Theorem breprexp 34610

Description: Express the 𝑆 th power of the finite series in terms of the number of representations of integers 𝑚 as sums of 𝑆 terms. This is a general formulation which allows logarithmic weighting of the sums (see https://mathoverflow.net/questions/253246) and a mix of different smoothing functions taken into account in 𝐿. See breprexpnat 34611 for the simple case presented in the proposition of [Nathanson] p. 123. (Contributed by Thierry Arnoux, 6-Dec-2021.)

Hypotheses

Ref	Expression
breprexp.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
breprexp.s	⊢ (𝜑 → 𝑆 ∈ ℕ0)
breprexp.z	⊢ (𝜑 → 𝑍 ∈ ℂ)
breprexp.h	⊢ (𝜑 → 𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))

Assertion

Ref	Expression
breprexp	⊢ (𝜑 → ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))

Distinct variable groups:   𝑁,𝑐,𝑚   𝑆,𝑎,𝑐,𝑚   𝑍,𝑐,𝑚,𝑏   𝜑,𝑐   𝐿,𝑐,𝑚,𝑎,𝑏   𝑁,𝑎,𝑏   𝑆,𝑏   𝑍,𝑎,𝑏   𝜑,𝑎,𝑏,𝑚

Proof of Theorem breprexp

Dummy variables 𝑠 𝑡 𝑖 𝑗 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breprexp.s	. 2 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
2		nn0ssre 12557	. . . . . 6 ⊢ ℕ0 ⊆ ℝ
3	2	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ0 ⊆ ℝ)
4	3	sselda 4008	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ ℕ0) → 𝑆 ∈ ℝ)
5		leid 11386	. . . 4 ⊢ (𝑆 ∈ ℝ → 𝑆 ≤ 𝑆)
6	4, 5	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ ℕ0) → 𝑆 ≤ 𝑆)
7		breq1 5169	. . . . 5 ⊢ (𝑡 = 0 → (𝑡 ≤ 𝑆 ↔ 0 ≤ 𝑆))
8		oveq2 7456	. . . . . . 7 ⊢ (𝑡 = 0 → (0..^𝑡) = (0..^0))
9	8	prodeq1d 15968	. . . . . 6 ⊢ (𝑡 = 0 → ∏𝑎 ∈ (0..^𝑡)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)))
10		oveq1 7455	. . . . . . . 8 ⊢ (𝑡 = 0 → (𝑡 · 𝑁) = (0 · 𝑁))
11	10	oveq2d 7464	. . . . . . 7 ⊢ (𝑡 = 0 → (0...(𝑡 · 𝑁)) = (0...(0 · 𝑁)))
12		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑡 = 0 → (repr‘𝑡) = (repr‘0))
13	12	oveqd 7465	. . . . . . . . 9 ⊢ (𝑡 = 0 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘0)𝑚))
14	8	prodeq1d 15968	. . . . . . . . . . 11 ⊢ (𝑡 = 0 → ∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)))
15	14	oveq1d 7463	. . . . . . . . . 10 ⊢ (𝑡 = 0 → (∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
16	15	adantr 480	. . . . . . . . 9 ⊢ ((𝑡 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)) → (∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
17	13, 16	sumeq12dv 15754	. . . . . . . 8 ⊢ (𝑡 = 0 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
18	17	adantr 480	. . . . . . 7 ⊢ ((𝑡 = 0 ∧ 𝑚 ∈ (0...(𝑡 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
19	11, 18	sumeq12dv 15754	. . . . . 6 ⊢ (𝑡 = 0 → Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑚 ∈ (0...(0 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
20	9, 19	eqeq12d 2756	. . . . 5 ⊢ (𝑡 = 0 → (∏𝑎 ∈ (0..^𝑡)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) ↔ ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(0 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚))))
21	7, 20	imbi12d 344	. . . 4 ⊢ (𝑡 = 0 → ((𝑡 ≤ 𝑆 → ∏𝑎 ∈ (0..^𝑡)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚))) ↔ (0 ≤ 𝑆 → ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(0 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))))
22		breq1 5169	. . . . 5 ⊢ (𝑡 = 𝑠 → (𝑡 ≤ 𝑆 ↔ 𝑠 ≤ 𝑆))
23		oveq2 7456	. . . . . . 7 ⊢ (𝑡 = 𝑠 → (0..^𝑡) = (0..^𝑠))
24	23	prodeq1d 15968	. . . . . 6 ⊢ (𝑡 = 𝑠 → ∏𝑎 ∈ (0..^𝑡)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = ∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)))
25		oveq1 7455	. . . . . . . 8 ⊢ (𝑡 = 𝑠 → (𝑡 · 𝑁) = (𝑠 · 𝑁))
26	25	oveq2d 7464	. . . . . . 7 ⊢ (𝑡 = 𝑠 → (0...(𝑡 · 𝑁)) = (0...(𝑠 · 𝑁)))
27		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑡 = 𝑠 → (repr‘𝑡) = (repr‘𝑠))
