MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fprodrev Structured version   Visualization version   GIF version

Theorem fprodrev 15326
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 13340 . 2 (𝜑 → ((𝐾𝑁)...(𝐾𝑀)) ∈ Fin)
3 ovex 7172 . . . . 5 (𝐾𝑗) ∈ V
4 eqid 2801 . . . . 5 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) = (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗))
53, 4fnmpti 6467 . . . 4 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn ((𝐾𝑁)...(𝐾𝑀))
65a1i 11 . . 3 (𝜑 → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn ((𝐾𝑁)...(𝐾𝑀)))
7 ovex 7172 . . . . 5 (𝐾𝑘) ∈ V
8 eqid 2801 . . . . 5 (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘)) = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘))
97, 8fnmpti 6467 . . . 4 (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘)) Fn (𝑀...𝑁)
10 simprr 772 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑘 = (𝐾𝑗))
11 simprl 770 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)))
12 fprodshft.2 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℤ)
1312adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑀 ∈ ℤ)
14 fprodshft.3 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
1514adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑁 ∈ ℤ)
16 fprodshft.1 . . . . . . . . . . . 12 (𝜑𝐾 ∈ ℤ)
1716adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝐾 ∈ ℤ)
18 elfzelz 12906 . . . . . . . . . . . 12 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) → 𝑗 ∈ ℤ)
1918ad2antrl 727 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 ∈ ℤ)
20 fzrev 12969 . . . . . . . . . . 11 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↔ (𝐾𝑗) ∈ (𝑀...𝑁)))
2113, 15, 17, 19, 20syl22anc 837 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↔ (𝐾𝑗) ∈ (𝑀...𝑁)))
2211, 21mpbid 235 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝐾𝑗) ∈ (𝑀...𝑁))
2310, 22eqeltrd 2893 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑘 ∈ (𝑀...𝑁))
24 oveq2 7147 . . . . . . . . . 10 (𝑘 = (𝐾𝑗) → (𝐾𝑘) = (𝐾 − (𝐾𝑗)))
2524ad2antll 728 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝐾𝑘) = (𝐾 − (𝐾𝑗)))
2616zcnd 12080 . . . . . . . . . . 11 (𝜑𝐾 ∈ ℂ)
2726adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝐾 ∈ ℂ)
2818zcnd 12080 . . . . . . . . . . 11 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) → 𝑗 ∈ ℂ)
2928ad2antrl 727 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 ∈ ℂ)
3027, 29nncand 10995 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝐾 − (𝐾𝑗)) = 𝑗)
3125, 30eqtr2d 2837 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 = (𝐾𝑘))
3223, 31jca 515 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘)))
33 simprr 772 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑗 = (𝐾𝑘))
34 simprl 770 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 ∈ (𝑀...𝑁))
3512adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑀 ∈ ℤ)
3614adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑁 ∈ ℤ)
3716adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝐾 ∈ ℤ)
38 elfzelz 12906 . . . . . . . . . . . 12 (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℤ)
3938ad2antrl 727 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 ∈ ℤ)
40 fzrev2 12970 . . . . . . . . . . 11 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝐾𝑘) ∈ ((𝐾𝑁)...(𝐾𝑀))))
4135, 36, 37, 39, 40syl22anc 837 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝐾𝑘) ∈ ((𝐾𝑁)...(𝐾𝑀))))
4234, 41mpbid 235 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝐾𝑘) ∈ ((𝐾𝑁)...(𝐾𝑀)))
4333, 42eqeltrd 2893 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)))
44 oveq2 7147 . . . . . . . . . 10 (𝑗 = (𝐾𝑘) → (𝐾𝑗) = (𝐾 − (𝐾𝑘)))
4544ad2antll 728 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝐾𝑗) = (𝐾 − (𝐾𝑘)))
4626adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝐾 ∈ ℂ)
4738zcnd 12080 . . . . . . . . . . 11 (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℂ)
4847ad2antrl 727 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 ∈ ℂ)
4946, 48nncand 10995 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝐾 − (𝐾𝑘)) = 𝑘)
5045, 49eqtr2d 2837 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 = (𝐾𝑗))
5143, 50jca 515 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗)))
5232, 51impbida 800 . . . . . 6 (𝜑 → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗)) ↔ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))))
5352mptcnv 5969 . . . . 5 (𝜑(𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘)))
5453fneq1d 6420 . . . 4 (𝜑 → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn (𝑀...𝑁) ↔ (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘)) Fn (𝑀...𝑁)))
559, 54mpbiri 261 . . 3 (𝜑(𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn (𝑀...𝑁))
56 dff1o4 6602 . . 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 586 . 2 (𝜑 → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)):((𝐾𝑁)...(𝐾𝑀))–1-1-onto→(𝑀...𝑁))
58 oveq2 7147 . . . 4 (𝑗 = 𝑘 → (𝐾𝑗) = (𝐾𝑘))
5958, 4, 7fvmpt 6749 . . 3 (𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀)) → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗))‘𝑘) = (𝐾𝑘))
6059adantl 485 . 2 ((𝜑𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀))) → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗))‘𝑘) = (𝐾𝑘))
61 fprodshft.4 . 2 ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
621, 2, 57, 60, 61fprodf1o 15295 1 (𝜑 → ∏𝑗 ∈ (𝑀...𝑁)𝐴 = ∏𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀))𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  cmpt 5113  ccnv 5522   Fn wfn 6323  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7139  cc 10528  cmin 10863  cz 11973  ...cfz 12889  cprod 15254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-fz 12890  df-fzo 13033  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-prod 15255
This theorem is referenced by:  fallfacval3  15361  bcprod  33078
  Copyright terms: Public domain W3C validator