Theorem fprodefsum 16071

Description: Move the exponential function from inside a finite product to outside a finite sum. (Contributed by Scott Fenton, 26-Dec-2017.)

Hypotheses

Ref	Expression
fprodefsum.1	⊢ 𝑍 = (ℤ≥‘𝑀)
fprodefsum.2	⊢ (𝜑 → 𝑁 ∈ 𝑍)
fprodefsum.3	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)

Assertion

Ref	Expression
fprodefsum	⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴) = (exp‘Σ𝑘 ∈ (𝑀...𝑁)𝐴))

Distinct variable groups:   𝜑,𝑘   𝑘,𝑀   𝑘,𝑁   𝑘,𝑍

Allowed substitution hint:   𝐴(𝑘)

Proof of Theorem fprodefsum

Dummy variables 𝑎 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fprodefsum.2	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
2		fprodefsum.1	. . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀)
3	1, 2	eleqtrdi 2839	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
4		oveq2 7428	. . . . . . 7 ⊢ (𝑎 = 𝑀 → (𝑀...𝑎) = (𝑀...𝑀))
5	4	prodeq1d 15897	. . . . . 6 ⊢ (𝑎 = 𝑀 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚))
6	4	sumeq1d 15679	. . . . . . 7 ⊢ (𝑎 = 𝑀 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
7	6	fveq2d 6901	. . . . . 6 ⊢ (𝑎 = 𝑀 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))
8	5, 7	eqeq12d 2744	. . . . 5 ⊢ (𝑎 = 𝑀 → (∏𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) ↔ ∏𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))))
9	8	imbi2d 340	. . . 4 ⊢ (𝑎 = 𝑀 → ((𝜑 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))) ↔ (𝜑 → ∏𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))))
10		oveq2 7428	. . . . . . 7 ⊢ (𝑎 = 𝑛 → (𝑀...𝑎) = (𝑀...𝑛))
11	10	prodeq1d 15897	. . . . . 6 ⊢ (𝑎 = 𝑛 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚))
12	10	sumeq1d 15679	. . . . . . 7 ⊢ (𝑎 = 𝑛 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
13	12	fveq2d 6901	. . . . . 6 ⊢ (𝑎 = 𝑛 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))
14	11, 13	eqeq12d 2744	. . . . 5 ⊢ (𝑎 = 𝑛 → (∏𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) ↔ ∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))))
15	14	imbi2d 340	. . . 4 ⊢ (𝑎 = 𝑛 → ((𝜑 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))) ↔ (𝜑 → ∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))))
16		oveq2 7428	. . . . . . 7 ⊢ (𝑎 = (𝑛 + 1) → (𝑀...𝑎) = (𝑀...(𝑛 + 1)))
17	16	prodeq1d 15897	. . . . . 6 ⊢ (𝑎 = (𝑛 + 1) → ∏𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚))
18	16	sumeq1d 15679	. . . . . . 7 ⊢ (𝑎 = (𝑛 + 1) → Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
19	18	fveq2d 6901	. . . . . 6 ⊢ (𝑎 = (𝑛 + 1) → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))
20	17, 19	eqeq12d 2744	. . . . 5 ⊢ (𝑎 = (𝑛 + 1) → (∏𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) ↔ ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))))
21	20	imbi2d 340	. . . 4 ⊢ (𝑎 = (𝑛 + 1) → ((𝜑 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))) ↔ (𝜑 → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))))
22		oveq2 7428	. . . . . . 7 ⊢ (𝑎 = 𝑁 → (𝑀...𝑎) = (𝑀...𝑁))
23	22	prodeq1d 15897	. . . . . 6 ⊢ (𝑎 = 𝑁 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚))
24	22	sumeq1d 15679	. . . . . . 7 ⊢ (𝑎 = 𝑁 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
25	24	fveq2d 6901	. . . . . 6 ⊢ (𝑎 = 𝑁 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))
26	23, 25	eqeq12d 2744	. . . . 5 ⊢ (𝑎 = 𝑁 → (∏𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) ↔ ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))))
27	26	imbi2d 340	. . . 4 ⊢ (𝑎 = 𝑁 → ((𝜑 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))) ↔ (𝜑 → ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))))
28		fzsn 13575	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
29	28	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (𝑀...𝑀) = {𝑀})
30	29	prodeq1d 15897	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ {𝑀} ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚))
31		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
32		uzid 12867	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
33	32, 2	eleqtrrdi 2840	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ 𝑍)
34		fprodefsum.3	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
35		efcl 16058	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
36	34, 35	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (exp‘𝐴) ∈ ℂ)
37	36	fmpttd 7125	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (exp‘𝐴)):𝑍⟶ℂ)
38	37	ffvelcdmda 7094	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑀) ∈ ℂ)
39	33, 38	sylan2 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑀) ∈ ℂ)
40		fveq2 6897	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑀))
41	40	prodsn 15938	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑀) ∈ ℂ) → ∏𝑚 ∈ {𝑀} ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑀))
