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Theorem fprodefsum 16035

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 2844	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
4		oveq2 7414	. . . . . . 7 ⊢ (𝑎 = 𝑀 → (𝑀...𝑎) = (𝑀...𝑀))
5	4	prodeq1d 15862	. . . . . 6 ⊢ (𝑎 = 𝑀 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚))
6	4	sumeq1d 15644	. . . . . . 7 ⊢ (𝑎 = 𝑀 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
7	6	fveq2d 6893	. . . . . 6 ⊢ (𝑎 = 𝑀 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))
8	5, 7	eqeq12d 2749	. . . . 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 341	. . . 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 7414	. . . . . . 7 ⊢ (𝑎 = 𝑛 → (𝑀...𝑎) = (𝑀...𝑛))
11	10	prodeq1d 15862	. . . . . 6 ⊢ (𝑎 = 𝑛 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚))
12	10	sumeq1d 15644	. . . . . . 7 ⊢ (𝑎 = 𝑛 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
13	12	fveq2d 6893	. . . . . 6 ⊢ (𝑎 = 𝑛 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))
14	11, 13	eqeq12d 2749	. . . . 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 341	. . . 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 7414	. . . . . . 7 ⊢ (𝑎 = (𝑛 + 1) → (𝑀...𝑎) = (𝑀...(𝑛 + 1)))
17	16	prodeq1d 15862	. . . . . 6 ⊢ (𝑎 = (𝑛 + 1) → ∏𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚))
18	16	sumeq1d 15644	. . . . . . 7 ⊢ (𝑎 = (𝑛 + 1) → Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...(𝑛 + 1))((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
19	18	fveq2d 6893	. . . . . 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 2749	. . . . 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 341	. . . 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 7414	. . . . . . 7 ⊢ (𝑎 = 𝑁 → (𝑀...𝑎) = (𝑀...𝑁))
23	22	prodeq1d 15862	. . . . . 6 ⊢ (𝑎 = 𝑁 → ∏𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚))
24	22	sumeq1d 15644	. . . . . . 7 ⊢ (𝑎 = 𝑁 → Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
25	24	fveq2d 6893	. . . . . 6 ⊢ (𝑎 = 𝑁 → (exp‘Σ𝑚 ∈ (𝑀...𝑎)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) = (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))
26	23, 25	eqeq12d 2749	. . . . 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 341	. . . 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 13540	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
29	28	adantl 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (𝑀...𝑀) = {𝑀})
30	29	prodeq1d 15862	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ {𝑀} ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚))
31		simpr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ)
32		uzid 12834	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
33	32, 2	eleqtrrdi 2845	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ 𝑍)
34		fprodefsum.3	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
35		efcl 16023	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ)
36	34, 35	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (exp‘𝐴) ∈ ℂ)
37	36	fmpttd 7112	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (exp‘𝐴)):𝑍⟶ℂ)
38	37	ffvelcdmda 7084	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑀) ∈ ℂ)
39	33, 38	sylan2 594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑀) ∈ ℂ)
40		fveq2 6889	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑀))
41	40	prodsn 15903	. . . . . . . 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 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑚 ∈ {𝑀} ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑀))
43	33	adantl 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ 𝑍)
44		fvex 6902	. . . . . . . 8 ⊢ (exp‘⦋𝑀 / 𝑘⦌𝐴) ∈ V
45		nfcv 2904	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑀
46		nfcv 2904	. . . . . . . . . 10 ⊢ Ⅎ𝑘exp
47		nfcsb1v 3918	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑀 / 𝑘⦌𝐴
48	46, 47	nffv 6899	. . . . . . . . 9 ⊢ Ⅎ𝑘(exp‘⦋𝑀 / 𝑘⦌𝐴)
49		csbeq1a 3907	. . . . . . . . . 10 ⊢ (𝑘 = 𝑀 → 𝐴 = ⦋𝑀 / 𝑘⦌𝐴)
50	49	fveq2d 6893	. . . . . . . . 9 ⊢ (𝑘 = 𝑀 → (exp‘𝐴) = (exp‘⦋𝑀 / 𝑘⦌𝐴))
51		eqid 2733	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝑍 ↦ (exp‘𝐴)) = (𝑘 ∈ 𝑍 ↦ (exp‘𝐴))
52	45, 48, 50, 51	fvmptf 7017	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝑍 ∧ (exp‘⦋𝑀 / 𝑘⦌𝐴) ∈ V) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑀) = (exp‘⦋𝑀 / 𝑘⦌𝐴))
53	43, 44, 52	sylancl 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑀) = (exp‘⦋𝑀 / 𝑘⦌𝐴))
54	30, 42, 53	3eqtrd 2777	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘⦋𝑀 / 𝑘⦌𝐴))
55	29	sumeq1d 15644	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → Σ𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑚 ∈ {𝑀} ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