28	27	oveqd 7465	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘𝑠)𝑚))
29	23	prodeq1d 15968	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑠 → ∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)))
30	29	oveq1d 7463	. . . . . . . . . 10 ⊢ (𝑡 = 𝑠 → (∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
31	30	adantr 480	. . . . . . . . 9 ⊢ ((𝑡 = 𝑠 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)) → (∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
32	28, 31	sumeq12dv 15754	. . . . . . . 8 ⊢ (𝑡 = 𝑠 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
33	32	adantr 480	. . . . . . 7 ⊢ ((𝑡 = 𝑠 ∧ 𝑚 ∈ (0...(𝑡 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
34	26, 33	sumeq12dv 15754	. . . . . 6 ⊢ (𝑡 = 𝑠 → Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
35	24, 34	eqeq12d 2756	. . . . 5 ⊢ (𝑡 = 𝑠 → (∏𝑎 ∈ (0..^𝑡)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) ↔ ∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚))))
36	22, 35	imbi12d 344	. . . 4 ⊢ (𝑡 = 𝑠 → ((𝑡 ≤ 𝑆 → ∏𝑎 ∈ (0..^𝑡)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚))) ↔ (𝑠 ≤ 𝑆 → ∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))))
37		breq1 5169	. . . . 5 ⊢ (𝑡 = (𝑠 + 1) → (𝑡 ≤ 𝑆 ↔ (𝑠 + 1) ≤ 𝑆))
38		oveq2 7456	. . . . . . 7 ⊢ (𝑡 = (𝑠 + 1) → (0..^𝑡) = (0..^(𝑠 + 1)))
39	38	prodeq1d 15968	. . . . . 6 ⊢ (𝑡 = (𝑠 + 1) → ∏𝑎 ∈ (0..^𝑡)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = ∏𝑎 ∈ (0..^(𝑠 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)))
40		oveq1 7455	. . . . . . . 8 ⊢ (𝑡 = (𝑠 + 1) → (𝑡 · 𝑁) = ((𝑠 + 1) · 𝑁))
41	40	oveq2d 7464	. . . . . . 7 ⊢ (𝑡 = (𝑠 + 1) → (0...(𝑡 · 𝑁)) = (0...((𝑠 + 1) · 𝑁)))
42		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑡 = (𝑠 + 1) → (repr‘𝑡) = (repr‘(𝑠 + 1)))
43	42	oveqd 7465	. . . . . . . . 9 ⊢ (𝑡 = (𝑠 + 1) → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘(𝑠 + 1))𝑚))
44	38	prodeq1d 15968	. . . . . . . . . . 11 ⊢ (𝑡 = (𝑠 + 1) → ∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿‘𝑎)‘(𝑐‘𝑎)))
45	44	oveq1d 7463	. . . . . . . . . 10 ⊢ (𝑡 = (𝑠 + 1) → (∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = (∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
46	45	adantr 480	. . . . . . . . 9 ⊢ ((𝑡 = (𝑠 + 1) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)) → (∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = (∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
47	43, 46	sumeq12dv 15754	. . . . . . . 8 ⊢ (𝑡 = (𝑠 + 1) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘(𝑠 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
48	47	adantr 480	. . . . . . 7 ⊢ ((𝑡 = (𝑠 + 1) ∧ 𝑚 ∈ (0...(𝑡 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘(𝑠 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
49	41, 48	sumeq12dv 15754	. . . . . 6 ⊢ (𝑡 = (𝑠 + 1) → Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑚 ∈ (0...((𝑠 + 1) · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘(𝑠 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
50	39, 49	eqeq12d 2756	. . . . 5 ⊢ (𝑡 = (𝑠 + 1) → (∏𝑎 ∈ (0..^𝑡)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) ↔ ∏𝑎 ∈ (0..^(𝑠 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...((𝑠 + 1) · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘(𝑠 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚))))
51	37, 50	imbi12d 344	. . . 4 ⊢ (𝑡 = (𝑠 + 1) → ((𝑡 ≤ 𝑆 → ∏𝑎 ∈ (0..^𝑡)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚))) ↔ ((𝑠 + 1) ≤ 𝑆 → ∏𝑎 ∈ (0..^(𝑠 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...((𝑠 + 1) · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘(𝑠 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))))
