Theorem breprexp 34790

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 34791 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 12405	. . . . . 6 ⊢ ℕ0 ⊆ ℝ
3	2	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ0 ⊆ ℝ)
4	3	sselda 3933	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ ℕ0) → 𝑆 ∈ ℝ)
5		leid 11229	. . . 4 ⊢ (𝑆 ∈ ℝ → 𝑆 ≤ 𝑆)
6	4, 5	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ ℕ0) → 𝑆 ≤ 𝑆)
7		breq1 5101	. . . . 5 ⊢ (𝑡 = 0 → (𝑡 ≤ 𝑆 ↔ 0 ≤ 𝑆))
8		oveq2 7366	. . . . . . 7 ⊢ (𝑡 = 0 → (0..^𝑡) = (0..^0))
9	8	prodeq1d 15843	. . . . . 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 7365	. . . . . . . 8 ⊢ (𝑡 = 0 → (𝑡 · 𝑁) = (0 · 𝑁))
11	10	oveq2d 7374	. . . . . . 7 ⊢ (𝑡 = 0 → (0...(𝑡 · 𝑁)) = (0...(0 · 𝑁)))
12		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑡 = 0 → (repr‘𝑡) = (repr‘0))
13	12	oveqd 7375	. . . . . . . . 9 ⊢ (𝑡 = 0 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘0)𝑚))
14	8	prodeq1d 15843	. . . . . . . . . . 11 ⊢ (𝑡 = 0 → ∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)))
15	14	oveq1d 7373	. . . . . . . . . 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 15629	. . . . . . . 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 15629	. . . . . 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 2752	. . . . 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 5101	. . . . 5 ⊢ (𝑡 = 𝑠 → (𝑡 ≤ 𝑆 ↔ 𝑠 ≤ 𝑆))
23		oveq2 7366	. . . . . . 7 ⊢ (𝑡 = 𝑠 → (0..^𝑡) = (0..^𝑠))
24	23	prodeq1d 15843	. . . . . 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 7365	. . . . . . . 8 ⊢ (𝑡 = 𝑠 → (𝑡 · 𝑁) = (𝑠 · 𝑁))
26	25	oveq2d 7374	. . . . . . 7 ⊢ (𝑡 = 𝑠 → (0...(𝑡 · 𝑁)) = (0...(𝑠 · 𝑁)))
27		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑡 = 𝑠 → (repr‘𝑡) = (repr‘𝑠))
28	27	oveqd 7375	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘𝑠)𝑚))
29	23	prodeq1d 15843	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑠 → ∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)))
30	29	oveq1d 7373	. . . . . . . . . 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 15629	. . . . . . . 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 15629	. . . . . 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 2752	. . . . 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 5101	. . . . 5 ⊢ (𝑡 = (𝑠 + 1) → (𝑡 ≤ 𝑆 ↔ (𝑠 + 1) ≤ 𝑆))
38		oveq2 7366	. . . . . . 7 ⊢ (𝑡 = (𝑠 + 1) → (0..^𝑡) = (0..^(𝑠 + 1)))
39	38	prodeq1d 15843	. . . . . 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 7365	. . . . . . . 8 ⊢ (𝑡 = (𝑠 + 1) → (𝑡 · 𝑁) = ((𝑠 + 1) · 𝑁))
41	40	oveq2d 7374	. . . . . . 7 ⊢ (𝑡 = (𝑠 + 1) → (0...(𝑡 · 𝑁)) = (0...((𝑠 + 1) · 𝑁)))
42		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑡 = (𝑠 + 1) → (repr‘𝑡) = (repr‘(𝑠 + 1)))
43	42	oveqd 7375	. . . . . . . . 9 ⊢ (𝑡 = (𝑠 + 1) → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘(𝑠 + 1))𝑚))
44	38	prodeq1d 15843	. . . . . . . . . . 11 ⊢ (𝑡 = (𝑠 + 1) → ∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿‘𝑎)‘(𝑐‘𝑎)))
45	44	oveq1d 7373	. . . . . . . . . 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 15629	. . . . . . . 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 15629	. . . . . 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 2752	. . . . 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 5101	. . . . 5 ⊢ (𝑡 = 𝑆 → (𝑡 ≤ 𝑆 ↔ 𝑆 ≤ 𝑆))
53		oveq2 7366	. . . . . . 7 ⊢ (𝑡 = 𝑆 → (0..^𝑡) = (0..^𝑆))
54	53	prodeq1d 15843	. . . . . 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 7365	. . . . . . . 8 ⊢ (𝑡 = 𝑆 → (𝑡 · 𝑁) = (𝑆 · 𝑁))
56	55	oveq2d 7374	. . . . . . 7 ⊢ (𝑡 = 𝑆 → (0...(𝑡 · 𝑁)) = (0...(𝑆 · 𝑁)))