42	31, 39, 41	syl2anc 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑚 ∈ {𝑀} ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑀))
43	33	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ 𝑍)
44		fvex 6910	. . . . . . . 8 ⊢ (exp‘⦋𝑀 / 𝑘⦌𝐴) ∈ V
45		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑀
46		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑘exp
47		nfcsb1v 3917	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑀 / 𝑘⦌𝐴
48	46, 47	nffv 6907	. . . . . . . . 9 ⊢ Ⅎ𝑘(exp‘⦋𝑀 / 𝑘⦌𝐴)
49		csbeq1a 3906	. . . . . . . . . 10 ⊢ (𝑘 = 𝑀 → 𝐴 = ⦋𝑀 / 𝑘⦌𝐴)
50	49	fveq2d 6901	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (exp‘𝐴) = (exp‘⦋𝑀 / 𝑘⦌𝐴))
51		eqid 2728	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑍 ↦ (exp‘𝐴)) = (𝑘 ∈ 𝑍 ↦ (exp‘𝐴))
52	45, 48, 50, 51	fvmptf 7026	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑍 ∧ (exp‘⦋𝑀 / 𝑘⦌𝐴) ∈ V) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑀) = (exp‘⦋𝑀 / 𝑘⦌𝐴))
53	43, 44, 52	sylancl 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑀) = (exp‘⦋𝑀 / 𝑘⦌𝐴))
54	30, 42, 53	3eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘⦋𝑀 / 𝑘⦌𝐴))
55	29	sumeq1d 15679	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → Σ𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑚 ∈ {𝑀} ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
56	34	fmpttd 7125	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐴):𝑍⟶ℂ)
57	56	ffvelcdmda 7094	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑀) ∈ ℂ)
58	33, 57	sylan2 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑀) ∈ ℂ)
59		fveq2 6897	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑀))
60	59	sumsn 15724	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑀) ∈ ℂ) → Σ𝑚 ∈ {𝑀} ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑀))
61	31, 58, 60	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → Σ𝑚 ∈ {𝑀} ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑀))
62	34	ralrimiva 3143	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ)
63	47	nfel1 2916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ
64	49	eleq1d 2814	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ))
65	63, 64	rspc 3597	. . . . . . . . . . 11 ⊢ (𝑀 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ))
66	65	impcom 407	. . . . . . . . . 10 ⊢ ((∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ ∧ 𝑀 ∈ 𝑍) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ)
67	62, 33, 66	syl2an 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ)
68		eqid 2728	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑍 ↦ 𝐴) = (𝑘 ∈ 𝑍 ↦ 𝐴)
69	68	fvmpts 7008	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑍 ∧ ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑀) = ⦋𝑀 / 𝑘⦌𝐴)
70	43, 67, 69	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑀) = ⦋𝑀 / 𝑘⦌𝐴)
71	55, 61, 70	3eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → Σ𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ⦋𝑀 / 𝑘⦌𝐴)
72	71	fveq2d 6901	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) = (exp‘⦋𝑀 / 𝑘⦌𝐴))
73	54, 72	eqtr4d 2771	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))
74	73	expcom 413	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝜑 → ∏𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))))
75		simp3 1136	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ ∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))) → ∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))
76	2	peano2uzs 12916	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → (𝑛 + 1) ∈ 𝑍)
77		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝑛 + 1) ∈ 𝑍)
78		nfcsb1v 3917	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋(𝑛 + 1) / 𝑘⦌𝐴
79	78	nfel1 2916	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ
80		csbeq1a 3906	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑛 + 1) → 𝐴 = ⦋(𝑛 + 1) / 𝑘⦌𝐴)
81	80	eleq1d 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑛 + 1) → (𝐴 ∈ ℂ ↔ ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ))
82	79, 81	rspc 3597	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ 𝑍 → (∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ → ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ))
83	62, 82	mpan9 506	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ)
84		efcl 16058	. . . . . . . . . . . . . . 15 ⊢ (⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ → (exp‘⦋(𝑛 + 1) / 𝑘⦌𝐴) ∈ ℂ)
85	83, 84	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (exp‘⦋(𝑛 + 1) / 𝑘⦌𝐴) ∈ ℂ)
86		nfcv 2899	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑛 + 1)
87	46, 78	nffv 6907	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(exp‘⦋(𝑛 + 1) / 𝑘⦌𝐴)
88	80	fveq2d 6901	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → (exp‘𝐴) = (exp‘⦋(𝑛 + 1) / 𝑘⦌𝐴))