52		breq1 5169	. . . . 5 ⊢ (𝑡 = 𝑆 → (𝑡 ≤ 𝑆 ↔ 𝑆 ≤ 𝑆))
53		oveq2 7456	. . . . . . 7 ⊢ (𝑡 = 𝑆 → (0..^𝑡) = (0..^𝑆))
54	53	prodeq1d 15968	. . . . . 6 ⊢ (𝑡 = 𝑆 → ∏𝑎 ∈ (0..^𝑡)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)))
55		oveq1 7455	. . . . . . . 8 ⊢ (𝑡 = 𝑆 → (𝑡 · 𝑁) = (𝑆 · 𝑁))
56	55	oveq2d 7464	. . . . . . 7 ⊢ (𝑡 = 𝑆 → (0...(𝑡 · 𝑁)) = (0...(𝑆 · 𝑁)))
57		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑡 = 𝑆 → (repr‘𝑡) = (repr‘𝑆))
58	57	oveqd 7465	. . . . . . . . 9 ⊢ (𝑡 = 𝑆 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘𝑆)𝑚))
59	53	prodeq1d 15968	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑆 → ∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)))
60	59	oveq1d 7463	. . . . . . . . . 10 ⊢ (𝑡 = 𝑆 → (∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
61	60	adantr 480	. . . . . . . . 9 ⊢ ((𝑡 = 𝑆 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)) → (∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
62	58, 61	sumeq12dv 15754	. . . . . . . 8 ⊢ (𝑡 = 𝑆 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
63	62	adantr 480	. . . . . . 7 ⊢ ((𝑡 = 𝑆 ∧ 𝑚 ∈ (0...(𝑡 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
64	56, 63	sumeq12dv 15754	. . . . . 6 ⊢ (𝑡 = 𝑆 → Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
65	54, 64	eqeq12d 2756	. . . . 5 ⊢ (𝑡 = 𝑆 → (∏𝑎 ∈ (0..^𝑡)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) ↔ ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚))))
66	52, 65	imbi12d 344	. . . 4 ⊢ (𝑡 = 𝑆 → ((𝑡 ≤ 𝑆 → ∏𝑎 ∈ (0..^𝑡)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚))) ↔ (𝑆 ≤ 𝑆 → ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))))
67		0nn0 12568	. . . . . . . 8 ⊢ 0 ∈ ℕ0
68		fz1ssnn 13615	. . . . . . . . . . . . 13 ⊢ (1...𝑁) ⊆ ℕ
69	68	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝑁) ⊆ ℕ)
70		0zd 12651	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℤ)
71		breprexp.n	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
72	69, 70, 71	repr0 34588	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...𝑁)(repr‘0)0) = if(0 = 0, {∅}, ∅))
73		eqid 2740	. . . . . . . . . . . 12 ⊢ 0 = 0
74	73	iftruei 4555	. . . . . . . . . . 11 ⊢ if(0 = 0, {∅}, ∅) = {∅}
75	72, 74	eqtrdi 2796	. . . . . . . . . 10 ⊢ (𝜑 → ((1...𝑁)(repr‘0)0) = {∅})
76		snfi 9109	. . . . . . . . . 10 ⊢ {∅} ∈ Fin
77	75, 76	eqeltrdi 2852	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝑁)(repr‘0)0) ∈ Fin)
78		fzo0 13740	. . . . . . . . . . . . . . . 16 ⊢ (0..^0) = ∅
79	78	prodeq1i 15964	. . . . . . . . . . . . . . 15 ⊢ ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ ∅ ((𝐿‘𝑎)‘(𝑐‘𝑎))
80		prod0 15991	. . . . . . . . . . . . . . 15 ⊢ ∏𝑎 ∈ ∅ ((𝐿‘𝑎)‘(𝑐‘𝑎)) = 1
81	79, 80	eqtri 2768	. . . . . . . . . . . . . 14 ⊢ ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) = 1
82	81	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) = 1)
83		breprexp.z	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ ℂ)
84		exp0 14116	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ ℂ → (𝑍↑0) = 1)
85	83, 84	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍↑0) = 1)
86	82, 85	oveq12d 7466	. . . . . . . . . . . 12 ⊢ (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) = (1 · 1))
87		ax-1cn 11242	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
88	87	mulridi 11294	. . . . . . . . . . . 12 ⊢ (1 · 1) = 1
89	86, 88	eqtrdi 2796	. . . . . . . . . . 11 ⊢ (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) = 1)
90	89, 87	eqeltrdi 2852	. . . . . . . . . 10 ⊢ (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) ∈ ℂ)
91	90	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)0)) → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) ∈ ℂ)
