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Theorem fsumrev 11482
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 9409 . . 3  |-  ( ph  ->  ( K  -  N
)  e.  ZZ )
5 fsumrev.2 . . . 4  |-  ( ph  ->  M  e.  ZZ )
62, 5zsubcld 9409 . . 3  |-  ( ph  ->  ( K  -  M
)  e.  ZZ )
74, 6fzfigd 10461 . 2  |-  ( ph  ->  ( ( K  -  N ) ... ( K  -  M )
)  e.  Fin )
8 eqid 2189 . . 3  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  |->  ( K  -  j ) )  =  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  |->  ( K  -  j ) )
92adantr 276 . . . 4  |-  ( (
ph  /\  j  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  K  e.  ZZ )
10 elfzelz 10054 . . . . 5  |-  ( j  e.  ( ( K  -  N ) ... ( K  -  M
) )  ->  j  e.  ZZ )
1110adantl 277 . . . 4  |-  ( (
ph  /\  j  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  j  e.  ZZ )
129, 11zsubcld 9409 . . 3  |-  ( (
ph  /\  j  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  ( K  -  j )  e.  ZZ )
132adantr 276 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  K  e.  ZZ )
14 elfzelz 10054 . . . . 5  |-  ( k  e.  ( M ... N )  ->  k  e.  ZZ )
1514adantl 277 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  k  e.  ZZ )
1613, 15zsubcld 9409 . . 3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( K  -  k )  e.  ZZ )
17 simprr 531 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
k  =  ( K  -  j ) )
18 simprl 529 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
195adantr 276 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  M  e.  ZZ )
203adantr 276 . . . . . . . 8  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  ->  N  e.  ZZ )
212adantr 276 . . . . . . . 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 10113 . . . . . . . 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 1250 . . . . . . 7  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  <->  ( K  -  j )  e.  ( M ... N
) ) )
2518, 24mpbid 147 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  j
)  e.  ( M ... N ) )
2617, 25eqeltrd 2266 . . . . 5  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
k  e.  ( M ... N ) )
2717oveq2d 5911 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  k
)  =  ( K  -  ( K  -  j ) ) )
28 zcn 9287 . . . . . . . 8  |-  ( K  e.  ZZ  ->  K  e.  CC )
29 zcn 9287 . . . . . . . 8  |-  ( j  e.  ZZ  ->  j  e.  CC )
30 nncan 8215 . . . . . . . 8  |-  ( ( K  e.  CC  /\  j  e.  CC )  ->  ( K  -  ( K  -  j )
)  =  j )
3128, 29, 30syl2an 289 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  j  e.  ZZ )  ->  ( K  -  ( K  -  j )
)  =  j )
3221, 22, 31syl2anc 411 . . . . . 6  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( K  -  ( K  -  j )
)  =  j )
3327, 32eqtr2d 2223 . . . . 5  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
j  =  ( K  -  k ) )
3426, 33jca 306 . . . 4  |-  ( (
ph  /\  ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) ) )  -> 
( k  e.  ( M ... N )  /\  j  =  ( K  -  k ) ) )
35 simprr 531 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
j  =  ( K  -  k ) )
36 simprl 529 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  e.  ( M ... N ) )
375adantr 276 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  M  e.  ZZ )
383adantr 276 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  ->  N  e.  ZZ )
392adantr 276 . . . . . . . 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 10114 . . . . . . . 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 1250 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( k  e.  ( M ... N )  <-> 
( K  -  k
)  e.  ( ( K  -  N ) ... ( K  -  M ) ) ) )
4336, 42mpbid 147 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  k
)  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
4435, 43eqeltrd 2266 . . . . 5  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
j  e.  ( ( K  -  N ) ... ( K  -  M ) ) )
4535oveq2d 5911 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  j
)  =  ( K  -  ( K  -  k ) ) )
46 zcn 9287 . . . . . . . 8  |-  ( k  e.  ZZ  ->  k  e.  CC )
47 nncan 8215 . . . . . . . 8  |-  ( ( K  e.  CC  /\  k  e.  CC )  ->  ( K  -  ( K  -  k )
)  =  k )
4828, 46, 47syl2an 289 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  k  e.  ZZ )  ->  ( K  -  ( K  -  k )
)  =  k )
4939, 40, 48syl2anc 411 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( K  -  ( K  -  k )
)  =  k )
5045, 49eqtr2d 2223 . . . . 5  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
k  =  ( K  -  j ) )
5144, 50jca 306 . . . 4  |-  ( (
ph  /\  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) )  -> 
( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  /\  k  =  ( K  -  j ) ) )
5234, 51impbida 596 . . 3  |-  ( ph  ->  ( ( j  e.  ( ( K  -  N ) ... ( K  -  M )
)  /\  k  =  ( K  -  j
) )  <->  ( k  e.  ( M ... N
)  /\  j  =  ( K  -  k
) ) ) )
538, 12, 16, 52f1od 6096 . 2  |-  ( ph  ->  ( j  e.  ( ( K  -  N
) ... ( K  -  M ) )  |->  ( K  -  j ) ) : ( ( K  -  N ) ... ( K  -  M ) ) -1-1-onto-> ( M ... N ) )
54 simpr 110 . . 3  |-  ( (
ph  /\  k  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  k  e.  ( ( K  -  N ) ... ( K  -  M )
) )
552adantr 276 . . . 4  |-  ( (
ph  /\  k  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  K  e.  ZZ )
56 elfzelz 10054 . . . . 5  |-  ( k  e.  ( ( K  -  N ) ... ( K  -  M
) )  ->  k  e.  ZZ )
5756adantl 277 . . . 4  |-  ( (
ph  /\  k  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  k  e.  ZZ )
5855, 57zsubcld 9409 . . 3  |-  ( (
ph  /\  k  e.  ( ( K  -  N ) ... ( K  -  M )
) )  ->  ( K  -  k )  e.  ZZ )
59 oveq2 5903 . . . 4  |-  ( j  =  k  ->  ( K  -  j )  =  ( K  -  k ) )
6059, 8fvmptg 5612 . . 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 411 . 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 11429 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 104    <-> wb 105    = wceq 1364    e. wcel 2160    |-> cmpt 4079   ` cfv 5235  (class class class)co 5895   CCcc 7838    - cmin 8157   ZZcz 9282   ...cfz 10037   sum_csu 11392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958  ax-arch 7959  ax-caucvg 7960
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-irdg 6394  df-frec 6415  df-1o 6440  df-oadd 6444  df-er 6558  df-en 6766  df-dom 6767  df-fin 6768  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568  df-div 8659  df-inn 8949  df-2 9007  df-3 9008  df-4 9009  df-n0 9206  df-z 9283  df-uz 9558  df-q 9649  df-rp 9683  df-fz 10038  df-fzo 10172  df-seqfrec 10476  df-exp 10550  df-ihash 10787  df-cj 10882  df-re 10883  df-im 10884  df-rsqrt 11038  df-abs 11039  df-clim 11318  df-sumdc 11393
This theorem is referenced by:  fisumrev2  11485
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