Theorem fsumgcl 12097

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 5619	. . . . . . 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 5817	. . . . 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 3188	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 = 𝐶)
9		fsum.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
10	9	ralrimiva 2617	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
11	10	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
12		nfcsb1v 3174	. . . . . . 7 ⊢ Ⅎ𝑘⦋(𝐹‘𝑛) / 𝑘⦌𝐵
13	12	nfel1 2397	. . . . . 6 ⊢ Ⅎ𝑘⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ℂ
14		csbeq1a 3150	. . . . . . 7 ⊢ (𝑘 = (𝐹‘𝑛) → 𝐵 = ⦋(𝐹‘𝑛) / 𝑘⦌𝐵)
15	14	eleq1d 2303	. . . . . 6 ⊢ (𝑘 = (𝐹‘𝑛) → (𝐵 ∈ ℂ ↔ ⦋(𝐹‘𝑛) / 𝑘⦌𝐵 ∈ ℂ))
16	13, 15	rspc 2917	. . . . 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 2312	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → 𝐶 ∈ ℂ)
19	1, 18	eqeltrd 2311	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → (𝐺‘𝑛) ∈ ℂ)
20	19	ralrimiva 2617	1 ⊢ (𝜑 → ∀𝑛 ∈ (1...𝑀)(𝐺‘𝑛) ∈ ℂ)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ⦋csb 3141  ⟶wf 5353  –1-1-onto→wf1o 5356  ‘cfv 5357  (class class class)co 6058  ℂcc 8141  1c1 8144  ℕcn 9254  ...cfz 10361

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-f1o 5364  df-fv 5365

This theorem is referenced by:  fsum3  12098  fprodseq  12294