92	77, 91	fsumcl 15781	. . . . . . . 8 ⊢ (𝜑 → Σ𝑐 ∈ ((1...𝑁)(repr‘0)0)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) ∈ ℂ)
93		oveq2 7456	. . . . . . . . . 10 ⊢ (𝑚 = 0 → ((1...𝑁)(repr‘0)𝑚) = ((1...𝑁)(repr‘0)0))
94		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑚 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)) → 𝑚 = 0)
95	94	oveq2d 7464	. . . . . . . . . . 11 ⊢ ((𝑚 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)) → (𝑍↑𝑚) = (𝑍↑0))
96	95	oveq2d 7464	. . . . . . . . . 10 ⊢ ((𝑚 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)) → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)))
97	93, 96	sumeq12dv 15754	. . . . . . . . 9 ⊢ (𝑚 = 0 → Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘0)0)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)))
98	97	sumsn 15794	. . . . . . . 8 ⊢ ((0 ∈ ℕ0 ∧ Σ𝑐 ∈ ((1...𝑁)(repr‘0)0)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) ∈ ℂ) → Σ𝑚 ∈ {0}Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘0)0)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)))
99	67, 92, 98	sylancr 586	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ {0}Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘0)0)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)))
100	75	sumeq1d 15748	. . . . . . 7 ⊢ (𝜑 → Σ𝑐 ∈ ((1...𝑁)(repr‘0)0)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) = Σ𝑐 ∈ {∅} (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)))
101		0ex 5325	. . . . . . . . 9 ⊢ ∅ ∈ V
102	78	prodeq1i 15964	. . . . . . . . . . . . 13 ⊢ ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) = ∏𝑎 ∈ ∅ ((𝐿‘𝑎)‘(∅‘𝑎))
103		prod0 15991	. . . . . . . . . . . . 13 ⊢ ∏𝑎 ∈ ∅ ((𝐿‘𝑎)‘(∅‘𝑎)) = 1
104	102, 103	eqtri 2768	. . . . . . . . . . . 12 ⊢ ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) = 1
105	104	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) = 1)
106	105, 87	eqeltrdi 2852	. . . . . . . . . 10 ⊢ (𝜑 → ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) ∈ ℂ)
107	85, 87	eqeltrdi 2852	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍↑0) ∈ ℂ)
108	106, 107	mulcld 11310	. . . . . . . . 9 ⊢ (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) · (𝑍↑0)) ∈ ℂ)
109		fveq1 6919	. . . . . . . . . . . . . 14 ⊢ (𝑐 = ∅ → (𝑐‘𝑎) = (∅‘𝑎))
110	109	fveq2d 6924	. . . . . . . . . . . . 13 ⊢ (𝑐 = ∅ → ((𝐿‘𝑎)‘(𝑐‘𝑎)) = ((𝐿‘𝑎)‘(∅‘𝑎)))
111	110	ralrimivw 3156	. . . . . . . . . . . 12 ⊢ (𝑐 = ∅ → ∀𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ((𝐿‘𝑎)‘(∅‘𝑎)))
112	111	prodeq2d 15969	. . . . . . . . . . 11 ⊢ (𝑐 = ∅ → ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)))
113	112	oveq1d 7463	. . . . . . . . . 10 ⊢ (𝑐 = ∅ → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) = (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) · (𝑍↑0)))
114	113	sumsn 15794	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) · (𝑍↑0)) ∈ ℂ) → Σ𝑐 ∈ {∅} (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) = (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) · (𝑍↑0)))
115	101, 108, 114	sylancr 586	. . . . . . . 8 ⊢ (𝜑 → Σ𝑐 ∈ {∅} (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) = (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) · (𝑍↑0)))
116	105, 85	oveq12d 7466	. . . . . . . . 9 ⊢ (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) · (𝑍↑0)) = (1 · 1))
117	116, 86, 89	3eqtr2d 2786	. . . . . . . 8 ⊢ (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) · (𝑍↑0)) = 1)
118	115, 117	eqtrd 2780	. . . . . . 7 ⊢ (𝜑 → Σ𝑐 ∈ {∅} (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) = 1)
119	99, 100, 118	3eqtrd 2784	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ {0}Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = 1)
