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Theorem fsumgcl 11349
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 5442 . . . . . . 7 (𝐹:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)⟶𝐴)
42, 3syl 14 . . . . . 6 (𝜑𝐹:(1...𝑀)⟶𝐴)
54ffvelrnda 5631 . . . . 5 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) ∈ 𝐴)
6 fsum.1 . . . . . 6 (𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)
76adantl 275 . . . . 5 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑘 = (𝐹𝑛)) → 𝐵 = 𝐶)
85, 7csbied 3095 . . . 4 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 = 𝐶)
9 fsum.4 . . . . . . 7 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
109ralrimiva 2543 . . . . . 6 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
1110adantr 274 . . . . 5 ((𝜑𝑛 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
12 nfcsb1v 3082 . . . . . . 7 𝑘(𝐹𝑛) / 𝑘𝐵
1312nfel1 2323 . . . . . 6 𝑘(𝐹𝑛) / 𝑘𝐵 ∈ ℂ
14 csbeq1a 3058 . . . . . . 7 (𝑘 = (𝐹𝑛) → 𝐵 = (𝐹𝑛) / 𝑘𝐵)
1514eleq1d 2239 . . . . . 6 (𝑘 = (𝐹𝑛) → (𝐵 ∈ ℂ ↔ (𝐹𝑛) / 𝑘𝐵 ∈ ℂ))
1613, 15rspc 2828 . . . . 5 ((𝐹𝑛) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐹𝑛) / 𝑘𝐵 ∈ ℂ))
175, 11, 16sylc 62 . . . 4 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 ∈ ℂ)
188, 17eqeltrrd 2248 . . 3 ((𝜑𝑛 ∈ (1...𝑀)) → 𝐶 ∈ ℂ)
191, 18eqeltrd 2247 . 2 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) ∈ ℂ)
2019ralrimiva 2543 1 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  wral 2448  csb 3049  wf 5194  1-1-ontowf1o 5197  cfv 5198  (class class class)co 5853  cc 7772  1c1 7775  cn 8878  ...cfz 9965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-f1o 5205  df-fv 5206
This theorem is referenced by:  fsum3  11350  fprodseq  11546
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