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Theorem fprodf1o 14601
Description: Re-index a finite product using a bijection. (Contributed by Scott Fenton, 7-Dec-2017.)
Hypotheses
Ref Expression
fprodf1o.1 (𝑘 = 𝐺𝐵 = 𝐷)
fprodf1o.2 (𝜑𝐶 ∈ Fin)
fprodf1o.3 (𝜑𝐹:𝐶1-1-onto𝐴)
fprodf1o.4 ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)
fprodf1o.5 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
Assertion
Ref Expression
fprodf1o (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷)
Distinct variable groups:   𝐴,𝑘,𝑛   𝐵,𝑛   𝐶,𝑛   𝐷,𝑘   𝑛,𝐹   𝑘,𝐺   𝑘,𝑛,𝜑
Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑘)   𝐷(𝑛)   𝐹(𝑘)   𝐺(𝑛)

Proof of Theorem fprodf1o
Dummy variables 𝑓 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prod0 14598 . . . 4 𝑘 ∈ ∅ 𝐵 = 1
2 fprodf1o.3 . . . . . . . . 9 (𝜑𝐹:𝐶1-1-onto𝐴)
32adantr 481 . . . . . . . 8 ((𝜑𝐶 = ∅) → 𝐹:𝐶1-1-onto𝐴)
4 f1oeq2 6085 . . . . . . . . 9 (𝐶 = ∅ → (𝐹:𝐶1-1-onto𝐴𝐹:∅–1-1-onto𝐴))
54adantl 482 . . . . . . . 8 ((𝜑𝐶 = ∅) → (𝐹:𝐶1-1-onto𝐴𝐹:∅–1-1-onto𝐴))
63, 5mpbid 222 . . . . . . 7 ((𝜑𝐶 = ∅) → 𝐹:∅–1-1-onto𝐴)
7 f1ofo 6101 . . . . . . 7 (𝐹:∅–1-1-onto𝐴𝐹:∅–onto𝐴)
86, 7syl 17 . . . . . 6 ((𝜑𝐶 = ∅) → 𝐹:∅–onto𝐴)
9 fo00 6129 . . . . . . 7 (𝐹:∅–onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
109simprbi 480 . . . . . 6 (𝐹:∅–onto𝐴𝐴 = ∅)
118, 10syl 17 . . . . 5 ((𝜑𝐶 = ∅) → 𝐴 = ∅)
1211prodeq1d 14576 . . . 4 ((𝜑𝐶 = ∅) → ∏𝑘𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
13 prodeq1 14564 . . . . . 6 (𝐶 = ∅ → ∏𝑛𝐶 𝐷 = ∏𝑛 ∈ ∅ 𝐷)
14 prod0 14598 . . . . . 6 𝑛 ∈ ∅ 𝐷 = 1
1513, 14syl6eq 2671 . . . . 5 (𝐶 = ∅ → ∏𝑛𝐶 𝐷 = 1)
1615adantl 482 . . . 4 ((𝜑𝐶 = ∅) → ∏𝑛𝐶 𝐷 = 1)
171, 12, 163eqtr4a 2681 . . 3 ((𝜑𝐶 = ∅) → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷)
1817ex 450 . 2 (𝜑 → (𝐶 = ∅ → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷))
19 fveq2 6148 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → (𝐹𝑚) = (𝐹‘(𝑓𝑛)))
2019fveq2d 6152 . . . . . . . 8 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘(𝐹𝑚)) = ((𝑘𝐴𝐵)‘(𝐹‘(𝑓𝑛))))
21 simprl 793 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) → (#‘𝐶) ∈ ℕ)
22 simprr 795 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) → 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)
23 f1of 6094 . . . . . . . . . . . 12 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶𝐴)
242, 23syl 17 . . . . . . . . . . 11 (𝜑𝐹:𝐶𝐴)
2524ffvelrnda 6315 . . . . . . . . . 10 ((𝜑𝑚𝐶) → (𝐹𝑚) ∈ 𝐴)
26 fprodf1o.5 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
27 eqid 2621 . . . . . . . . . . . 12 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
2826, 27fmptd 6340 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
2928ffvelrnda 6315 . . . . . . . . . 10 ((𝜑 ∧ (𝐹𝑚) ∈ 𝐴) → ((𝑘𝐴𝐵)‘(𝐹𝑚)) ∈ ℂ)
3025, 29syldan 487 . . . . . . . . 9 ((𝜑𝑚𝐶) → ((𝑘𝐴𝐵)‘(𝐹𝑚)) ∈ ℂ)
3130adantlr 750 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) ∧ 𝑚𝐶) → ((𝑘𝐴𝐵)‘(𝐹𝑚)) ∈ ℂ)