120	71	nn0cnd 12615	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ)
121	120	mul02d 11488	. . . . . . . . 9 ⊢ (𝜑 → (0 · 𝑁) = 0)
122	121	oveq2d 7464	. . . . . . . 8 ⊢ (𝜑 → (0...(0 · 𝑁)) = (0...0))
123		fz0sn 13684	. . . . . . . 8 ⊢ (0...0) = {0}
124	122, 123	eqtrdi 2796	. . . . . . 7 ⊢ (𝜑 → (0...(0 · 𝑁)) = {0})
125	124	sumeq1d 15748	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ (0...(0 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑚 ∈ {0}Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
126	78	prodeq1i 15964	. . . . . . . 8 ⊢ ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = ∏𝑎 ∈ ∅ Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏))
127		prod0 15991	. . . . . . . 8 ⊢ ∏𝑎 ∈ ∅ Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = 1
128	126, 127	eqtri 2768	. . . . . . 7 ⊢ ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = 1
129	128	a1i 11	. . . . . 6 ⊢ (𝜑 → ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = 1)
130	119, 125, 129	3eqtr4rd 2791	. . . . 5 ⊢ (𝜑 → ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(0 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
131	130	a1d 25	. . . 4 ⊢ (𝜑 → (0 ≤ 𝑆 → ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(0 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚))))
132		simpll 766	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))) ∧ (𝑠 + 1) ≤ 𝑆) → (𝜑 ∧ 𝑠 ∈ ℕ0))
133		simplr 768	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))) ∧ (𝑠 + 1) ≤ 𝑆) → (𝑠 ≤ 𝑆 → ∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚))))
134		oveq2 7456	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((1...𝑁)(repr‘𝑠)𝑚) = ((1...𝑁)(repr‘𝑠)𝑛))
135		oveq2 7456	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑍↑𝑚) = (𝑍↑𝑛))
136	135	oveq2d 7464	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑛)))
137	136	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑛 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)) → (∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑛)))
138	134, 137	sumeq12dv 15754	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑛)))
139	138	cbvsumv 15744	. . . . . . . . . 10 ⊢ Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑛))
140	139	eqeq2i 2753	. . . . . . . . 9 ⊢ (∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) ↔ ∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑛)))
141		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 𝑖 ∧ 𝑏 ∈ (1...𝑁)) → 𝑎 = 𝑖)
142	141	fveq2d 6924	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑖 ∧ 𝑏 ∈ (1...𝑁)) → (𝐿‘𝑎) = (𝐿‘𝑖))
143	142	fveq1d 6922	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑖 ∧ 𝑏 ∈ (1...𝑁)) → ((𝐿‘𝑎)‘𝑏) = ((𝐿‘𝑖)‘𝑏))
144	143	oveq1d 7463	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑖 ∧ 𝑏 ∈ (1...𝑁)) → (((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = (((𝐿‘𝑖)‘𝑏) · (𝑍↑𝑏)))
145	144	sumeq2dv 15750	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑖 → Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑏) · (𝑍↑𝑏)))
146	145	cbvprodv 15962	. . . . . . . . . . 11 ⊢ ∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = ∏𝑖 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑏) · (𝑍↑𝑏))
147		fveq2 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑗 → ((𝐿‘𝑖)‘𝑏) = ((𝐿‘𝑖)‘𝑗))
148		oveq2 7456	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑗 → (𝑍↑𝑏) = (𝑍↑𝑗))
149	147, 148	oveq12d 7466	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑗 → (((𝐿‘𝑖)‘𝑏) · (𝑍↑𝑏)) = (((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)))