57		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑡 = 𝑆 → (repr‘𝑡) = (repr‘𝑆))
58	57	oveqd 7375	. . . . . . . . 9 ⊢ (𝑡 = 𝑆 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘𝑆)𝑚))
59	53	prodeq1d 15843	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑆 → ∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)))
60	59	oveq1d 7373	. . . . . . . . . 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 15629	. . . . . . . 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 15629	. . . . . 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 2752	. . . . 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 12416	. . . . . . . 8 ⊢ 0 ∈ ℕ0
68		fz1ssnn 13471	. . . . . . . . . . . . 13 ⊢ (1...𝑁) ⊆ ℕ
69	68	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝑁) ⊆ ℕ)
70		0zd 12500	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℤ)
71		breprexp.n	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
72	69, 70, 71	repr0 34768	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...𝑁)(repr‘0)0) = if(0 = 0, {∅}, ∅))
73		eqid 2736	. . . . . . . . . . . 12 ⊢ 0 = 0
74	73	iftruei 4486	. . . . . . . . . . 11 ⊢ if(0 = 0, {∅}, ∅) = {∅}
75	72, 74	eqtrdi 2787	. . . . . . . . . 10 ⊢ (𝜑 → ((1...𝑁)(repr‘0)0) = {∅})
76		snfi 8980	. . . . . . . . . 10 ⊢ {∅} ∈ Fin
77	75, 76	eqeltrdi 2844	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝑁)(repr‘0)0) ∈ Fin)
78		fzo0 13599	. . . . . . . . . . . . . . . 16 ⊢ (0..^0) = ∅
79	78	prodeq1i 15839	. . . . . . . . . . . . . . 15 ⊢ ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ ∅ ((𝐿‘𝑎)‘(𝑐‘𝑎))
80		prod0 15866	. . . . . . . . . . . . . . 15 ⊢ ∏𝑎 ∈ ∅ ((𝐿‘𝑎)‘(𝑐‘𝑎)) = 1
81	79, 80	eqtri 2759	. . . . . . . . . . . . . 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 13988	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ ℂ → (𝑍↑0) = 1)
85	83, 84	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍↑0) = 1)
86	82, 85	oveq12d 7376	. . . . . . . . . . . 12 ⊢ (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) = (1 · 1))
87		ax-1cn 11084	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
88	87	mulridi 11136	. . . . . . . . . . . 12 ⊢ (1 · 1) = 1
89	86, 88	eqtrdi 2787	. . . . . . . . . . 11 ⊢ (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) = 1)
90	89, 87	eqeltrdi 2844	. . . . . . . . . 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 15656	. . . . . . . 8 ⊢ (𝜑 → Σ𝑐 ∈ ((1...𝑁)(repr‘0)0)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) ∈ ℂ)
93		oveq2 7366	. . . . . . . . . 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 7374	. . . . . . . . . . 11 ⊢ ((𝑚 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)) → (𝑍↑𝑚) = (𝑍↑0))
96	95	oveq2d 7374	. . . . . . . . . 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 15629	. . . . . . . . 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 15669	. . . . . . . 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 587	. . . . . . 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 15623	. . . . . . 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 5252	. . . . . . . . 9 ⊢ ∅ ∈ V
102	78	prodeq1i 15839	. . . . . . . . . . . . 13 ⊢ ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) = ∏𝑎 ∈ ∅ ((𝐿‘𝑎)‘(∅‘𝑎))
103		prod0 15866	. . . . . . . . . . . . 13 ⊢ ∏𝑎 ∈ ∅ ((𝐿‘𝑎)‘(∅‘𝑎)) = 1
104	102, 103	eqtri 2759	. . . . . . . . . . . 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 2844	. . . . . . . . . 10 ⊢ (𝜑 → ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) ∈ ℂ)
107	85, 87	eqeltrdi 2844	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍↑0) ∈ ℂ)
108	106, 107	mulcld 11152	. . . . . . . . 9 ⊢ (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) · (𝑍↑0)) ∈ ℂ)