56	34	fmpttd 7112	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐴):𝑍⟶ℂ)
57	56	ffvelcdmda 7084	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑀 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑀) ∈ ℂ)
58	33, 57	sylan2 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑀) ∈ ℂ)
59		fveq2 6889	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑀))
60	59	sumsn 15689	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑀) ∈ ℂ) → Σ𝑚 ∈ {𝑀} ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑀))
61	31, 58, 60	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → Σ𝑚 ∈ {𝑀} ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑀))
62	34	ralrimiva 3147	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ)
63	47	nfel1 2920	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ
64	49	eleq1d 2819	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑀 → (𝐴 ∈ ℂ ↔ ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ))
65	63, 64	rspc 3601	. . . . . . . . . . 11 ⊢ (𝑀 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ))
66	65	impcom 409	. . . . . . . . . 10 ⊢ ((∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ ∧ 𝑀 ∈ 𝑍) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ)
67	62, 33, 66	syl2an 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ)
68		eqid 2733	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝑍 ↦ 𝐴) = (𝑘 ∈ 𝑍 ↦ 𝐴)
69	68	fvmpts 6999	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑍 ∧ ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑀) = ⦋𝑀 / 𝑘⦌𝐴)
70	43, 67, 69	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑀) = ⦋𝑀 / 𝑘⦌𝐴)
71	55, 61, 70	3eqtrd 2777	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → Σ𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ⦋𝑀 / 𝑘⦌𝐴)
72	71	fveq2d 6893	. . . . . 6 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) = (exp‘⦋𝑀 / 𝑘⦌𝐴))
73	54, 72	eqtr4d 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → ∏𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)))
74	73	expcom 415	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝜑 → ∏𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = (exp‘Σ𝑚 ∈ (𝑀...𝑀)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))))
75		simp3 1139	. . . . . . . . . 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 12883	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → (𝑛 + 1) ∈ 𝑍)
77		simpr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (𝑛 + 1) ∈ 𝑍)
78		nfcsb1v 3918	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑘⦋(𝑛 + 1) / 𝑘⦌𝐴
79	78	nfel1 2920	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ
80		csbeq1a 3907	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑛 + 1) → 𝐴 = ⦋(𝑛 + 1) / 𝑘⦌𝐴)
81	80	eleq1d 2819	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑛 + 1) → (𝐴 ∈ ℂ ↔ ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ))
82	79, 81	rspc 3601	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 + 1) ∈ 𝑍 → (∀𝑘 ∈ 𝑍 𝐴 ∈ ℂ → ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ))
83	62, 82	mpan9 508	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ)
84		efcl 16023	. . . . . . . . . . . . . . 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 2904	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝑛 + 1)
87	46, 78	nffv 6899	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(exp‘⦋(𝑛 + 1) / 𝑘⦌𝐴)
88	80	fveq2d 6893	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑛 + 1) → (exp‘𝐴) = (exp‘⦋(𝑛 + 1) / 𝑘⦌𝐴))
89	86, 87, 88, 51	fvmptf 7017	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘⦋(𝑛 + 1) / 𝑘⦌𝐴))
91	68	fvmpts 6999	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 + 1) ∈ 𝑍 ∧ ⦋(𝑛 + 1) / 𝑘⦌𝐴 ∈ ℂ) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)) = ⦋(𝑛 + 1) / 𝑘⦌𝐴)
92	77, 83, 91	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)) = ⦋(𝑛 + 1) / 𝑘⦌𝐴)
93	92	fveq2d 6893	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → (exp‘((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1))) = (exp‘⦋(𝑛 + 1) / 𝑘⦌𝐴))
94	90, 93	eqtr4d 2776	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1))))
95	76, 94	sylan2 594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)) = (exp‘((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1))))
96	95	3adant3 1133	. . . . . . . . . 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 7424	. . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
99	98, 2	eleqtrdi 2844	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (ℤ_≥‘𝑀))
100		elfzuz 13494	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (𝑀...(𝑛 + 1)) → 𝑚 ∈ (ℤ_≥‘𝑀))
101	100, 2	eleqtrrdi 2845	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (𝑀...(𝑛 + 1)) → 𝑚 ∈ 𝑍)