150	149	cbvsumv 15744	. . . . . . . . . . . . 13 ⊢ Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑏) · (𝑍↑𝑏)) = Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗))
151	150	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^𝑠) → Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑏) · (𝑍↑𝑏)) = Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)))
152	151	prodeq2i 15966	. . . . . . . . . . 11 ⊢ ∏𝑖 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑏) · (𝑍↑𝑏)) = ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗))
153	146, 152	eqtri 2768	. . . . . . . . . 10 ⊢ ∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗))
154		fveq2 6920	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑖 → (𝐿‘𝑎) = (𝐿‘𝑖))
155		fveq2 6920	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑖 → (𝑐‘𝑎) = (𝑐‘𝑖))
156	154, 155	fveq12d 6927	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑖 → ((𝐿‘𝑎)‘(𝑐‘𝑎)) = ((𝐿‘𝑖)‘(𝑐‘𝑖)))
157	156	cbvprodv 15962	. . . . . . . . . . . . . . . 16 ⊢ ∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑐‘𝑖))
158	157	oveq1i 7458	. . . . . . . . . . . . . . 15 ⊢ (∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑛)) = (∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑐‘𝑖)) · (𝑍↑𝑛))
159	158	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛) → (∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑛)) = (∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑐‘𝑖)) · (𝑍↑𝑛)))
160	159	sumeq2i 15746	. . . . . . . . . . . . 13 ⊢ Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑛)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑐‘𝑖)) · (𝑍↑𝑛))
161		simpl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 = 𝑘 ∧ 𝑖 ∈ (0..^𝑠)) → 𝑐 = 𝑘)
162	161	fveq1d 6922	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 = 𝑘 ∧ 𝑖 ∈ (0..^𝑠)) → (𝑐‘𝑖) = (𝑘‘𝑖))
163	162	fveq2d 6924	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 = 𝑘 ∧ 𝑖 ∈ (0..^𝑠)) → ((𝐿‘𝑖)‘(𝑐‘𝑖)) = ((𝐿‘𝑖)‘(𝑘‘𝑖)))
164	163	prodeq2dv 15970	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑘 → ∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑐‘𝑖)) = ∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)))
165	164	oveq1d 7463	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑘 → (∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑐‘𝑖)) · (𝑍↑𝑛)) = (∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))
166	165	cbvsumv 15744	. . . . . . . . . . . . 13 ⊢ Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑐‘𝑖)) · (𝑍↑𝑛)) = Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛))
167	160, 166	eqtri 2768	. . . . . . . . . . . 12 ⊢ Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑛)) = Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛))
168	167	a1i 11	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (0...(𝑠 · 𝑁)) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑛)) = Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))
169	168	sumeq2i 15746	. . . . . . . . . 10 ⊢ Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑛)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛))
170	153, 169	eqeq12i 2758	. . . . . . . . 9 ⊢ (∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑛)) ↔ ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))
171	140, 170	bitri 275	. . . . . . . 8 ⊢ (∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) ↔ ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))
172	171	imbi2i 336	. . . . . . 7 ⊢ ((𝑠 ≤ 𝑆 → ∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚))) ↔ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛))))
173	133, 172	sylib 218	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))) ∧ (𝑠 + 1) ≤ 𝑆) → (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛))))