89	86, 87, 88, 51	fvmptf 7026	. . . . . . . . . . . . . 14 ⊢ (((𝑛 + 1) ∈ 𝑍 ∧ (exp‘⦋(𝑛 + 1) / 𝑘⦌𝐴) ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘⦋(𝑛 + 1) / 𝑘⦌𝐴))
90	77, 85, 89	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘⦋(𝑛 + 1) / 𝑘⦌𝐴))
91	68	fvmpts 7008	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 + 1) ∈ 𝑍 ∧ ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)) = ⦋(𝑛 + 1) / 𝑘⦌𝐴)
92	77, 83, 91	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)) = ⦋(𝑛 + 1) / 𝑘⦌𝐴)
93	92	fveq2d 6901	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (exp‘((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1))) = (exp‘⦋(𝑛 + 1) / 𝑘⦌𝐴))
94	90, 93	eqtr4d 2771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1))))
95	76, 94	sylan2 592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1))))
96	95	3adant3 1130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ ∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1))))
97	75, 96	oveq12d 7438	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ ∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))) → (∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) · ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1))) = ((exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) · (exp‘((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)))))
98		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
99	98, 2	eleqtrdi 2839	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (ℤ≥‘𝑀))
100		elfzuz 13529	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (𝑀...(𝑛 + 1)) → 𝑚 ∈ (ℤ≥‘𝑀))
101	100, 2	eleqtrrdi 2840	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (𝑀...(𝑛 + 1)) → 𝑚 ∈ 𝑍)
102	37	ffvelcdmda 7094	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) ∈ ℂ)
103	101, 102	sylan2 592	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) ∈ ℂ)
104	103	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) ∈ ℂ)
105		fveq2 6897	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)))
106	99, 104, 105	fprodp1 15945	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) · ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1))))
107	106	3adant3 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ ∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))) → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) · ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1))))
108	56	ffvelcdmda 7094	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
109	101, 108	sylan2 592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
110	109	adantlr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (𝑀...(𝑛 + 1))) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
111		fveq2 6897	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)))
112	99, 110, 111	fsump1 15734	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = (Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) + ((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1))))
113	112	fveq2d 6901	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) = (exp‘(Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) + ((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)))))
114		fzfid 13970	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀...𝑛) ∈ Fin)
115		elfzuz 13529	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (𝑀...𝑛) → 𝑚 ∈ (ℤ≥‘𝑀))
116	115, 2	eleqtrrdi 2840	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (𝑀...𝑛) → 𝑚 ∈ 𝑍)
117	116, 108	sylan2 592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀...𝑛)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
118	117	adantlr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (𝑀...𝑛)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
119	114, 118	fsumcl 15711	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
120	56	ffvelcdmda 7094	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)) ∈ ℂ)
121	76, 120	sylan2 592	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)) ∈ ℂ)
122		efadd 16070	. . . . . . . . . . . 12 ⊢ ((Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ ∧ ((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)) ∈ ℂ) → (exp‘(Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) + ((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)))) = ((exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) · (exp‘((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)))))
123	119, 121, 122	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (exp‘(Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) + ((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)))) = ((exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) · (exp‘((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)))))
