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Theorem fsumgcl 11106
 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 5333 . . . . . . 7 (𝐹:(1...𝑀)–1-1-onto𝐴𝐹:(1...𝑀)⟶𝐴)
42, 3syl 14 . . . . . 6 (𝜑𝐹:(1...𝑀)⟶𝐴)
54ffvelrnda 5521 . . . . 5 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) ∈ 𝐴)
6 fsum.1 . . . . . 6 (𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)
76adantl 273 . . . . 5 (((𝜑𝑛 ∈ (1...𝑀)) ∧ 𝑘 = (𝐹𝑛)) → 𝐵 = 𝐶)
85, 7csbied 3014 . . . 4 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 = 𝐶)
9 fsum.4 . . . . . . 7 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
109ralrimiva 2480 . . . . . 6 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
1110adantr 272 . . . . 5 ((𝜑𝑛 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
12 nfcsb1v 3003 . . . . . . 7 𝑘(𝐹𝑛) / 𝑘𝐵
1312nfel1 2267 . . . . . 6 𝑘(𝐹𝑛) / 𝑘𝐵 ∈ ℂ
14 csbeq1a 2981 . . . . . . 7 (𝑘 = (𝐹𝑛) → 𝐵 = (𝐹𝑛) / 𝑘𝐵)
1514eleq1d 2184 . . . . . 6 (𝑘 = (𝐹𝑛) → (𝐵 ∈ ℂ ↔ (𝐹𝑛) / 𝑘𝐵 ∈ ℂ))
1613, 15rspc 2755 . . . . 5 ((𝐹𝑛) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝐹𝑛) / 𝑘𝐵 ∈ ℂ))
175, 11, 16sylc 62 . . . 4 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐹𝑛) / 𝑘𝐵 ∈ ℂ)
188, 17eqeltrrd 2193 . . 3 ((𝜑𝑛 ∈ (1...𝑀)) → 𝐶 ∈ ℂ)
191, 18eqeltrd 2192 . 2 ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) ∈ ℂ)
2019ralrimiva 2480 1 (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺𝑛) ∈ ℂ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1314   ∈ wcel 1463  ∀wral 2391  ⦋csb 2973  ⟶wf 5087  –1-1-onto→wf1o 5090  ‘cfv 5091  (class class class)co 5740  ℂcc 7582  1c1 7585  ℕcn 8680  ...cfz 9741 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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-f1o 5098  df-fv 5099 This theorem is referenced by:  fsum3  11107
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