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Theorem fprodrev 16010
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 14011 . 2 (𝜑 → ((𝐾𝑁)...(𝐾𝑀)) ∈ Fin)
3 eqid 2735 . . 3 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)) = (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗))
4 fprodshft.1 . . . . 5 (𝜑𝐾 ∈ ℤ)
54adantr 480 . . . 4 ((𝜑𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀))) → 𝐾 ∈ ℤ)
6 elfzelz 13561 . . . . 5 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) → 𝑗 ∈ ℤ)
76adantl 481 . . . 4 ((𝜑𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀))) → 𝑗 ∈ ℤ)
85, 7zsubcld 12725 . . 3 ((𝜑𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀))) → (𝐾𝑗) ∈ ℤ)
94adantr 480 . . . 4 ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐾 ∈ ℤ)
10 elfzelz 13561 . . . . 5 (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℤ)
1110adantl 481 . . . 4 ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ ℤ)
129, 11zsubcld 12725 . . 3 ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐾𝑘) ∈ ℤ)
13 simprr 773 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑘 = (𝐾𝑗))
14 simprl 771 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)))
15 fprodshft.2 . . . . . . . . 9 (𝜑𝑀 ∈ ℤ)
1615adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑀 ∈ ℤ)
17 fprodshft.3 . . . . . . . . 9 (𝜑𝑁 ∈ ℤ)
1817adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑁 ∈ ℤ)
194adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝐾 ∈ ℤ)
206ad2antrl 728 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 ∈ ℤ)
21 fzrev 13624 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑗 ∈ ℤ)) → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↔ (𝐾𝑗) ∈ (𝑀...𝑁)))
2216, 18, 19, 20, 21syl22anc 839 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↔ (𝐾𝑗) ∈ (𝑀...𝑁)))
2314, 22mpbid 232 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝐾𝑗) ∈ (𝑀...𝑁))
2413, 23eqeltrd 2839 . . . . 5 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑘 ∈ (𝑀...𝑁))
25 oveq2 7439 . . . . . . 7 (𝑘 = (𝐾𝑗) → (𝐾𝑘) = (𝐾 − (𝐾𝑗)))
2625ad2antll 729 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝐾𝑘) = (𝐾 − (𝐾𝑗)))
274zcnd 12721 . . . . . . . 8 (𝜑𝐾 ∈ ℂ)
2827adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝐾 ∈ ℂ)
296zcnd 12721 . . . . . . . 8 (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) → 𝑗 ∈ ℂ)
3029ad2antrl 728 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 ∈ ℂ)
3128, 30nncand 11623 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝐾 − (𝐾𝑗)) = 𝑗)
3226, 31eqtr2d 2776 . . . . 5 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → 𝑗 = (𝐾𝑘))
3324, 32jca 511 . . . 4 ((𝜑 ∧ (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗))) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘)))
34 simprr 773 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑗 = (𝐾𝑘))
35 simprl 771 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 ∈ (𝑀...𝑁))
3615adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑀 ∈ ℤ)
3717adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑁 ∈ ℤ)
384adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝐾 ∈ ℤ)
3910ad2antrl 728 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 ∈ ℤ)
40 fzrev2 13625 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ ℤ ∧ 𝑘 ∈ ℤ)) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝐾𝑘) ∈ ((𝐾𝑁)...(𝐾𝑀))))
4136, 37, 38, 39, 40syl22anc 839 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝐾𝑘) ∈ ((𝐾𝑁)...(𝐾𝑀))))
4235, 41mpbid 232 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝐾𝑘) ∈ ((𝐾𝑁)...(𝐾𝑀)))
4334, 42eqeltrd 2839 . . . . 5 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)))
44 oveq2 7439 . . . . . . 7 (𝑗 = (𝐾𝑘) → (𝐾𝑗) = (𝐾 − (𝐾𝑘)))
4544ad2antll 729 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝐾𝑗) = (𝐾 − (𝐾𝑘)))
4627adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝐾 ∈ ℂ)
4710zcnd 12721 . . . . . . . 8 (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℂ)
4847ad2antrl 728 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 ∈ ℂ)
4946, 48nncand 11623 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝐾 − (𝐾𝑘)) = 𝑘)
5045, 49eqtr2d 2776 . . . . 5 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → 𝑘 = (𝐾𝑗))
5143, 50jca 511 . . . 4 ((𝜑 ∧ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))) → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗)))
5233, 51impbida 801 . . 3 (𝜑 → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ∧ 𝑘 = (𝐾𝑗)) ↔ (𝑘 ∈ (𝑀...𝑁) ∧ 𝑗 = (𝐾𝑘))))
533, 8, 12, 52f1od 7685 . 2 (𝜑 → (𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗)):((𝐾𝑁)...(𝐾𝑀))–1-1-onto→(𝑀...𝑁))
54 oveq2 7439 . . . 4 (𝑗 = 𝑘 → (𝐾𝑗) = (𝐾𝑘))
55 ovex 7464 . . . 4 (𝐾𝑘) ∈ V
5654, 3, 55fvmpt 7016 . . 3 (𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀)) → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗))‘𝑘) = (𝐾𝑘))
5756adantl 481 . 2 ((𝜑𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀))) → ((𝑗 ∈ ((𝐾𝑁)...(𝐾𝑀)) ↦ (𝐾𝑗))‘𝑘) = (𝐾𝑘))
58 fprodshft.4 . 2 ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
591, 2, 53, 57, 58fprodf1o 15979 1 (𝜑 → ∏𝑗 ∈ (𝑀...𝑁)𝐴 = ∏𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀))𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cmpt 5231  cfv 6563  (class class class)co 7431  cc 11151  cmin 11490  cz 12611  ...cfz 13544  cprod 15936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-prod 15937
This theorem is referenced by:  fallfacval3  16045  bcprod  35718
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