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Theorem fprodf1o 15914
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 15911 . . . 4 𝑘 ∈ ∅ 𝐵 = 1
2 fprodf1o.3 . . . . . . . . 9 (𝜑𝐹:𝐶1-1-onto𝐴)
32adantr 480 . . . . . . . 8 ((𝜑𝐶 = ∅) → 𝐹:𝐶1-1-onto𝐴)
4 f1oeq2 6822 . . . . . . . . 9 (𝐶 = ∅ → (𝐹:𝐶1-1-onto𝐴𝐹:∅–1-1-onto𝐴))
54adantl 481 . . . . . . . 8 ((𝜑𝐶 = ∅) → (𝐹:𝐶1-1-onto𝐴𝐹:∅–1-1-onto𝐴))
63, 5mpbid 231 . . . . . . 7 ((𝜑𝐶 = ∅) → 𝐹:∅–1-1-onto𝐴)
7 f1ofo 6840 . . . . . . 7 (𝐹:∅–1-1-onto𝐴𝐹:∅–onto𝐴)
86, 7syl 17 . . . . . 6 ((𝜑𝐶 = ∅) → 𝐹:∅–onto𝐴)
9 fo00 6869 . . . . . . 7 (𝐹:∅–onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
109simprbi 496 . . . . . 6 (𝐹:∅–onto𝐴𝐴 = ∅)
118, 10syl 17 . . . . 5 ((𝜑𝐶 = ∅) → 𝐴 = ∅)
1211prodeq1d 15889 . . . 4 ((𝜑𝐶 = ∅) → ∏𝑘𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
13 prodeq1 15877 . . . . . 6 (𝐶 = ∅ → ∏𝑛𝐶 𝐷 = ∏𝑛 ∈ ∅ 𝐷)
14 prod0 15911 . . . . . 6 𝑛 ∈ ∅ 𝐷 = 1
1513, 14eqtrdi 2783 . . . . 5 (𝐶 = ∅ → ∏𝑛𝐶 𝐷 = 1)
1615adantl 481 . . . 4 ((𝜑𝐶 = ∅) → ∏𝑛𝐶 𝐷 = 1)
171, 12, 163eqtr4a 2793 . . 3 ((𝜑𝐶 = ∅) → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷)
1817ex 412 . 2 (𝜑 → (𝐶 = ∅ → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷))
19 2fveq3 6896 . . . . . . . 8 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘(𝐹𝑚)) = ((𝑘𝐴𝐵)‘(𝐹‘(𝑓𝑛))))
20 simprl 770 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → (♯‘𝐶) ∈ ℕ)
21 simprr 772 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)
22 f1of 6833 . . . . . . . . . . . 12 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶𝐴)
232, 22syl 17 . . . . . . . . . . 11 (𝜑𝐹:𝐶𝐴)
2423ffvelcdmda 7088 . . . . . . . . . 10 ((𝜑𝑚𝐶) → (𝐹𝑚) ∈ 𝐴)
25 fprodf1o.5 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
2625fmpttd 7119 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
2726ffvelcdmda 7088 . . . . . . . . . 10 ((𝜑 ∧ (𝐹𝑚) ∈ 𝐴) → ((𝑘𝐴𝐵)‘(𝐹𝑚)) ∈ ℂ)
2824, 27syldan 590 . . . . . . . . 9 ((𝜑𝑚𝐶) → ((𝑘𝐴𝐵)‘(𝐹𝑚)) ∈ ℂ)
2928adantlr 714 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) ∧ 𝑚𝐶) → ((𝑘𝐴𝐵)‘(𝐹𝑚)) ∈ ℂ)
30 simpr 484 . . . . . . . . . . . 12 (((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶) → 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)
31 f1oco 6856 . . . . . . . . . . . 12 ((𝐹:𝐶1-1-onto𝐴𝑓:(1...(♯‘𝐶))–1-1-onto𝐶) → (𝐹𝑓):(1...(♯‘𝐶))–1-1-onto𝐴)
322, 30, 31syl2an 595 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → (𝐹𝑓):(1...(♯‘𝐶))–1-1-onto𝐴)
33 f1of 6833 . . . . . . . . . . 11 ((𝐹𝑓):(1...(♯‘𝐶))–1-1-onto𝐴 → (𝐹𝑓):(1...(♯‘𝐶))⟶𝐴)
3432, 33syl 17 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → (𝐹𝑓):(1...(♯‘𝐶))⟶𝐴)
