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Theorem fsumrev2 13370
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 13319 . . . . 5  |-  sum_ j  e.  (/)  A  =  0
2 sum0 13319 . . . . 5  |-  sum_ k  e.  (/)  B  =  0
31, 2eqtr4i 2486 . . . 4  |-  sum_ j  e.  (/)  A  =  sum_ k  e.  (/)  B
4 sumeq1 13287 . . . 4  |-  ( ( M ... N )  =  (/)  ->  sum_ j  e.  ( M ... N
) A  =  sum_ j  e.  (/)  A )
5 sumeq1 13287 . . . 4  |-  ( ( M ... N )  =  (/)  ->  sum_ k  e.  ( M ... N
) B  =  sum_ k  e.  (/)  B )
63, 4, 53eqtr4a 2521 . . 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 11584 . . 3  |-  ( ( M ... N )  =/=  (/)  <->  N  e.  ( ZZ>=
`  M ) )
9 eluzel2 10980 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
109adantl 466 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  M  e.  ZZ )
11 eluzelz 10984 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
1211adantl 466 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  N  e.  ZZ )
1310, 12zaddcld 10865 . . . . 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 13367 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  sum_ j  e.  ( M ... N
) A  =  sum_ k  e.  ( (
( M  +  N
)  -  N ) ... ( ( M  +  N )  -  M ) ) B )
1810zcnd 10862 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  M  e.  CC )
1912zcnd 10862 . . . . . . 7  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  N  e.  CC )
2018, 19pncand 9834 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  ( ( M  +  N )  -  N )  =  M )
2118, 19pncan2d 9835 . . . . . 6  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  ( ( M  +  N )  -  M )  =  N )
2220, 21oveq12d 6221 . . . . 5  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  ( (
( M  +  N
)  -  N ) ... ( ( M  +  N )  -  M ) )  =  ( M ... N
) )
2322sumeq1d 13299 . . . 4  |-  ( (
ph  /\  N  e.  ( ZZ>= `  M )
)  ->  sum_ k  e.  ( ( ( M  +  N )  -  N ) ... (
( M  +  N
)  -  M ) ) B  =  sum_ k  e.  ( M ... N ) B )
2417, 23eqtrd 2495 . . 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 2770 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 1370    e. wcel 1758    =/= wne 2648   (/)c0 3748   ` cfv 5529  (class class class)co 6203   CCcc 9394   0cc0 9396    + caddc 9399    - cmin 9709   ZZcz 10760   ZZ>=cuz 10975   ...cfz 11557   sum_csu 13284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-oi 7838  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-fz 11558  df-fzo 11669  df-seq 11927  df-exp 11986  df-hash 12224  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-clim 13087  df-sum 13285
This theorem is referenced by:  fsum0diag2  13371  efaddlem  13499  aareccl  21928
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