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Theorem fsumrev 14353
 Description: Reversal of a finite sum. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
Hypotheses
Ref Expression
fsumrev.1 (𝜑𝐾 ∈ ℤ)
fsumrev.2 (𝜑𝑀 ∈ ℤ)
fsumrev.3 (𝜑𝑁 ∈ ℤ)
fsumrev.4 ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
fsumrev.5 (𝑗 = (𝐾𝑘) → 𝐴 = 𝐵)
Assertion
Ref Expression
fsumrev (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀))𝐵)
Distinct variable groups:   𝐴,𝑘   𝐵,𝑗   𝑗,𝑘,𝐾   𝑗,𝑀,𝑘   𝑗,𝑁,𝑘   𝜑,𝑗,𝑘
Allowed substitution hints:   𝐴(𝑗)   𝐵(𝑘)

Proof of Theorem fsumrev
StepHypRef Expression
1 fsumrev.5 . 2 (𝑗 = (𝐾𝑘) → 𝐴 = 𝐵)
2 fzfid 12634 . 2 (𝜑 → ((𝐾𝑁)...(𝐾𝑀)) ∈ Fin)
3 ovex 6577 . . . . 5 (𝐾𝑗) ∈ V
4 eqid 2610 . . . . 5 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) = (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗))
53, 4fnmpti 5935 . . . 4 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn ((𝐾𝑁)...(𝐾𝑀))
65a1i 11 . . 3 (𝜑 → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn ((𝐾𝑁)...(𝐾𝑀)))
7 ovex 6577 . . . . 5 (𝐾𝑘) ∈ V
8 eqid 2610 . . . . 5 (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘)) = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘))
97, 8fnmpti 5935 . . . 4 (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘)) Fn (𝑀...𝑁)
10 simprr 792 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑘 = (𝐾𝑗))
11 simprl 790 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)))
12 fsumrev.2 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℤ)
1312adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑀 ∈ ℤ)
14 fsumrev.3 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
1514adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑁 ∈ ℤ)
16 fsumrev.1 . . . . . . . . . . . 12 (𝜑𝐾 ∈ ℤ)
1716adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝐾 ∈ ℤ)
18 elfzelz 12213 . . . . . . . . . . . 12 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) → 𝑗 ∈ ℤ)
1911, 18syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 ∈ ℤ)
20 fzrev 12273 . . . . . . . . . . 11 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↔ (𝐾𝑗) ∈ (𝑀...𝑁)))
2113, 15, 17, 19, 20syl22anc 1319 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↔ (𝐾𝑗) ∈ (𝑀...𝑁)))
2211, 21mpbid 221 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝐾𝑗) ∈ (𝑀...𝑁))
2310, 22eqeltrd 2688 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑘 ∈ (𝑀...𝑁))
2410oveq2d 6565 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝐾𝑘) = (𝐾 − (𝐾𝑗)))
25 zcn 11259 . . . . . . . . . . 11 (𝐾 ∈ ℤ → 𝐾 ∈ ℂ)
26 zcn 11259 . . . . . . . . . . 11 (𝑗 ∈ ℤ → 𝑗 ∈ ℂ)
27 nncan 10189 . . . . . . . . . . 11 ((𝐾 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝐾 − (𝐾𝑗)) = 𝑗)
2825, 26, 27syl2an 493 . . . . . . . . . 10 ((𝐾 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝐾 − (𝐾𝑗)) = 𝑗)
2917, 19, 28syl2anc 691 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝐾 − (𝐾𝑗)) = 𝑗)
3024, 29eqtr2d 2645 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 = (𝐾𝑘))
3123, 30jca 553 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘)))
32 simprr 792 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑗 = (𝐾𝑘))
33 simprl 790 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 ∈ (𝑀...𝑁))
3412adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑀 ∈ ℤ)
3514adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑁 ∈ ℤ)
3616adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝐾 ∈ ℤ)
37 elfzelz 12213 . . . . . . . . . . . 12 (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℤ)
3833, 37syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 ∈ ℤ)
39 fzrev2 12274 . . . . . . . . . . 11 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝐾𝑘) ∈ ((𝐾𝑁)...(𝐾𝑀))))
4034, 35, 36, 38, 39syl22anc 1319 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝐾𝑘) ∈ ((𝐾𝑁)...(𝐾𝑀))))
4133, 40mpbid 221 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝐾𝑘) ∈ ((𝐾𝑁)...(𝐾𝑀)))
4232, 41eqeltrd 2688 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)))
4332oveq2d 6565 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝐾𝑗) = (𝐾 − (𝐾𝑘)))
44 zcn 11259 . . . . . . . . . . 11 (𝑘 ∈ ℤ → 𝑘 ∈ ℂ)
45 nncan 10189 . . . . . . . . . . 11 ((𝐾 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐾 − (𝐾𝑘)) = 𝑘)
4625, 44, 45syl2an 493 . . . . . . . . . 10 ((𝐾 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝐾 − (𝐾𝑘)) = 𝑘)
4736, 38, 46syl2anc 691 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝐾 − (𝐾𝑘)) = 𝑘)
4843, 47eqtr2d 2645 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 = (𝐾𝑗))
4942, 48jca 553 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗)))
5031, 49impbida 873 . . . . . 6 (𝜑 → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗)) ↔ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))))
5150mptcnv 5453 . . . . 5 (𝜑(𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘)))
5251fneq1d 5895 . . . 4 (𝜑 → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn (𝑀...𝑁) ↔ (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘)) Fn (𝑀...𝑁)))
539, 52mpbiri 247 . . 3 (𝜑(𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn (𝑀...𝑁))
54 dff1o4 6058 . . 3 ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)):((𝐾𝑁)...(𝐾𝑀))–1-1-onto→(𝑀...𝑁) ↔ ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn ((𝐾𝑁)...(𝐾𝑀)) ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn (𝑀...𝑁)))
556, 53, 54sylanbrc 695 . 2 (𝜑 → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)):((𝐾𝑁)...(𝐾𝑀))–1-1-onto→(𝑀...𝑁))
56 oveq2 6557 . . . 4 (𝑗 = 𝑘 → (𝐾𝑗) = (𝐾𝑘))
5756, 4, 7fvmpt 6191 . . 3 (𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀)) → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗))‘𝑘) = (𝐾𝑘))
5857adantl 481 . 2 ((𝜑𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀))) → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗))‘𝑘) = (𝐾𝑘))
59 fsumrev.4 . 2 ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
601, 2, 55, 58, 59fsumf1o 14301 1 (𝜑 → Σ𝑗 ∈ (𝑀...𝑁)𝐴 = Σ𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀))𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ↦ cmpt 4643  ◡ccnv 5037   Fn wfn 5799  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ℂcc 9813   − cmin 10145  ℤcz 11254  ...cfz 12197  Σcsu 14264 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265 This theorem is referenced by:  fsumrev2  14356  birthdaylem2  24479
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