174		simpr 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))) ∧ (𝑠 + 1) ≤ 𝑆) → (𝑠 + 1) ≤ 𝑆)
175	71	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑁 ∈ ℕ0)
176	1	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑆 ∈ ℕ0)
177	83	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑍 ∈ ℂ)
178		breprexp.h	. . . . . . . 8 ⊢ (𝜑 → 𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
179	178	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
180		simpllr 775	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑠 ∈ ℕ0)
181		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → (𝑠 + 1) ≤ 𝑆)
182	2, 180	sselid 4006	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑠 ∈ ℝ)
183		1red 11291	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 1 ∈ ℝ)
184	182, 183	readdcld 11319	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → (𝑠 + 1) ∈ ℝ)
185	2, 176	sselid 4006	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑆 ∈ ℝ)
186	182	ltp1d 12225	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑠 < (𝑠 + 1))
187	182, 184, 186	ltled 11438	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑠 ≤ (𝑠 + 1))
188	182, 184, 185, 187, 181	letrd 11447	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑠 ≤ 𝑆)
189		simplr 768	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛))))
190	189, 172	sylibr 234	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → (𝑠 ≤ 𝑆 → ∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚))))
191	188, 190	mpd 15	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → ∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
192	175, 176, 177, 179, 180, 181, 191	breprexplemc 34609	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑖 ∈ (0..^𝑠)Σ𝑗 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → ∏𝑎 ∈ (0..^(𝑠 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...((𝑠 + 1) · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘(𝑠 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
193	132, 173, 174, 192	syl21anc 837	. . . . 5 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))) ∧ (𝑠 + 1) ≤ 𝑆) → ∏𝑎 ∈ (0..^(𝑠 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...((𝑠 + 1) · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘(𝑠 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
194	193	ex 412	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ (𝑠 ≤ 𝑆 → ∏𝑎 ∈ (0..^𝑠)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))) → ((𝑠 + 1) ≤ 𝑆 → ∏𝑎 ∈ (0..^(𝑠 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...((𝑠 + 1) · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘(𝑠 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚))))
195	21, 36, 51, 66, 131, 194	nn0indd 12740	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ ℕ0) → (𝑆 ≤ 𝑆 → ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚))))
196	6, 195	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑆 ∈ ℕ0) → ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))
197	1, 196	mpdan 686	1 ⊢ (𝜑 → ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  ifcif 4548  {csn 4648   class class class wbr 5166  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884  Fincfn 9003  ℂcc 11182  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187   · cmul 11189   ≤ cle 11325  ℕcn 12293  ℕ₀cn0 12553  ...cfz 13567  ..^cfzo 13711  ↑cexp 14112  Σcsu 15734  ∏cprod 15951  reprcrepr 34585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-disj 5134  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-ico 13413  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735  df-prod 15952  df-repr 34586

This theorem is referenced by:  breprexpnat  34611  vtsprod  34616