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Theorem fsumgcl 11425
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 5480 . . . . . . 7 (𝐹:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)⟶𝐴)
42, 3syl 14 . . . . . 6 (𝜑𝐹:(1...𝑀)⟶𝐴)
54ffvelcdmda 5671 . . . . 5 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) ∈ 𝐴)
6 fsum.1 . . . . . 6 (𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)
76adantl 277 . . . . 5 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑘 = (𝐹𝑛)) → 𝐵 = 𝐶)
85, 7csbied 3118 . . . 4 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 = 𝐶)
9 fsum.4 . . . . . . 7 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
109ralrimiva 2563 . . . . . 6 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
1110adantr 276 . . . . 5 ((𝜑𝑛 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
12 nfcsb1v 3105 . . . . . . 7 𝑘(𝐹𝑛) / 𝑘𝐵
1312nfel1 2343 . . . . . 6 𝑘(𝐹𝑛) / 𝑘𝐵 ∈ ℂ
14 csbeq1a 3081 . . . . . . 7 (𝑘 = (𝐹𝑛) → 𝐵 = (𝐹𝑛) / 𝑘𝐵)
1514eleq1d 2258 . . . . . 6 (𝑘 = (𝐹𝑛) → (𝐵 ∈ ℂ ↔ (𝐹𝑛) / 𝑘𝐵 ∈ ℂ))
1613, 15rspc 2850 . . . . 5 ((𝐹𝑛) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐹𝑛) / 𝑘𝐵 ∈ ℂ))
175, 11, 16sylc 62 . . . 4 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 ∈ ℂ)
188, 17eqeltrrd 2267 . . 3 ((𝜑𝑛 ∈ (1...𝑀)) → 𝐶 ∈ ℂ)
191, 18eqeltrd 2266 . 2 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) ∈ ℂ)
2019ralrimiva 2563 1 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  wral 2468  csb 3072  wf 5231  1-1-ontowf1o 5234  cfv 5235  (class class class)co 5895  cc 7838  1c1 7841  cn 8948  ...cfz 10037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-f1o 5242  df-fv 5243
This theorem is referenced by:  fsum3  11426  fprodseq  11622
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