124	113, 123	eqtrd 2768	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) = ((exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) · (exp‘((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)))))
125	124	3adant3 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ ∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))) → (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) = ((exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) · (exp‘((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)))))
126	97, 107, 125	3eqtr4d 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ ∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))) → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))
127	126	3exp 1117	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ 𝑍 → (∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))))
128	127	com12 32	. . . . . 6 ⊢ (𝑛 ∈ 𝑍 → (𝜑 → (∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))))
129	128	a2d 29	. . . . 5 ⊢ (𝑛 ∈ 𝑍 → ((𝜑 → ∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))) → (𝜑 → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))))
130	2	eqcomi 2737	. . . . 5 ⊢ (ℤ≥‘𝑀) = 𝑍
131	129, 130	eleq2s 2847	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → ((𝜑 → ∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))) → (𝜑 → ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))))
132	9, 15, 21, 27, 74, 131	uzind4 12920	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))))
133	3, 132	mpcom 38	. 2 ⊢ (𝜑 → ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))
134		fvres 6916	. . . . 5 ⊢ (𝑚 ∈ (𝑀...𝑁) → (((𝑘 ∈ 𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚))
135		fzssuz 13574	. . . . . . . 8 ⊢ (𝑀...𝑁) ⊆ (ℤ≥‘𝑀)
136	135, 2	sseqtrri 4017	. . . . . . 7 ⊢ (𝑀...𝑁) ⊆ 𝑍
137		resmpt 6041	. . . . . . 7 ⊢ ((𝑀...𝑁) ⊆ 𝑍 → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁)) = (𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴)))
138	136, 137	ax-mp 5	. . . . . 6 ⊢ ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁)) = (𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))
139	138	fveq1i 6898	. . . . 5 ⊢ (((𝑘 ∈ 𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚)
140	134, 139	eqtr3di 2783	. . . 4 ⊢ (𝑚 ∈ (𝑀...𝑁) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚))
141	140	prodeq2i 15895	. . 3 ⊢ ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚)
142		prodfc 15921	. . 3 ⊢ ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚) = ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴)
143	141, 142	eqtri 2756	. 2 ⊢ ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴)
144		fvres 6916	. . . . . 6 ⊢ (𝑚 ∈ (𝑀...𝑁) → (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
145		resmpt 6041	. . . . . . . 8 ⊢ ((𝑀...𝑁) ⊆ 𝑍 → ((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑁)) = (𝑘 ∈ (𝑀...𝑁) ↦ 𝐴))
146	136, 145	ax-mp 5	. . . . . . 7 ⊢ ((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑁)) = (𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)
147	146	fveq1i 6898	. . . . . 6 ⊢ (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚)
148	144, 147	eqtr3di 2783	. . . . 5 ⊢ (𝑚 ∈ (𝑀...𝑁) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚))
149	148	sumeq2i 15677	. . . 4 ⊢ Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚)
150		sumfc 15687	. . . 4 ⊢ Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑁)𝐴
151	149, 150	eqtri 2756	. . 3 ⊢ Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑁)𝐴
152	151	fveq2i 6900	. 2 ⊢ (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) = (exp‘Σ𝑘 ∈ (𝑀...𝑁)𝐴)
153	133, 143, 152	3eqtr3g 2791	1 ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴) = (exp‘Σ𝑘 ∈ (𝑀...𝑁)𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3058  Vcvv 3471  ⦋csb 3892   ⊆ wss 3947  {csn 4629   ↦ cmpt 5231   ↾ cres 5680  ‘cfv 6548  (class class class)co 7420  ℂcc 11136  1c1 11139   + caddc 11141   · cmul 11143  ℤcz 12588  ℤ_≥cuz 12852  ...cfz 13516  Σcsu 15664  ∏cprod 15881  expce 16037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-inf2 9664  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-pre-sup 11216

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-er 8724  df-pm 8847  df-en 8964  df-dom 8965  df-sdom 8966  df-fin 8967  df-sup 9465  df-inf 9466  df-oi 9533  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-div 11902  df-nn 12243  df-2 12305  df-3 12306  df-n0 12503  df-z 12589  df-uz 12853  df-rp 13007  df-ico 13362  df-fz 13517  df-fzo 13660  df-fl 13789  df-seq 13999  df-exp 14059  df-fac 14265  df-bc 14294  df-hash 14322  df-shft 15046  df-cj 15078  df-re 15079  df-im 15080  df-sqrt 15214  df-abs 15215  df-limsup 15447  df-clim 15464  df-rlim 15465  df-sum 15665  df-prod 15882  df-ef 16043

This theorem is referenced by:  aks4d1p1p1  41534