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Theorem fsumrev 11051
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 9030 . . 3  |-  ( ph  ->  ( K  -  N
)  e.  ZZ )
5 fsumrev.2 . . . 4  |-  ( ph  ->  M  e.  ZZ )
62, 5zsubcld 9030 . . 3  |-  ( ph  ->  ( K  -  M
)  e.  ZZ )
74, 6fzfigd 10045 . 2  |-  ( ph  ->  ( ( K  -  N ) ... ( K  -  M )
)  e.  Fin )
8 eqid 2100 . . 3  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) )  =  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  |->  ( K  -  j ) )
92adantr 272 . . . 4  |-  ( (
ph  /\  j  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  K  e.  ZZ )
10 elfzelz 9647 . . . . 5  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  ->  j  e.  ZZ )
1110adantl 273 . . . 4  |-  ( (
ph  /\  j  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  j  e.  ZZ )
129, 11zsubcld 9030 . . 3  |-  ( (
ph  /\  j  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  ( K  -  j )  e.  ZZ )
132adantr 272 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  K  e.  ZZ )
14 elfzelz 9647 . . . . 5  |-  ( k  e.  ( M ... N )  ->  k  e.  ZZ )
1514adantl 273 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  k  e.  ZZ )
1613, 15zsubcld 9030 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( K  -  k )  e.  ZZ )
17 simprr 502 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
k  =  ( K  -  j ) )
18 simprl 501 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
195adantr 272 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  M  e.  ZZ )
203adantr 272 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  N  e.  ZZ )
212adantr 272 . . . . . . . 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 9705 . . . . . . . 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 1185 . . . . . . 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 2176 . . . . 5  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
k  e.  ( M ... N ) )
2717oveq2d 5722 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  k
)  =  ( K  -  ( K  -  j ) ) )
28 zcn 8911 . . . . . . . 8  |-  ( K  e.  ZZ  ->  K  e.  CC )
29 zcn 8911 . . . . . . . 8  |-  ( j  e.  ZZ  ->  j  e.  CC )
30 nncan 7862 . . . . . . . 8  |-  ( ( K  e.  CC  /\  j  e.  CC )  ->  ( K  -  ( K  -  j )
)  =  j )
3128, 29, 30syl2an 285 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  j  e.  ZZ )  ->  ( K  -  ( K  -  j )
)  =  j )
3221, 22, 31syl2anc 406 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  ( K  -  j )
)  =  j )
3327, 32eqtr2d 2133 . . . . 5  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  =  ( K  -  k ) )
3426, 33jca 302 . . . 4  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( k  e.  ( M ... N )  /\  j  =  ( K  -  k ) ) )
35 simprr 502 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
j  =  ( K  -  k ) )
36 simprl 501 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  e.  ( M ... N ) )
375adantr 272 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  M  e.  ZZ )
383adantr 272 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  N  e.  ZZ )
392adantr 272 . . . . . . . 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 9706 . . . . . . . 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 1185 . . . . . . 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 2176 . . . . 5  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
j  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
4535oveq2d 5722 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  j
)  =  ( K  -  ( K  -  k ) ) )
46 zcn 8911 . . . . . . . 8  |-  ( k  e.  ZZ  ->  k  e.  CC )
47 nncan 7862 . . . . . . . 8  |-  ( ( K  e.  CC  /\  k  e.  CC )  ->  ( K  -  ( K  -  k )
)  =  k )
4828, 46, 47syl2an 285 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  k  e.  ZZ )  ->  ( K  -  ( K  -  k )
)  =  k )
4939, 40, 48syl2anc 406 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  ( K  -  k )
)  =  k )
5045, 49eqtr2d 2133 . . . . 5  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  =  ( K  -  j ) )
5144, 50jca 302 . . . 4  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  /\  k  =  ( K  -  j ) ) )
5234, 51impbida 566 . . 3  |-  ( ph  ->  ( ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) )  <->  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) ) )
538, 12, 16, 52f1od 5905 . 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 272 . . . 4  |-  ( (
ph  /\  k  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  K  e.  ZZ )
56 elfzelz 9647 . . . . 5  |-  ( k  e.  ( ( K  -  N ) ... ( K  -  M
) )  ->  k  e.  ZZ )
5756adantl 273 . . . 4  |-  ( (
ph  /\  k  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  k  e.  ZZ )
5855, 57zsubcld 9030 . . 3  |-  ( (
ph  /\  k  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  ( K  -  k )  e.  ZZ )
59 oveq2 5714 . . . 4  |-  ( j  =  k  ->  ( K  -  j )  =  ( K  -  k ) )
6059, 8fvmptg 5429 . . 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 406 . 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 10998 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 1299    e. wcel 1448    |-> cmpt 3929   ` cfv 5059  (class class class)co 5706   CCcc 7498    - cmin 7804   ZZcz 8906   ...cfz 9631   sum_csu 10961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-isom 5068  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-frec 6218  df-1o 6243  df-oadd 6247  df-er 6359  df-en 6565  df-dom 6566  df-fin 6567  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-q 9262  df-rp 9292  df-fz 9632  df-fzo 9761  df-seqfrec 10060  df-exp 10134  df-ihash 10363  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-clim 10887  df-sumdc 10962
This theorem is referenced by:  fisumrev2  11054
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