Theorem fsumgcl 11551

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 5504	. . . . . . 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	ffvelcdmda 5697	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐹‘𝑛) ∈ 𝐴)
6		fsum.1	. . . . . 6 ⊢ (𝑘 = (𝐹‘𝑛) → 𝐵 = 𝐶)
7	6	adantl 277	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑀)) ∧ 𝑘 = (𝐹‘𝑛)) → 𝐵 = 𝐶)
8	5, 7	csbied 3131	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐶)
9		fsum.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
10	9	ralrimiva 2570	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
11	10	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
12		nfcsb1v 3117	. . . . . . 7 ⊢ Ⅎ𝑘⦋(𝐹‘𝑛) / 𝑘⦌𝐵
13	12	nfel1 2350	. . . . . 6 ⊢ Ⅎ𝑘⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ℂ
14		csbeq1a 3093	. . . . . . 7 ⊢ (𝑘 = (𝐹‘𝑛) → 𝐵 = ⦋(𝐹‘𝑛) / 𝑘⦌𝐵)
15	14	eleq1d 2265	. . . . . 6 ⊢ (𝑘 = (𝐹‘𝑛) → (𝐵 ∈ ℂ ↔ ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ℂ))
16	13, 15	rspc 2862	. . . . 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 2274	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → 𝐶 ∈ ℂ)
19	1, 18	eqeltrd 2273	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) ∈ ℂ)
20	19	ralrimiva 2570	1 ⊢ (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺‘𝑛) ∈ ℂ)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ⦋csb 3084  ⟶wf 5254  –1-1-onto→wf1o 5257  ‘cfv 5258  (class class class)co 5922  ℂcc 7877  1c1 7880  ℕcn 8990  ...cfz 10083

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-f1o 5265  df-fv 5266

This theorem is referenced by:  fsum3  11552  fprodseq  11748