32 simpr 477 . . . . . . . . . . . 12 (((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶) → 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)
33 f1oco 6116 . . . . . . . . . . . 12 ((𝐹:𝐶1-1-onto𝐴𝑓:(1...(#‘𝐶))–1-1-onto𝐶) → (𝐹𝑓):(1...(#‘𝐶))–1-1-onto𝐴)
342, 32, 33syl2an 494 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) → (𝐹𝑓):(1...(#‘𝐶))–1-1-onto𝐴)
35 f1of 6094 . . . . . . . . . . 11 ((𝐹𝑓):(1...(#‘𝐶))–1-1-onto𝐴 → (𝐹𝑓):(1...(#‘𝐶))⟶𝐴)
3634, 35syl 17 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) → (𝐹𝑓):(1...(#‘𝐶))⟶𝐴)
37 fvco3 6232 . . . . . . . . . 10 (((𝐹𝑓):(1...(#‘𝐶))⟶𝐴𝑛 ∈ (1...(#‘𝐶))) → (((𝑘𝐴𝐵) ∘ (𝐹𝑓))‘𝑛) = ((𝑘𝐴𝐵)‘((𝐹𝑓)‘𝑛)))
3836, 37sylan 488 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) ∧ 𝑛 ∈ (1...(#‘𝐶))) → (((𝑘𝐴𝐵) ∘ (𝐹𝑓))‘𝑛) = ((𝑘𝐴𝐵)‘((𝐹𝑓)‘𝑛)))
39 f1of 6094 . . . . . . . . . . . . 13 (𝑓:(1...(#‘𝐶))–1-1-onto𝐶𝑓:(1...(#‘𝐶))⟶𝐶)
4039adantl 482 . . . . . . . . . . . 12 (((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶) → 𝑓:(1...(#‘𝐶))⟶𝐶)
4140adantl 482 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) → 𝑓:(1...(#‘𝐶))⟶𝐶)
42 fvco3 6232 . . . . . . . . . . 11 ((𝑓:(1...(#‘𝐶))⟶𝐶𝑛 ∈ (1...(#‘𝐶))) → ((𝐹𝑓)‘𝑛) = (𝐹‘(𝑓𝑛)))
4341, 42sylan 488 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) ∧ 𝑛 ∈ (1...(#‘𝐶))) → ((𝐹𝑓)‘𝑛) = (𝐹‘(𝑓𝑛)))
4443fveq2d 6152 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) ∧ 𝑛 ∈ (1...(#‘𝐶))) → ((𝑘𝐴𝐵)‘((𝐹𝑓)‘𝑛)) = ((𝑘𝐴𝐵)‘(𝐹‘(𝑓𝑛))))
4538, 44eqtrd 2655 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) ∧ 𝑛 ∈ (1...(#‘𝐶))) → (((𝑘𝐴𝐵) ∘ (𝐹𝑓))‘𝑛) = ((𝑘𝐴𝐵)‘(𝐹‘(𝑓𝑛))))
4620, 21, 22, 31, 45fprod 14596 . . . . . . 7 ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) → ∏𝑚𝐶 ((𝑘𝐴𝐵)‘(𝐹𝑚)) = (seq1( · , ((𝑘𝐴𝐵) ∘ (𝐹𝑓)))‘(#‘𝐶)))
47 fprodf1o.4 . . . . . . . . . . . . . 14 ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)
4824ffvelrnda 6315 . . . . . . . . . . . . . 14 ((𝜑𝑛𝐶) → (𝐹𝑛) ∈ 𝐴)
4947, 48eqeltrrd 2699 . . . . . . . . . . . . 13 ((𝜑𝑛𝐶) → 𝐺𝐴)
50 fprodf1o.1 . . . . . . . . . . . . . 14 (𝑘 = 𝐺𝐵 = 𝐷)
5150, 27fvmpti 6238 . . . . . . . . . . . . 13 (𝐺𝐴 → ((𝑘𝐴𝐵)‘𝐺) = ( I ‘𝐷))
5249, 51syl 17 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → ((𝑘𝐴𝐵)‘𝐺) = ( I ‘𝐷))
5347fveq2d 6152 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → ((𝑘𝐴𝐵)‘(𝐹𝑛)) = ((𝑘𝐴𝐵)‘𝐺))
54 eqid 2621 . . . . . . . . . . . . . 14 (𝑛𝐶𝐷) = (𝑛𝐶𝐷)
5554fvmpt2i 6247 . . . . . . . . . . . . 13 (𝑛𝐶 → ((𝑛𝐶𝐷)‘𝑛) = ( I ‘𝐷))
5655adantl 482 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → ((𝑛𝐶𝐷)‘𝑛) = ( I ‘𝐷))
5752, 53, 563eqtr4rd 2666 . . . . . . . . . . 11 ((𝜑𝑛𝐶) → ((𝑛𝐶𝐷)‘𝑛) = ((𝑘𝐴𝐵)‘(𝐹𝑛)))
5857ralrimiva 2960 . . . . . . . . . 10 (𝜑 → ∀𝑛𝐶 ((𝑛𝐶𝐷)‘𝑛) = ((𝑘𝐴𝐵)‘(𝐹𝑛)))
59 nffvmpt1 6156 . . . . . . . . . . . 12 𝑛((𝑛𝐶𝐷)‘𝑚)
6059nfeq1 2774 . . . . . . . . . . 11 𝑛((𝑛𝐶𝐷)‘𝑚) = ((𝑘𝐴𝐵)‘(𝐹𝑚))
61 fveq2 6148 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((𝑛𝐶𝐷)‘𝑛) = ((𝑛𝐶𝐷)‘𝑚))
