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Theorem fprodrev 14926
 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 12986 . 2 (𝜑 → ((𝐾𝑁)...(𝐾𝑀)) ∈ Fin)
3 ovex 6842 . . . . 5 (𝐾𝑗) ∈ V
4 eqid 2760 . . . . 5 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) = (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗))
53, 4fnmpti 6183 . . . 4 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn ((𝐾𝑁)...(𝐾𝑀))
65a1i 11 . . 3 (𝜑 → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn ((𝐾𝑁)...(𝐾𝑀)))
7 ovex 6842 . . . . 5 (𝐾𝑘) ∈ V
8 eqid 2760 . . . . 5 (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘)) = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘))
97, 8fnmpti 6183 . . . 4 (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘)) Fn (𝑀...𝑁)
10 simprr 813 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑘 = (𝐾𝑗))
11 simprl 811 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)))
12 fprodshft.2 . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℤ)
1312adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑀 ∈ ℤ)
14 fprodshft.3 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
1514adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑁 ∈ ℤ)
16 fprodshft.1 . . . . . . . . . . . 12 (𝜑𝐾 ∈ ℤ)
1716adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝐾 ∈ ℤ)
18 elfzelz 12555 . . . . . . . . . . . 12 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) → 𝑗 ∈ ℤ)
1918ad2antrl 766 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 ∈ ℤ)
20 fzrev 12616 . . . . . . . . . . 11 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↔ (𝐾𝑗) ∈ (𝑀...𝑁)))
2113, 15, 17, 19, 20syl22anc 1478 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↔ (𝐾𝑗) ∈ (𝑀...𝑁)))
2211, 21mpbid 222 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝐾𝑗) ∈ (𝑀...𝑁))
2310, 22eqeltrd 2839 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑘 ∈ (𝑀...𝑁))
24 oveq2 6822 . . . . . . . . . 10 (𝑘 = (𝐾𝑗) → (𝐾𝑘) = (𝐾 − (𝐾𝑗)))
2524ad2antll 767 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝐾𝑘) = (𝐾 − (𝐾𝑗)))
2616zcnd 11695 . . . . . . . . . . 11 (𝜑𝐾 ∈ ℂ)
2726adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝐾 ∈ ℂ)
2818zcnd 11695 . . . . . . . . . . 11 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) → 𝑗 ∈ ℂ)
2928ad2antrl 766 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 ∈ ℂ)
3027, 29nncand 10609 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝐾 − (𝐾𝑗)) = 𝑗)
3125, 30eqtr2d 2795 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 = (𝐾𝑘))
3223, 31jca 555 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘)))
33 simprr 813 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑗 = (𝐾𝑘))
34 simprl 811 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 ∈ (𝑀...𝑁))
3512adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑀 ∈ ℤ)
3614adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑁 ∈ ℤ)
3716adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝐾 ∈ ℤ)
38 elfzelz 12555 . . . . . . . . . . . 12 (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℤ)
3938ad2antrl 766 . . . . . . . . . . 11 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 ∈ ℤ)
40 fzrev2 12617 . . . . . . . . . . 11 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝐾𝑘) ∈ ((𝐾𝑁)...(𝐾𝑀))))
4135, 36, 37, 39, 40syl22anc 1478 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝐾𝑘) ∈ ((𝐾𝑁)...(𝐾𝑀))))
4234, 41mpbid 222 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝐾𝑘) ∈ ((𝐾𝑁)...(𝐾𝑀)))
4333, 42eqeltrd 2839 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)))
44 oveq2 6822 . . . . . . . . . 10 (𝑗 = (𝐾𝑘) → (𝐾𝑗) = (𝐾 − (𝐾𝑘)))
4544ad2antll 767 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝐾𝑗) = (𝐾 − (𝐾𝑘)))
4626adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝐾 ∈ ℂ)
4738zcnd 11695 . . . . . . . . . . 11 (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℂ)
4847ad2antrl 766 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 ∈ ℂ)
4946, 48nncand 10609 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝐾 − (𝐾𝑘)) = 𝑘)
5045, 49eqtr2d 2795 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 = (𝐾𝑗))
5143, 50jca 555 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗)))
5232, 51impbida 913 . . . . . 6 (𝜑 → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗)) ↔ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))))
5352mptcnv 5692 . . . . 5 (𝜑(𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘)))
5453fneq1d 6142 . . . 4 (𝜑 → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn (𝑀...𝑁) ↔ (𝑘 ∈ (𝑀...𝑁) ↦ (𝐾𝑘)) Fn (𝑀...𝑁)))
559, 54mpbiri 248 . . 3 (𝜑(𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) Fn (𝑀...𝑁))
56 dff1o4 6307 . . 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 701 . 2 (𝜑 → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)):((𝐾𝑁)...(𝐾𝑀))–1-1-onto→(𝑀...𝑁))
58 oveq2 6822 . . . 4 (𝑗 = 𝑘 → (𝐾𝑗) = (𝐾𝑘))
5958, 4, 7fvmpt 6445 . . 3 (𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀)) → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗))‘𝑘) = (𝐾𝑘))
6059adantl 473 . 2 ((𝜑𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀))) → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗))‘𝑘) = (𝐾𝑘))
61 fprodshft.4 . 2 ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
621, 2, 57, 60, 61fprodf1o 14895 1 (𝜑 → ∏𝑗 ∈ (𝑀...𝑁)𝐴 = ∏𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀))𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139   ↦ cmpt 4881  ◡ccnv 5265   Fn wfn 6044  –1-1-onto→wf1o 6048  ‘cfv 6049  (class class class)co 6814  ℂcc 10146   − cmin 10478  ℤcz 11589  ...cfz 12539  ∏cprod 14854 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-oi 8582  df-card 8975  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-n0 11505  df-z 11590  df-uz 11900  df-rp 12046  df-fz 12540  df-fzo 12680  df-seq 13016  df-exp 13075  df-hash 13332  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-clim 14438  df-prod 14855 This theorem is referenced by:  fallfacval3  14962  bcprod  31952
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