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Theorem fsumrev 11243
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 fsumrev.1 . . . 4  |-  ( ph  ->  K  e.  ZZ )
3 fsumrev.3 . . . 4  |-  ( ph  ->  N  e.  ZZ )
42, 3zsubcld 9201 . . 3  |-  ( ph  ->  ( K  -  N
)  e.  ZZ )
5 fsumrev.2 . . . 4  |-  ( ph  ->  M  e.  ZZ )
62, 5zsubcld 9201 . . 3  |-  ( ph  ->  ( K  -  M
)  e.  ZZ )
74, 6fzfigd 10234 . 2  |-  ( ph  ->  ( ( K  -  N ) ... ( K  -  M )
)  e.  Fin )
8 eqid 2140 . . 3  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) )  =  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  |->  ( K  -  j ) )
92adantr 274 . . . 4  |-  ( (
ph  /\  j  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  K  e.  ZZ )
10 elfzelz 9836 . . . . 5  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  ->  j  e.  ZZ )
1110adantl 275 . . . 4  |-  ( (
ph  /\  j  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  j  e.  ZZ )
129, 11zsubcld 9201 . . 3  |-  ( (
ph  /\  j  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  ( K  -  j )  e.  ZZ )
132adantr 274 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  K  e.  ZZ )
14 elfzelz 9836 . . . . 5  |-  ( k  e.  ( M ... N )  ->  k  e.  ZZ )
1514adantl 275 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  k  e.  ZZ )
1613, 15zsubcld 9201 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( K  -  k )  e.  ZZ )
17 simprr 522 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
k  =  ( K  -  j ) )
18 simprl 521 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
195adantr 274 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  M  e.  ZZ )
203adantr 274 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  N  e.  ZZ )
212adantr 274 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  K  e.  ZZ )
2218, 10syl 14 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  e.  ZZ )
23 fzrev 9894 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  j  e.  ZZ ) )  -> 
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  <->  ( K  -  j )  e.  ( M ... N
) ) )
2419, 20, 21, 22, 23syl22anc 1218 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  <->  ( K  -  j )  e.  ( M ... N
) ) )
2518, 24mpbid 146 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  j
)  e.  ( M ... N ) )
2617, 25eqeltrd 2217 . . . . 5  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
k  e.  ( M ... N ) )
2717oveq2d 5797 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  k
)  =  ( K  -  ( K  -  j ) ) )
28 zcn 9082 . . . . . . . 8  |-  ( K  e.  ZZ  ->  K  e.  CC )
29 zcn 9082 . . . . . . . 8  |-  ( j  e.  ZZ  ->  j  e.  CC )
30 nncan 8014 . . . . . . . 8  |-  ( ( K  e.  CC  /\  j  e.  CC )  ->  ( K  -  ( K  -  j )
)  =  j )
3128, 29, 30syl2an 287 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  j  e.  ZZ )  ->  ( K  -  ( K  -  j )
)  =  j )
3221, 22, 31syl2anc 409 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  ( K  -  j )
)  =  j )
3327, 32eqtr2d 2174 . . . . 5  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  =  ( K  -  k ) )
3426, 33jca 304 . . . 4  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( k  e.  ( M ... N )  /\  j  =  ( K  -  k ) ) )
35 simprr 522 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
j  =  ( K  -  k ) )
36 simprl 521 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  e.  ( M ... N ) )
375adantr 274 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  M  e.  ZZ )
383adantr 274 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  N  e.  ZZ )
392adantr 274 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  K  e.  ZZ )
4036, 14syl 14 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  e.  ZZ )
41 fzrev2 9895 . . . . . . . 8  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  k  e.  ZZ ) )  -> 
( k  e.  ( M ... N )  <-> 
( K  -  k
)  e.  ( ( K  -  N ) ... ( K  -  M ) ) ) )
4237, 38, 39, 40, 41syl22anc 1218 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( k  e.  ( M ... N )  <-> 
( K  -  k
)  e.  ( ( K  -  N ) ... ( K  -  M ) ) ) )
4336, 42mpbid 146 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  k
)  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
4435, 43eqeltrd 2217 . . . . 5  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
j  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
4535oveq2d 5797 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  j
)  =  ( K  -  ( K  -  k ) ) )
46 zcn 9082 . . . . . . . 8  |-  ( k  e.  ZZ  ->  k  e.  CC )
47 nncan 8014 . . . . . . . 8  |-  ( ( K  e.  CC  /\  k  e.  CC )  ->  ( K  -  ( K  -  k )
)  =  k )
4828, 46, 47syl2an 287 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  k  e.  ZZ )  ->  ( K  -  ( K  -  k )
)  =  k )
4939, 40, 48syl2anc 409 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  ( K  -  k )
)  =  k )
5045, 49eqtr2d 2174 . . . . 5  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  =  ( K  -  j ) )
5144, 50jca 304 . . . 4  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  /\  k  =  ( K  -  j ) ) )
5234, 51impbida 586 . . 3  |-  ( ph  ->  ( ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) )  <->  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) ) )
538, 12, 16, 52f1od 5980 . 2  |-  ( ph  ->  ( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  |->  ( K  -  j ) ) : ( ( K  -  N ) ... ( K  -  M ) ) -1-1-onto-> ( M ... N ) )
54 simpr 109 . . 3  |-  ( (
ph  /\  k  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  k  e.  ( ( K  -  N ) ... ( K  -  M )
) )
552adantr 274 . . . 4  |-  ( (
ph  /\  k  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  K  e.  ZZ )
56 elfzelz 9836 . . . . 5  |-  ( k  e.  ( ( K  -  N ) ... ( K  -  M
) )  ->  k  e.  ZZ )
5756adantl 275 . . . 4  |-  ( (
ph  /\  k  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  k  e.  ZZ )
5855, 57zsubcld 9201 . . 3  |-  ( (
ph  /\  k  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  ( K  -  k )  e.  ZZ )
59 oveq2 5789 . . . 4  |-  ( j  =  k  ->  ( K  -  j )  =  ( K  -  k ) )
6059, 8fvmptg 5504 . . 3  |-  ( ( k  e.  ( ( K  -  N ) ... ( K  -  M ) )  /\  ( K  -  k
)  e.  ZZ )  ->  ( ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) ) `
 k )  =  ( K  -  k
) )
6154, 58, 60syl2anc 409 . 2  |-  ( (
ph  /\  k  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  (
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  |->  ( K  -  j ) ) `  k )  =  ( K  -  k ) )
62 fsumrev.4 . 2  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )
631, 7, 53, 61, 62fsumf1o 11190 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    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481    |-> cmpt 3996   ` cfv 5130  (class class class)co 5781   CCcc 7641    - cmin 7956   ZZcz 9077   ...cfz 9820   sum_csu 11153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761  ax-arch 7762  ax-caucvg 7763
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-isom 5139  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-frec 6295  df-1o 6320  df-oadd 6324  df-er 6436  df-en 6642  df-dom 6643  df-fin 6644  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-inn 8744  df-2 8802  df-3 8803  df-4 8804  df-n0 9001  df-z 9078  df-uz 9350  df-q 9438  df-rp 9470  df-fz 9821  df-fzo 9950  df-seqfrec 10249  df-exp 10323  df-ihash 10553  df-cj 10645  df-re 10646  df-im 10647  df-rsqrt 10801  df-abs 10802  df-clim 11079  df-sumdc 11154
This theorem is referenced by:  fisumrev2  11246
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