109		fveq1 6833	. . . . . . . . . . . . . 14 ⊢ (𝑐 = ∅ → (𝑐‘𝑎) = (∅‘𝑎))
110	109	fveq2d 6838	. . . . . . . . . . . . 13 ⊢ (𝑐 = ∅ → ((𝐿‘𝑎)‘(𝑐‘𝑎)) = ((𝐿‘𝑎)‘(∅‘𝑎)))
111	110	ralrimivw 3132	. . . . . . . . . . . 12 ⊢ (𝑐 = ∅ → ∀𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ((𝐿‘𝑎)‘(∅‘𝑎)))
112	111	prodeq2d 15844	. . . . . . . . . . 11 ⊢ (𝑐 = ∅ → ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)))
113	112	oveq1d 7373	. . . . . . . . . 10 ⊢ (𝑐 = ∅ → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) = (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) · (𝑍↑0)))
114	113	sumsn 15669	. . . . . . . . 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 587	. . . . . . . 8 ⊢ (𝜑 → Σ𝑐 ∈ {∅} (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) = (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) · (𝑍↑0)))
116	105, 85	oveq12d 7376	. . . . . . . . 9 ⊢ (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) · (𝑍↑0)) = (1 · 1))
117	116, 86, 89	3eqtr2d 2777	. . . . . . . 8 ⊢ (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) · (𝑍↑0)) = 1)
118	115, 117	eqtrd 2771	. . . . . . 7 ⊢ (𝜑 → Σ𝑐 ∈ {∅} (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) = 1)
119	99, 100, 118	3eqtrd 2775	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ {0}Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = 1)
120	71	nn0cnd 12464	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ)
121	120	mul02d 11331	. . . . . . . . 9 ⊢ (𝜑 → (0 · 𝑁) = 0)
122	121	oveq2d 7374	. . . . . . . 8 ⊢ (𝜑 → (0...(0 · 𝑁)) = (0...0))
123		fz0sn 13543	. . . . . . . 8 ⊢ (0...0) = {0}
124	122, 123	eqtrdi 2787	. . . . . . 7 ⊢ (𝜑 → (0...(0 · 𝑁)) = {0})
125	124	sumeq1d 15623	. . . . . 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 15839	. . . . . . . 8 ⊢ ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = ∏𝑎 ∈ ∅ Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏))
127		prod0 15866	. . . . . . . 8 ⊢ ∏𝑎 ∈ ∅ Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = 1
128	126, 127	eqtri 2759	. . . . . . 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 2782	. . . . 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 7366	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((1...𝑁)(repr‘𝑠)𝑚) = ((1...𝑁)(repr‘𝑠)𝑛))
135		oveq2 7366	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑍↑𝑚) = (𝑍↑𝑛))
136	135	oveq2d 7374	. . . . . . . . . . . . 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 15629	. . . . . . . . . . 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 15619	. . . . . . . . . 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 2749	. . . . . . . . 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 6838	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑖 ∧ 𝑏 ∈ (1...𝑁)) → (𝐿‘𝑎) = (𝐿‘𝑖))
143	142	fveq1d 6836	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑖 ∧ 𝑏 ∈ (1...𝑁)) → ((𝐿‘𝑎)‘𝑏) = ((𝐿‘𝑖)‘𝑏))
144	143	oveq1d 7373	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑖 ∧ 𝑏 ∈ (1...𝑁)) → (((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = (((𝐿‘𝑖)‘𝑏) · (𝑍↑𝑏)))
145	144	sumeq2dv 15625	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑖 → Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑏) · (𝑍↑𝑏)))
146	145	cbvprodv 15837	. . . . . . . . . . 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 6834	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑗 → ((𝐿‘𝑖)‘𝑏) = ((𝐿‘𝑖)‘𝑗))
148		oveq2 7366	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑗 → (𝑍↑𝑏) = (𝑍↑𝑗))
149	147, 148	oveq12d 7376	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑗 → (((𝐿‘𝑖)‘𝑏) · (𝑍↑𝑏)) = (((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)))
