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Theorem fprodf1o 15279
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 15276 . . . 4 𝑘 ∈ ∅ 𝐵 = 1
2 fprodf1o.3 . . . . . . . . 9 (𝜑𝐹:𝐶1-1-onto𝐴)
32adantr 484 . . . . . . . 8 ((𝜑𝐶 = ∅) → 𝐹:𝐶1-1-onto𝐴)
4 f1oeq2 6578 . . . . . . . . 9 (𝐶 = ∅ → (𝐹:𝐶1-1-onto𝐴𝐹:∅–1-1-onto𝐴))
54adantl 485 . . . . . . . 8 ((𝜑𝐶 = ∅) → (𝐹:𝐶1-1-onto𝐴𝐹:∅–1-1-onto𝐴))
63, 5mpbid 235 . . . . . . 7 ((𝜑𝐶 = ∅) → 𝐹:∅–1-1-onto𝐴)
7 f1ofo 6595 . . . . . . 7 (𝐹:∅–1-1-onto𝐴𝐹:∅–onto𝐴)
86, 7syl 17 . . . . . 6 ((𝜑𝐶 = ∅) → 𝐹:∅–onto𝐴)
9 fo00 6623 . . . . . . 7 (𝐹:∅–onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
109simprbi 500 . . . . . 6 (𝐹:∅–onto𝐴𝐴 = ∅)
118, 10syl 17 . . . . 5 ((𝜑𝐶 = ∅) → 𝐴 = ∅)
1211prodeq1d 15254 . . . 4 ((𝜑𝐶 = ∅) → ∏𝑘𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
13 prodeq1 15242 . . . . . 6 (𝐶 = ∅ → ∏𝑛𝐶 𝐷 = ∏𝑛 ∈ ∅ 𝐷)
14 prod0 15276 . . . . . 6 𝑛 ∈ ∅ 𝐷 = 1
1513, 14syl6eq 2872 . . . . 5 (𝐶 = ∅ → ∏𝑛𝐶 𝐷 = 1)
1615adantl 485 . . . 4 ((𝜑𝐶 = ∅) → ∏𝑛𝐶 𝐷 = 1)
171, 12, 163eqtr4a 2882 . . 3 ((𝜑𝐶 = ∅) → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷)
1817ex 416 . 2 (𝜑 → (𝐶 = ∅ → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷))
19 2fveq3 6648 . . . . . . . 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 6588 . . . . . . . . . . . 12 (𝐹:𝐶1-1-onto𝐴𝐹:𝐶𝐴)
232, 22syl 17 . . . . . . . . . . 11 (𝜑𝐹:𝐶𝐴)
2423ffvelrnda 6824 . . . . . . . . . 10 ((𝜑𝑚𝐶) → (𝐹𝑚) ∈ 𝐴)
25 fprodf1o.5 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
2625fmpttd 6852 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
2726ffvelrnda 6824 . . . . . . . . . 10 ((𝜑 ∧ (𝐹𝑚) ∈ 𝐴) → ((𝑘𝐴𝐵)‘(𝐹𝑚)) ∈ ℂ)
2824, 27syldan 594 . . . . . . . . 9 ((𝜑𝑚𝐶) → ((𝑘𝐴𝐵)‘(𝐹𝑚)) ∈ ℂ)
2928adantlr 714 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) ∧ 𝑚𝐶) → ((𝑘𝐴𝐵)‘(𝐹𝑚)) ∈ ℂ)
30 simpr 488 . . . . . . . . . . . 12 (((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶) → 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)
31 f1oco 6610 . . . . . . . . . . . 12 ((𝐹:𝐶1-1-onto𝐴𝑓:(1...(♯‘𝐶))–1-1-onto𝐶) → (𝐹𝑓):(1...(♯‘𝐶))–1-1-onto𝐴)
322, 30, 31syl2an 598 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → (𝐹𝑓):(1...(♯‘𝐶))–1-1-onto𝐴)
33 f1of 6588 . . . . . . . . . . 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 6733 . . . . . . . . . 10 (((𝐹𝑓):(1...(♯‘𝐶))⟶𝐴𝑛 ∈ (1...(♯‘𝐶))) → (((𝑘𝐴𝐵) ∘ (𝐹𝑓))‘𝑛) = ((𝑘𝐴𝐵)‘((𝐹𝑓)‘𝑛)))
3634, 35sylan 583 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → (((𝑘𝐴𝐵) ∘ (𝐹𝑓))‘𝑛) = ((𝑘𝐴𝐵)‘((𝐹𝑓)‘𝑛)))
37 f1of 6588 . . . . . . . . . . . . 13 (𝑓:(1...(♯‘𝐶))–1-1-onto𝐶𝑓:(1...(♯‘𝐶))⟶𝐶)
