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Theorem fprodrev 15319
Description: Reversal of a finite product. (Contributed by Scott Fenton, 5-Jan-2018.)
Hypotheses
Ref Expression
fprodshft.1 (𝜑𝐾 ∈ ℤ)
fprodshft.2 (𝜑𝑀 ∈ ℤ)
fprodshft.3 (𝜑𝑁 ∈ ℤ)
fprodshft.4 ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
fprodrev.5 (𝑗 = (𝐾𝑘) → 𝐴 = 𝐵)
Assertion
Ref Expression
fprodrev (𝜑 → ∏𝑗 ∈ (𝑀...𝑁)𝐴 = ∏𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀))𝐵)
Distinct variable groups:   𝐴,𝑘   𝐵,𝑗   𝑗,𝑘,𝜑   𝑗,𝐾,𝑘   𝜑,𝑘   𝑗,𝑀,𝑘   𝑗,𝑁,𝑘
Allowed substitution hints:   𝐴(𝑗)   𝐵(𝑘)

Proof of Theorem fprodrev
StepHypRef Expression
1 fprodrev.5 . 2 (𝑗 = (𝐾𝑘) → 𝐴 = 𝐵)
2 fzfid 13329 . 2 (𝜑 → ((𝐾𝑁)...(𝐾𝑀)) ∈ Fin)
3 ovex 7178 . . . . 5 (𝐾𝑗) ∈ V
4 eqid 2818 . . . . 5 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) = (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗))
53, 4fnmpti 6484 . . . 4 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn ((𝐾𝑁)...(𝐾𝑀))
65a1i 11 . . 3 (𝜑 → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn ((𝐾𝑁)...(𝐾𝑀)))
7 ovex 7178 . . . . 5 (𝐾𝑘) ∈ V
8 eqid 2818 . . . . 5 (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘)) = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘))
97, 8fnmpti 6484 . . . 4 (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘)) Fn (𝑀...𝑁)
10 simprr 769 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑘 = (𝐾𝑗))
11 simprl 767 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)))
12 fprodshft.2 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℤ)
1312adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑀 ∈ ℤ)
14 fprodshft.3 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
1514adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑁 ∈ ℤ)
16 fprodshft.1 . . . . . . . . . . . 12 (𝜑𝐾 ∈ ℤ)
1716adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝐾 ∈ ℤ)
18 elfzelz 12896 . . . . . . . . . . . 12 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) → 𝑗 ∈ ℤ)
1918ad2antrl 724 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 ∈ ℤ)
20 fzrev 12958 . . . . . . . . . . 11 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↔ (𝐾𝑗) ∈ (𝑀...𝑁)))
2113, 15, 17, 19, 20syl22anc 834 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↔ (𝐾𝑗) ∈ (𝑀...𝑁)))
2211, 21mpbid 233 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝐾𝑗) ∈ (𝑀...𝑁))
2310, 22eqeltrd 2910 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑘 ∈ (𝑀...𝑁))
24 oveq2 7153 . . . . . . . . . 10 (𝑘 = (𝐾𝑗) → (𝐾𝑘) = (𝐾 − (𝐾𝑗)))
2524ad2antll 725 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝐾𝑘) = (𝐾 − (𝐾𝑗)))
2616zcnd 12076 . . . . . . . . . . 11 (𝜑𝐾 ∈ ℂ)
2726adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝐾 ∈ ℂ)
2818zcnd 12076 . . . . . . . . . . 11 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) → 𝑗 ∈ ℂ)
2928ad2antrl 724 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 ∈ ℂ)
3027, 29nncand 10990 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝐾 − (𝐾𝑗)) = 𝑗)
3125, 30eqtr2d 2854 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 = (𝐾𝑘))
3223, 31jca 512 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘)))
33 simprr 769 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑗 = (𝐾𝑘))
34 simprl 767 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 ∈ (𝑀...𝑁))
3512adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑀 ∈ ℤ)
3614adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑁 ∈ ℤ)
3716adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝐾 ∈ ℤ)
38 elfzelz 12896 . . . . . . . . . . . 12 (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℤ)
3938ad2antrl 724 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 ∈ ℤ)
40 fzrev2 12959 . . . . . . . . . . 11 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝐾𝑘) ∈ ((𝐾𝑁)...(𝐾𝑀))))
4135, 36, 37, 39, 40syl22anc 834 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝐾𝑘) ∈ ((𝐾𝑁)...(𝐾𝑀))))
4234, 41mpbid 233 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝐾𝑘) ∈ ((𝐾𝑁)...(𝐾𝑀)))
4333, 42eqeltrd 2910 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)))
44 oveq2 7153 . . . . . . . . . 10 (𝑗 = (𝐾𝑘) → (𝐾𝑗) = (𝐾 − (𝐾𝑘)))
4544ad2antll 725 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝐾𝑗) = (𝐾 − (𝐾𝑘)))
4626adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝐾 ∈ ℂ)
4738zcnd 12076 . . . . . . . . . . 11 (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℂ)
4847ad2antrl 724 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 ∈ ℂ)
4946, 48nncand 10990 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝐾 − (𝐾𝑘)) = 𝑘)
5045, 49eqtr2d 2854 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 = (𝐾𝑗))
5143, 50jca 512 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗)))
5232, 51impbida 797 . . . . . 6 (𝜑 → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗)) ↔ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))))
5352mptcnv 5991 . . . . 5 (𝜑(𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘)))
5453fneq1d 6439 . . . 4 (𝜑 → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn (𝑀...𝑁) ↔ (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘)) Fn (𝑀...𝑁)))
559, 54mpbiri 259 . . 3 (𝜑(𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn (𝑀...𝑁))
56 dff1o4 6616 . . 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)))
576, 55, 56sylanbrc 583 . 2 (𝜑 → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)):((𝐾𝑁)...(𝐾𝑀))–1-1-onto→(𝑀...𝑁))
58 oveq2 7153 . . . 4 (𝑗 = 𝑘 → (𝐾𝑗) = (𝐾𝑘))
5958, 4, 7fvmpt 6761 . . 3 (𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀)) → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗))‘𝑘) = (𝐾𝑘))
6059adantl 482 . 2 ((𝜑𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀))) → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗))‘𝑘) = (𝐾𝑘))
61 fprodshft.4 . 2 ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
621, 2, 57, 60, 61fprodf1o 15288 1 (𝜑 → ∏𝑗 ∈ (𝑀...𝑁)𝐴 = ∏𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀))𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  cmpt 5137  ccnv 5547   Fn wfn 6343  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  cc 10523  cmin 10858  cz 11969  ...cfz 12880  cprod 15247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-prod 15248
This theorem is referenced by:  fallfacval3  15354  bcprod  32867
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