150	149	cbvsumv 15619	. . . . . . . . . . . . 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 15841	. . . . . . . . . . 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 2759	. . . . . . . . . 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 6834	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑖 → (𝐿‘𝑎) = (𝐿‘𝑖))
155		fveq2 6834	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑖 → (𝑐‘𝑎) = (𝑐‘𝑖))
156	154, 155	fveq12d 6841	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑖 → ((𝐿‘𝑎)‘(𝑐‘𝑎)) = ((𝐿‘𝑖)‘(𝑐‘𝑖)))
157	156	cbvprodv 15837	. . . . . . . . . . . . . . . 16 ⊢ ∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑐‘𝑖))
158	157	oveq1i 7368	. . . . . . . . . . . . . . 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 15621	. . . . . . . . . . . . 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 6836	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 = 𝑘 ∧ 𝑖 ∈ (0..^𝑠)) → (𝑐‘𝑖) = (𝑘‘𝑖))
163	162	fveq2d 6838	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 = 𝑘 ∧ 𝑖 ∈ (0..^𝑠)) → ((𝐿‘𝑖)‘(𝑐‘𝑖)) = ((𝐿‘𝑖)‘(𝑘‘𝑖)))
164	163	prodeq2dv 15845	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑘 → ∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑐‘𝑖)) = ∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)))
165	164	oveq1d 7373	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑘 → (∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑐‘𝑖)) · (𝑍↑𝑛)) = (∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))
166	165	cbvsumv 15619	. . . . . . . . . . . . 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 2759	. . . . . . . . . . . 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 15621	. . . . . . . . . 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 2754	. . . . . . . . 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 730	. . . . . . 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 730	. . . . . . 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 730	. . . . . . 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 730	. . . . . . 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 3931	. . . . . . . . 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 11133	. . . . . . . . . 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 11161	. . . . . . . . 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 3931	. . . . . . . . 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 12072	. . . . . . . . . 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 11281	. . . . . . . . 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 11290	. . . . . . . 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 34789	. . . . . 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 12589	. . 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 687	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 1541   ∈ wcel 2113  Vcvv 3440   ⊆ wss 3901  ∅c0 4285  ifcif 4479  {csn 4580   class class class wbr 5098  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763  Fincfn 8883  ℂcc 11024  ℝcr 11025  0cc0 11026  1c1 11027   + caddc 11029   · cmul 11031   ≤ cle 11167  ℕcn 12145  ℕ₀cn0 12401  ...cfz 13423  ..^cfzo 13570  ↑cexp 13984  Σcsu 15609  ∏cprod 15826  reprcrepr 34765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-disj 5066  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-n0 12402  df-z 12489  df-uz 12752  df-rp 12906  df-ico 13267  df-fz 13424  df-fzo 13571  df-seq 13925  df-exp 13985  df-hash 14254  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-clim 15411  df-sum 15610  df-prod 15827  df-repr 34766

This theorem is referenced by:  breprexpnat  34791  vtsprod  34796