3837adantl 485 . . . . . . . . . . . 12 (((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶) → 𝑓:(1...(♯‘𝐶))⟶𝐶)
3938adantl 485 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → 𝑓:(1...(♯‘𝐶))⟶𝐶)
40 fvco3 6733 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐶))⟶𝐶𝑛 ∈ (1...(♯‘𝐶))) → ((𝐹𝑓)‘𝑛) = (𝐹‘(𝑓𝑛)))
4139, 40sylan 583 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → ((𝐹𝑓)‘𝑛) = (𝐹‘(𝑓𝑛)))
4241fveq2d 6647 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → ((𝑘𝐴𝐵)‘((𝐹𝑓)‘𝑛)) = ((𝑘𝐴𝐵)‘(𝐹‘(𝑓𝑛))))
4336, 42eqtrd 2856 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → (((𝑘𝐴𝐵) ∘ (𝐹𝑓))‘𝑛) = ((𝑘𝐴𝐵)‘(𝐹‘(𝑓𝑛))))
4419, 20, 21, 29, 43fprod 15274 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → ∏𝑚𝐶 ((𝑘𝐴𝐵)‘(𝐹𝑚)) = (seq1( · , ((𝑘𝐴𝐵) ∘ (𝐹𝑓)))‘(♯‘𝐶)))
45 fprodf1o.4 . . . . . . . . . . . . . 14 ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)
4623ffvelrnda 6824 . . . . . . . . . . . . . 14 ((𝜑𝑛𝐶) → (𝐹𝑛) ∈ 𝐴)
4745, 46eqeltrrd 2913 . . . . . . . . . . . . 13 ((𝜑𝑛𝐶) → 𝐺𝐴)
48 fprodf1o.1 . . . . . . . . . . . . . 14 (𝑘 = 𝐺𝐵 = 𝐷)
49 eqid 2821 . . . . . . . . . . . . . 14 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
5048, 49fvmpti 6740 . . . . . . . . . . . . 13 (𝐺𝐴 → ((𝑘𝐴𝐵)‘𝐺) = ( I ‘𝐷))
5147, 50syl 17 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → ((𝑘𝐴𝐵)‘𝐺) = ( I ‘𝐷))
5245fveq2d 6647 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → ((𝑘𝐴𝐵)‘(𝐹𝑛)) = ((𝑘𝐴𝐵)‘𝐺))
53 eqid 2821 . . . . . . . . . . . . . 14 (𝑛𝐶𝐷) = (𝑛𝐶𝐷)
5453fvmpt2i 6751 . . . . . . . . . . . . 13 (𝑛𝐶 → ((𝑛𝐶𝐷)‘𝑛) = ( I ‘𝐷))
5554adantl 485 . . . . . . . . . . . 12 ((𝜑𝑛𝐶) → ((𝑛𝐶𝐷)‘𝑛) = ( I ‘𝐷))
5651, 52, 553eqtr4rd 2867 . . . . . . . . . . 11 ((𝜑𝑛𝐶) → ((𝑛𝐶𝐷)‘𝑛) = ((𝑘𝐴𝐵)‘(𝐹𝑛)))
5756ralrimiva 3170 . . . . . . . . . 10 (𝜑 → ∀𝑛𝐶 ((𝑛𝐶𝐷)‘𝑛) = ((𝑘𝐴𝐵)‘(𝐹𝑛)))
58 nffvmpt1 6654 . . . . . . . . . . . 12 𝑛((𝑛𝐶𝐷)‘𝑚)
5958nfeq1 2989 . . . . . . . . . . 11 𝑛((𝑛𝐶𝐷)‘𝑚) = ((𝑘𝐴𝐵)‘(𝐹𝑚))
60 fveq2 6643 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((𝑛𝐶𝐷)‘𝑛) = ((𝑛𝐶𝐷)‘𝑚))
61 2fveq3 6648 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((𝑘𝐴𝐵)‘(𝐹𝑛)) = ((𝑘𝐴𝐵)‘(𝐹𝑚)))
6260, 61eqeq12d 2837 . . . . . . . . . . 11 (𝑛 = 𝑚 → (((𝑛𝐶𝐷)‘𝑛) = ((𝑘𝐴𝐵)‘(𝐹𝑛)) ↔ ((𝑛𝐶𝐷)‘𝑚) = ((𝑘𝐴𝐵)‘(𝐹𝑚))))
6359, 62rspc 3588 . . . . . . . . . 10 (𝑚𝐶 → (∀𝑛𝐶 ((𝑛𝐶𝐷)‘𝑛) = ((𝑘𝐴𝐵)‘(𝐹𝑛)) → ((𝑛𝐶𝐷)‘𝑚) = ((𝑘𝐴𝐵)‘(𝐹𝑚))))
6457, 63mpan9 510 . . . . . . . . 9 ((𝜑𝑚𝐶) → ((𝑛𝐶𝐷)‘𝑚) = ((𝑘𝐴𝐵)‘(𝐹𝑚)))
6564adantlr 714 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) ∧ 𝑚𝐶) → ((𝑛𝐶𝐷)‘𝑚) = ((𝑘𝐴𝐵)‘(𝐹𝑚)))
6665prodeq2dv 15256 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → ∏𝑚𝐶 ((𝑛𝐶𝐷)‘𝑚) = ∏𝑚𝐶 ((𝑘𝐴𝐵)‘(𝐹𝑚)))
