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Theorem fsumrev2 13241
Description: Reversal of a finite sum. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 13-Apr-2016.)
Hypotheses
Ref Expression
fsumrev2.1  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )
fsumrev2.2  |-  ( j  =  ( ( M  +  N )  -  k )  ->  A  =  B )
Assertion
Ref Expression
fsumrev2  |-  ( ph  -> 
sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( M ... N
) B )
Distinct variable groups:    A, k    B, j    j, k, M   
j, N, k    ph, j,
k
Allowed substitution hints:    A( j)    B( k)

Proof of Theorem fsumrev2
StepHypRef Expression
1 sum0 13190 . . . . 5  |-  sum_ j  e.  (/)  A  =  0
2 sum0 13190 . . . . 5  |-  sum_ k  e.  (/)  B  =  0
31, 2eqtr4i 2461 . . . 4  |-  sum_ j  e.  (/)  A  =  sum_ k  e.  (/)  B
4 sumeq1 13158 . . . 4  |-  ( ( M ... N )  =  (/)  ->  sum_ j  e.  ( M ... N
) A  =  sum_ j  e.  (/)  A )
5 sumeq1 13158 . . . 4  |-  ( ( M ... N )  =  (/)  ->  sum_ k  e.  ( M ... N
) B  =  sum_ k  e.  (/)  B )
63, 4, 53eqtr4a 2496 . . 3  |-  ( ( M ... N )  =  (/)  ->  sum_ j  e.  ( M ... N
) A  =  sum_ k  e.  ( M ... N ) B )
76adantl 466 . 2  |-  ( (
ph  /\  ( M ... N )  =  (/) )  ->  sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( M ... N
) B )
8 fzn0 11456 . . 3  |-  ( ( M ... N )  =/=  (/)  <->  N  e.  ( ZZ>=
`  M ) )
9 eluzel2 10858 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
109adantl 466 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  M  e.  ZZ )
11 eluzelz 10862 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
1211adantl 466 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  N  e.  ZZ )
1310, 12zaddcld 10743 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  ( M  +  N )  e.  ZZ )
14 fsumrev2.1 . . . . . 6  |-  ( (
ph  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )
1514adantlr 714 . . . . 5  |-  ( ( ( ph  /\  N  e.  ( ZZ>= `  M )
)  /\  j  e.  ( M ... N ) )  ->  A  e.  CC )
16 fsumrev2.2 . . . . 5  |-  ( j  =  ( ( M  +  N )  -  k )  ->  A  =  B )
1713, 10, 12, 15, 16fsumrev 13238 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  sum_ j  e.  ( M ... N
) A  =  sum_ k  e.  ( (
( M  +  N
)  -  N ) ... ( ( M  +  N )  -  M ) ) B )
1810zcnd 10740 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  M  e.  CC )
1912zcnd 10740 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  N  e.  CC )
2018, 19pncand 9712 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  ( ( M  +  N )  -  N )  =  M )
2118, 19pncan2d 9713 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  ( ( M  +  N )  -  M )  =  N )
2220, 21oveq12d 6104 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  ( (
( M  +  N
)  -  N ) ... ( ( M  +  N )  -  M ) )  =  ( M ... N
) )
2322sumeq1d 13170 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  sum_ k  e.  ( ( ( M  +  N )  -  N ) ... (
( M  +  N
)  -  M ) ) B  =  sum_ k  e.  ( M ... N ) B )
2417, 23eqtrd 2470 . . 3  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  sum_ j  e.  ( M ... N
) A  =  sum_ k  e.  ( M ... N ) B )
258, 24sylan2b 475 . 2  |-  ( (
ph  /\  ( M ... N )  =/=  (/) )  ->  sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( M ... N ) B )
267, 25pm2.61dane 2684 1  |-  ( ph  -> 
sum_ j  e.  ( M ... N ) A  =  sum_ k  e.  ( M ... N
) B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601   (/)c0 3632   ` cfv 5413  (class class class)co 6086   CCcc 9272   0cc0 9274    + caddc 9277    - cmin 9587   ZZcz 10638   ZZ>=cuz 10853   ...cfz 11429   sum_csu 13155
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156
This theorem is referenced by:  fsum0diag2  13242  efaddlem  13370  aareccl  21767
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