35 fvco3 6991 . . . . . . . . . 10 (((𝐹𝑓):(1...(♯‘𝐶))⟶𝐴𝑛 ∈ (1...(♯‘𝐶))) → (((𝑘𝐴𝐵) ∘ (𝐹𝑓))‘𝑛) = ((𝑘𝐴𝐵)‘((𝐹𝑓)‘𝑛)))
3634, 35sylan 579 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → (((𝑘𝐴𝐵) ∘ (𝐹𝑓))‘𝑛) = ((𝑘𝐴𝐵)‘((𝐹𝑓)‘𝑛)))
37 f1of 6833 . . . . . . . . . . . . 13 (𝑓:(1...(♯‘𝐶))–1-1-onto𝐶𝑓:(1...(♯‘𝐶))⟶𝐶)
3837adantl 481 . . . . . . . . . . . 12 (((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶) → 𝑓:(1...(♯‘𝐶))⟶𝐶)
3938adantl 481 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → 𝑓:(1...(♯‘𝐶))⟶𝐶)
40 fvco3 6991 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐶))⟶𝐶𝑛 ∈ (1...(♯‘𝐶))) → ((𝐹𝑓)‘𝑛) = (𝐹‘(𝑓𝑛)))
4139, 40sylan 579 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → ((𝐹𝑓)‘𝑛) = (𝐹‘(𝑓𝑛)))
4241fveq2d 6895 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → ((𝑘𝐴𝐵)‘((𝐹𝑓)‘𝑛)) = ((𝑘𝐴𝐵)‘(𝐹‘(𝑓𝑛))))
4336, 42eqtrd 2767 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → (((𝑘𝐴𝐵) ∘ (𝐹𝑓))‘𝑛) = ((𝑘𝐴𝐵)‘(𝐹‘(𝑓𝑛))))
4419, 20, 21, 29, 43fprod 15909 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → ∏𝑚𝐶 ((𝑘𝐴𝐵)‘(𝐹𝑚)) = (seq1( · , ((𝑘𝐴𝐵) ∘ (𝐹𝑓)))‘(♯‘𝐶)))
45 fprodf1o.4 . . . . . . . . . . . . . 14 ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)
4623ffvelcdmda 7088 . . . . . . . . . . . . . 14 ((𝜑𝑛𝐶) → (𝐹𝑛) ∈ 𝐴)
4745, 46eqeltrrd 2829 . . . . . . . . . . . . 13 ((𝜑𝑛𝐶) → 𝐺𝐴)
48 fprodf1o.1 . . . . . . . . . . . . . 14 (𝑘 = 𝐺𝐵 = 𝐷)
49 eqid 2727 . . . . . . . . . . . . . 14 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
5048, 49fvmpti 6998 . . . . . . . . . . . . 13 (𝐺𝐴 → ((𝑘𝐴𝐵)‘𝐺) = ( I ‘𝐷))
5147, 50syl 17 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → ((𝑘𝐴𝐵)‘𝐺) = ( I ‘𝐷))
5245fveq2d 6895 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → ((𝑘𝐴𝐵)‘(𝐹𝑛)) = ((𝑘𝐴𝐵)‘𝐺))
53 eqid 2727 . . . . . . . . . . . . . 14 (𝑛𝐶𝐷) = (𝑛𝐶𝐷)
5453fvmpt2i 7009 . . . . . . . . . . . . 13 (𝑛𝐶 → ((𝑛𝐶𝐷)‘𝑛) = ( I ‘𝐷))
5554adantl 481 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → ((𝑛𝐶𝐷)‘𝑛) = ( I ‘𝐷))
5651, 52, 553eqtr4rd 2778 . . . . . . . . . . 11 ((𝜑𝑛𝐶) → ((𝑛𝐶𝐷)‘𝑛) = ((𝑘𝐴𝐵)‘(𝐹𝑛)))
5756ralrimiva 3141 . . . . . . . . . 10 (𝜑 → ∀𝑛𝐶 ((𝑛𝐶𝐷)‘𝑛) = ((𝑘𝐴𝐵)‘(𝐹𝑛)))
58 nffvmpt1 6902 . . . . . . . . . . . 12 𝑛((𝑛𝐶𝐷)‘𝑚)
5958nfeq1 2913 . . . . . . . . . . 11 𝑛((𝑛𝐶𝐷)‘𝑚) = ((𝑘𝐴𝐵)‘(𝐹𝑚))
60 fveq2 6891 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((𝑛𝐶𝐷)‘𝑛) = ((𝑛𝐶𝐷)‘𝑚))
61 2fveq3 6896 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((𝑘𝐴𝐵)‘(𝐹𝑛)) = ((𝑘𝐴𝐵)‘(𝐹𝑚)))
6260, 61eqeq12d 2743 . . . . . . . . . . 11 (𝑛 = 𝑚 → (((𝑛𝐶𝐷)‘𝑛) = ((𝑘𝐴𝐵)‘(𝐹𝑛)) ↔ ((𝑛𝐶𝐷)‘𝑚) = ((𝑘𝐴𝐵)‘(𝐹𝑚))))
6359, 62rspc 3595 . . . . . . . . . 10 (𝑚𝐶 → (∀𝑛𝐶 ((𝑛𝐶𝐷)‘𝑛) = ((𝑘𝐴𝐵)‘(𝐹𝑛)) → ((𝑛𝐶𝐷)‘𝑚) = ((𝑘𝐴𝐵)‘(𝐹𝑚))))
