Theorem fsumgcl 11155

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 = (𝐹‘𝑛) → 𝐵 = 𝐶)
fsum.2	⊢ (𝜑 → 𝑀 ∈ ℕ)
fsum.3	⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)
fsum.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
fsum.5	⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) = 𝐶)

Assertion

Ref	Expression
fsumgcl	⊢ (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺‘𝑛) ∈ 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

Step	Hyp	Ref	Expression
1		fsum.5	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) = 𝐶)
2		fsum.3	. . . . . . 7 ⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→𝐴)
3		f1of 5367	. . . . . . 7 ⊢ (𝐹:(1...𝑀)–1-1-onto→𝐴 → 𝐹:(1...𝑀)⟶𝐴)
4	2, 3	syl 14	. . . . . 6 ⊢ (𝜑 → 𝐹:(1...𝑀)⟶𝐴)
5	4	ffvelrnda 5555	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐹‘𝑛) ∈ 𝐴)
6		fsum.1	. . . . . 6 ⊢ (𝑘 = (𝐹‘𝑛) → 𝐵 = 𝐶)
7	6	adantl 275	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑀)) ∧ 𝑘 = (𝐹‘𝑛)) → 𝐵 = 𝐶)
8	5, 7	csbied 3046	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐶)
9		fsum.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
10	9	ralrimiva 2505	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
11	10	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
12		nfcsb1v 3035	. . . . . . 7 ⊢ Ⅎ𝑘⦋(𝐹‘𝑛) / 𝑘⦌𝐵
13	12	nfel1 2292	. . . . . 6 ⊢ Ⅎ𝑘⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ℂ
14		csbeq1a 3012	. . . . . . 7 ⊢ (𝑘 = (𝐹‘𝑛) → 𝐵 = ⦋(𝐹‘𝑛) / 𝑘⦌𝐵)
15	14	eleq1d 2208	. . . . . 6 ⊢ (𝑘 = (𝐹‘𝑛) → (𝐵 ∈ ℂ ↔ ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ℂ))
16	13, 15	rspc 2783	. . . . 5 ⊢ ((𝐹‘𝑛) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ℂ))
17	5, 11, 16	sylc 62	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ℂ)
18	8, 17	eqeltrrd 2217	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → 𝐶 ∈ ℂ)
19	1, 18	eqeltrd 2216	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) ∈ ℂ)
20	19	ralrimiva 2505	1 ⊢ (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺‘𝑛) ∈ ℂ)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  ∀wral 2416  ⦋csb 3003  ⟶wf 5119  –1-1-onto→wf1o 5122  ‘cfv 5123  (class class class)co 5774  ℂcc 7618  1c1 7621  ℕcn 8720  ...cfz 9790

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-f1o 5130  df-fv 5131

This theorem is referenced by:  fsum3  11156