62 fveq2 6148 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
6362fveq2d 6152 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((𝑘𝐴𝐵)‘(𝐹𝑛)) = ((𝑘𝐴𝐵)‘(𝐹𝑚)))
6461, 63eqeq12d 2636 . . . . . . . . . . 11 (𝑛 = 𝑚 → (((𝑛𝐶𝐷)‘𝑛) = ((𝑘𝐴𝐵)‘(𝐹𝑛)) ↔ ((𝑛𝐶𝐷)‘𝑚) = ((𝑘𝐴𝐵)‘(𝐹𝑚))))
6560, 64rspc 3289 . . . . . . . . . 10 (𝑚𝐶 → (∀𝑛𝐶 ((𝑛𝐶𝐷)‘𝑛) = ((𝑘𝐴𝐵)‘(𝐹𝑛)) → ((𝑛𝐶𝐷)‘𝑚) = ((𝑘𝐴𝐵)‘(𝐹𝑚))))
6658, 65mpan9 486 . . . . . . . . 9 ((𝜑𝑚𝐶) → ((𝑛𝐶𝐷)‘𝑚) = ((𝑘𝐴𝐵)‘(𝐹𝑚)))
6766adantlr 750 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) ∧ 𝑚𝐶) → ((𝑛𝐶𝐷)‘𝑚) = ((𝑘𝐴𝐵)‘(𝐹𝑚)))
6867prodeq2dv 14578 . . . . . . 7 ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) → ∏𝑚𝐶 ((𝑛𝐶𝐷)‘𝑚) = ∏𝑚𝐶 ((𝑘𝐴𝐵)‘(𝐹𝑚)))
69 fveq2 6148 . . . . . . . 8 (𝑚 = ((𝐹𝑓)‘𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘((𝐹𝑓)‘𝑛)))
7028adantr 481 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
7170ffvelrnda 6315 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
7269, 21, 34, 71, 38fprod 14596 . . . . . . 7 ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) → ∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( · , ((𝑘𝐴𝐵) ∘ (𝐹𝑓)))‘(#‘𝐶)))
7346, 68, 723eqtr4rd 2666 . . . . . 6 ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) → ∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = ∏𝑚𝐶 ((𝑛𝐶𝐷)‘𝑚))
74 prodfc 14600 . . . . . 6 𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = ∏𝑘𝐴 𝐵
75 prodfc 14600 . . . . . 6 𝑚𝐶 ((𝑛𝐶𝐷)‘𝑚) = ∏𝑛𝐶 𝐷
7673, 74, 753eqtr3g 2678 . . . . 5 ((𝜑 ∧ ((#‘𝐶) ∈ ℕ ∧ 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)) → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷)
7776expr 642 . . . 4 ((𝜑 ∧ (#‘𝐶) ∈ ℕ) → (𝑓:(1...(#‘𝐶))–1-1-onto𝐶 → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷))
7877exlimdv 1858 . . 3 ((𝜑 ∧ (#‘𝐶) ∈ ℕ) → (∃𝑓 𝑓:(1...(#‘𝐶))–1-1-onto𝐶 → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷))
7978expimpd 628 . 2 (𝜑 → (((#‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐶))–1-1-onto𝐶) → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷))
80 fprodf1o.2 . . 3 (𝜑𝐶 ∈ Fin)
81 fz1f1o 14374 . . 3 (𝐶 ∈ Fin → (𝐶 = ∅ ∨ ((#‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)))
8280, 81syl 17 . 2 (𝜑 → (𝐶 = ∅ ∨ ((#‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐶))–1-1-onto𝐶)))
8318, 79, 82mpjaod 396 1 (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1987  wral 2907  c0 3891  cmpt 4673   I cid 4984  ccom 5078  wf 5843  ontowfo 5845  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  Fincfn 7899  cc 9878  1c1 9881   · cmul 9885  cn 10964  ...cfz 12268  seqcseq 12741  #chash 13057  cprod 14560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-prod 14561
This theorem is referenced by:  fprodss  14603  fprodshft  14631  fprodrev  14632  fprod2dlem  14635  fprodcnv  14638  gausslemma2dlem1  24991
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