102	37	ffvelcdmda 7084	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) ∈ ℂ)
103	101, 102	sylan2 594	. . . . . . . . . . . 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 6889	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘(𝑛 + 1)))
106	99, 104, 105	fprodp1 15910	. . . . . . . . . 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 1133	. . . . . . . . 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 7084	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
109	101, 108	sylan2 594	. . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)))
112	99, 110, 111	fsump1 15699	. . . . . . . . . . . 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 6893	. . . . . . . . . . 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 13935	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑀...𝑛) ∈ Fin)
115		elfzuz 13494	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (𝑀...𝑛) → 𝑚 ∈ (ℤ_≥‘𝑀))
116	115, 2	eleqtrrdi 2845	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (𝑀...𝑛) → 𝑚 ∈ 𝑍)
117	116, 108	sylan2 594	. . . . . . . . . . . . . 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 15676	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → Σ𝑚 ∈ (𝑀...𝑛)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ)
120	56	ffvelcdmda 7084	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 + 1) ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)) ∈ ℂ)
121	76, 120	sylan2 594	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘(𝑛 + 1)) ∈ ℂ)
122		efadd 16034	. . . . . . . . . . . 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 585	. . . . . . . . . . 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 2773	. . . . . . . . . 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 1133	. . . . . . . . 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 2783	. . . . . . . 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 1120	. . . . . . 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 2742	. . . . 5 ⊢ (ℤ_≥‘𝑀) = 𝑍
131	129, 130	eleq2s 2852	. . . 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 12887	. . 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 6908	. . . . 5 ⊢ (𝑚 ∈ (𝑀...𝑁) → (((𝑘 ∈ 𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚))
135		fzssuz 13539	. . . . . . . 8 ⊢ (𝑀...𝑁) ⊆ (ℤ_≥‘𝑀)
136	135, 2	sseqtrri 4019	. . . . . . 7 ⊢ (𝑀...𝑁) ⊆ 𝑍
137		resmpt 6036	. . . . . . 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 6890	. . . . 5 ⊢ (((𝑘 ∈ 𝑍 ↦ (exp‘𝐴)) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚)
140	134, 139	eqtr3di 2788	. . . 4 ⊢ (𝑚 ∈ (𝑀...𝑁) → ((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚))
141	140	prodeq2i 15860	. . 3 ⊢ ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚)
142		prodfc 15886	. . 3 ⊢ ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ (exp‘𝐴))‘𝑚) = ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴)
143	141, 142	eqtri 2761	. 2 ⊢ ∏𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ (exp‘𝐴))‘𝑚) = ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴)
144		fvres 6908	. . . . . 6 ⊢ (𝑚 ∈ (𝑀...𝑁) → (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚))
145		resmpt 6036	. . . . . . . 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 6890	. . . . . 6 ⊢ (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑁))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚)
148	144, 147	eqtr3di 2788	. . . . 5 ⊢ (𝑚 ∈ (𝑀...𝑁) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚))
149	148	sumeq2i 15642	. . . 4 ⊢ Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚)
150		sumfc 15652	. . . 4 ⊢ Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑁)𝐴
151	149, 150	eqtri 2761	. . 3 ⊢ Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = Σ𝑘 ∈ (𝑀...𝑁)𝐴
152	151	fveq2i 6892	. 2 ⊢ (exp‘Σ𝑚 ∈ (𝑀...𝑁)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) = (exp‘Σ𝑘 ∈ (𝑀...𝑁)𝐴)
153	133, 143, 152	3eqtr3g 2796	1 ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)(exp‘𝐴) = (exp‘Σ𝑘 ∈ (𝑀...𝑁)𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 ⦋csb 3893 ⊆ wss 3948 {csn 4628 ↦ cmpt 5231 ↾ cres 5678 ‘cfv 6541 (class class class)co 7406 ℂcc 11105 1c1 11108 + caddc 11110 · cmul 11112 ℤcz 12555 ℤ_≥cuz 12819 ...cfz 13481 Σcsu 15629 ∏cprod 15846 expce 16002

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-ico 13327 df-fz 13482 df-fzo 13625 df-fl 13754 df-seq 13964 df-exp 14025 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15011 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-limsup 15412 df-clim 15429 df-rlim 15430 df-sum 15630 df-prod 15847 df-ef 16008

This theorem is referenced by: aks4d1p1p1 40917

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