67 fveq2 6643 . . . . . . . 8 (𝑚 = ((𝐹𝑓)‘𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘((𝐹𝑓)‘𝑛)))
6826adantr 484 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
6968ffvelrnda 6824 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
7067, 20, 32, 69, 36fprod 15274 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → ∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( · , ((𝑘𝐴𝐵) ∘ (𝐹𝑓)))‘(♯‘𝐶)))
7144, 66, 703eqtr4rd 2867 . . . . . 6 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → ∏𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = ∏𝑚𝐶 ((𝑛𝐶𝐷)‘𝑚))
72 prodfc 15278 . . . . . 6 𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = ∏𝑘𝐴 𝐵
73 prodfc 15278 . . . . . 6 𝑚𝐶 ((𝑛𝐶𝐷)‘𝑚) = ∏𝑛𝐶 𝐷
7471, 72, 733eqtr3g 2879 . . . . 5 ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶)) → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷)
7574expr 460 . . . 4 ((𝜑 ∧ (♯‘𝐶) ∈ ℕ) → (𝑓:(1...(♯‘𝐶))–1-1-onto𝐶 → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷))
7675exlimdv 1935 . . 3 ((𝜑 ∧ (♯‘𝐶) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶 → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷))
7776expimpd 457 . 2 (𝜑 → (((♯‘𝐶) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto𝐶) → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷))
78 fprodf1o.2 . . 3 (𝜑𝐶 ∈ Fin)
79 fz1f1o 15046 . . 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 857 1 (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wex 1781  wcel 2115  wral 3126  c0 4266  cmpt 5119   I cid 5432  ccom 5532  wf 6324  ontowfo 6326  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7130  Fincfn 8484  cc 10512  1c1 10515   · cmul 10519  cn 11615  ...cfz 12875  seqcseq 13352  chash 13674  cprod 15238
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-inf2 9080  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591  ax-pre-sup 10592
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  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 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-sup 8882  df-oi 8950  df-card 9344  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-div 11275  df-nn 11616  df-2 11678  df-3 11679  df-n0 11876  df-z 11960  df-uz 12222  df-rp 12368  df-fz 12876  df-fzo 13017  df-seq 13353  df-exp 13414  df-hash 13675  df-cj 14437  df-re 14438  df-im 14439  df-sqrt 14573  df-abs 14574  df-clim 14824  df-prod 15239
This theorem is referenced by:  fprodss  15281  fprodshft  15309  fprodrev  15310  fprod2dlem  15313  fprodcnv  15316  gausslemma2dlem1  25929  hgt750lemg  31933
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