )
4942, 48jca 518 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  /\  k  =  ( K  -  j ) ) )
5031, 49impbida 805 . . . . . . 7  |-  ( ph  ->  ( ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) )  <->  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) ) )
5150opabbidv 4098 . . . . . 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 4095 . . . . . . . 8  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) )  =  { <. j ,  k >.  |  ( j  e.  ( ( K  -  N ) ... ( K  -  M ) )  /\  k  =  ( K  -  j ) ) }
5352cnveqi 4872 . . . . . . 7  |-  `' ( j  e.  ( ( K  -  N ) ... ( K  -  M ) )  |->  ( K  -  j ) )  =  `' { <. j ,  k >.  |  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) }
54 cnvopab 5099 . . . . . . 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 2316 . . . . . 6  |-  `' ( j  e.  ( ( K  -  N ) ... ( K  -  M ) )  |->  ( K  -  j ) )  =  { <. k ,  j >.  |  ( j  e.  ( ( K  -  N ) ... ( K  -  M ) )  /\  k  =  ( K  -  j ) ) }
56 df-mpt 4095 . . . . . 6  |-  ( k  e.  ( M ... N )  |->  ( K  -  k ) )  =  { <. k ,  j >.  |  ( k  e.  ( M ... N )  /\  j  =  ( K  -  k ) ) }
5751, 55, 563eqtr4g 2353 . . . . 5  |-  ( ph  ->  `' ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  |->  ( K  -  j ) )  =  ( k  e.  ( M ... N ) 
|->  ( K  -  k
) ) )
5857fneq1d 5351 . . . 4  |-  ( ph  ->  ( `' ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) )  Fn  ( M ... N )  <->  ( k  e.  ( M ... N
)  |->  ( K  -  k ) )  Fn  ( M ... N
) ) )
599, 58mpbiri 224 . . 3  |-  ( ph  ->  `' ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  |->  ( K  -  j ) )  Fn  ( M ... N
) )
60 dff1o4 5496 . . 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 645 . 2  |-  ( ph  ->  ( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  |->  ( K  -  j ) ) : ( ( K  -  N ) ... ( K  -  M ) ) -1-1-onto-> ( M ... N ) )
62 oveq2 5882 . . . 4  |-  ( j  =  k  ->  ( K  -  j )  =  ( K  -  k ) )
6362, 4, 7fvmpt 5618 . . 3  |-  ( k  e.  ( ( K  -  N ) ... ( K  -  M
) )  ->  (
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  |->  ( K  -  j ) ) `  k )  =  ( K  -  k ) )
6463adantl 452 . 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 12212 1  |-  ( ph  -> 
sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( ( K  -  N ) ... ( K  -  M )
) B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {copab 4092    e. cmpt 4093   `'ccnv 4704    Fn wfn 5266   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   CCcc 8751    - cmin 9053   ZZcz 10040   ...cfz 10798   sum_csu 12174
This theorem is referenced by:  fsumrev2  12260  birthdaylem2  20263
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
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
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