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Theorem fsumgcl 10777
Description: Closure for a function used to describe a sum over a nonempty finite set. (Contributed by Jim Kingdon, 10-Oct-2022.)
Hypotheses
Ref Expression
fsum.1 (𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)
fsum.2 (𝜑𝑀 ∈ ℕ)
fsum.3 (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)
fsum.4 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
fsum.5 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)
Assertion
Ref Expression
fsumgcl (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
Distinct variable groups:   𝐴,𝑘,𝑛   𝐵,𝑛   𝐶,𝑘   𝑘,𝐹,𝑛   𝑘,𝐺,𝑛   𝑘,𝑀,𝑛   𝜑,𝑘,𝑛
Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑛)

Proof of Theorem fsumgcl
StepHypRef Expression
1 fsum.5 . . 3 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)
2 fsum.3 . . . . . . 7 (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)
3 f1of 5253 . . . . . . 7 (𝐹:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)⟶𝐴)
42, 3syl 14 . . . . . 6 (𝜑𝐹:(1...𝑀)⟶𝐴)
54ffvelrnda 5434 . . . . 5 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) ∈ 𝐴)
6 fsum.1 . . . . . 6 (𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)
76adantl 271 . . . . 5 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑘 = (𝐹𝑛)) → 𝐵 = 𝐶)
85, 7csbied 2974 . . . 4 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 = 𝐶)
9 fsum.4 . . . . . . 7 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
109ralrimiva 2446 . . . . . 6 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
1110adantr 270 . . . . 5 ((𝜑𝑛 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
12 nfcsb1v 2963 . . . . . . 7 𝑘(𝐹𝑛) / 𝑘𝐵
1312nfel1 2239 . . . . . 6 𝑘(𝐹𝑛) / 𝑘𝐵 ∈ ℂ
14 csbeq1a 2941 . . . . . . 7 (𝑘 = (𝐹𝑛) → 𝐵 = (𝐹𝑛) / 𝑘𝐵)
1514eleq1d 2156 . . . . . 6 (𝑘 = (𝐹𝑛) → (𝐵 ∈ ℂ ↔ (𝐹𝑛) / 𝑘𝐵 ∈ ℂ))
1613, 15rspc 2716 . . . . 5 ((𝐹𝑛) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐹𝑛) / 𝑘𝐵 ∈ ℂ))
175, 11, 16sylc 61 . . . 4 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 ∈ ℂ)
188, 17eqeltrrd 2165 . . 3 ((𝜑𝑛 ∈ (1...𝑀)) → 𝐶 ∈ ℂ)
191, 18eqeltrd 2164 . 2 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) ∈ ℂ)
2019ralrimiva 2446 1 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  wral 2359  csb 2933  wf 5011  1-1-ontowf1o 5014  cfv 5015  (class class class)co 5652  cc 7348  1c1 7351  cn 8422  ...cfz 9424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-f1o 5022  df-fv 5023
This theorem is referenced by:  fisum  10778  fsum3  10779
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