6457, 63mpan9 506 . . . . . . . . 9 ((𝜑𝑚𝐶) → ((𝑛𝐶𝐷)‘𝑚) = ((𝑘𝐴𝐵)‘(𝐹𝑚)))
6564adantlr 714 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) ∧ 𝑚𝐶) → ((𝑛𝐶𝐷)‘𝑚) = ((𝑘𝐴𝐵)‘(𝐹𝑚)))
6665prodeq2dv 15891 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → ∏𝑚𝐶 ((𝑛𝐶𝐷)‘𝑚) = ∏𝑚𝐶 ((𝑘𝐴𝐵)‘(𝐹𝑚)))
67 fveq2 6891 . . . . . . . 8 (𝑚 = ((𝐹𝑓)‘𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘((𝐹𝑓)‘𝑛)))
6826adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
6968ffvelcdmda 7088 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
7067, 20, 32, 69, 36fprod 15909 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → ∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( · , ((𝑘𝐴𝐵) ∘ (𝐹𝑓)))‘(♯‘𝐶)))
7144, 66, 703eqtr4rd 2778 . . . . . 6 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → ∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = ∏𝑚𝐶 ((𝑛𝐶𝐷)‘𝑚))
72 prodfc 15913 . . . . . 6 𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = ∏𝑘𝐴 𝐵
73 prodfc 15913 . . . . . 6 𝑚𝐶 ((𝑛𝐶𝐷)‘𝑚) = ∏𝑛𝐶 𝐷
7471, 72, 733eqtr3g 2790 . . . . 5 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷)
7574expr 456 . . . 4 ((𝜑 ∧ (♯‘𝐶) ∈ ℕ) → (𝑓:(1...(♯‘𝐶))–1-1-onto𝐶 → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷))
7675exlimdv 1929 . . 3 ((𝜑 ∧ (♯‘𝐶) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶 → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷))
7776expimpd 453 . 2 (𝜑 → (((♯‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶) → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷))
78 fprodf1o.2 . . 3 (𝜑𝐶 ∈ Fin)
79 fz1f1o 15680 . . 3 (𝐶 ∈ Fin → (𝐶 = ∅ ∨ ((♯‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)))
8078, 79syl 17 . 2 (𝜑 → (𝐶 = ∅ ∨ ((♯‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)))
8118, 77, 80mpjaod 859 1 (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wo 846   = wceq 1534  wex 1774  wcel 2099  wral 3056  c0 4318  cmpt 5225   I cid 5569  ccom 5676  wf 6538  ontowfo 6540  1-1-ontowf1o 6541  cfv 6542  (class class class)co 7414  Fincfn 8955  cc 11128  1c1 11131   · cmul 11135  cn 12234  ...cfz 13508  seqcseq 13990  chash 14313  cprod 15873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-oi 9525  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-z 12581  df-uz 12845  df-rp 12999  df-fz 13509  df-fzo 13652  df-seq 13991  df-exp 14051  df-hash 14314  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-clim 15456  df-prod 15874
This theorem is referenced by:  fprodss  15916  fprodshft  15944  fprodrev  15945  fprod2dlem  15948  fprodcnv  15951  gausslemma2dlem1  27286  hgt750lemg  34222
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