Theorem breprexp 34624

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 34625 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 12446	. . . . . 6 ⊢ ℕ0 ⊆ ℝ
3	2	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ0 ⊆ ℝ)
4	3	sselda 3946	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ ℕ0) → 𝑆 ∈ ℝ)
5		leid 11270	. . . 4 ⊢ (𝑆 ∈ ℝ → 𝑆 ≤ 𝑆)
6	4, 5	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ ℕ0) → 𝑆 ≤ 𝑆)
7		breq1 5110	. . . . 5 ⊢ (𝑡 = 0 → (𝑡 ≤ 𝑆 ↔ 0 ≤ 𝑆))
8		oveq2 7395	. . . . . . 7 ⊢ (𝑡 = 0 → (0..^𝑡) = (0..^0))
9	8	prodeq1d 15886	. . . . . 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 7394	. . . . . . . 8 ⊢ (𝑡 = 0 → (𝑡 · 𝑁) = (0 · 𝑁))
11	10	oveq2d 7403	. . . . . . 7 ⊢ (𝑡 = 0 → (0...(𝑡 · 𝑁)) = (0...(0 · 𝑁)))
12		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑡 = 0 → (repr‘𝑡) = (repr‘0))
13	12	oveqd 7404	. . . . . . . . 9 ⊢ (𝑡 = 0 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘0)𝑚))
14	8	prodeq1d 15886	. . . . . . . . . . 11 ⊢ (𝑡 = 0 → ∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)))
15	14	oveq1d 7402	. . . . . . . . . 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 15672	. . . . . . . 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 15672	. . . . . 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 2745	. . . . 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 5110	. . . . 5 ⊢ (𝑡 = 𝑠 → (𝑡 ≤ 𝑆 ↔ 𝑠 ≤ 𝑆))
23		oveq2 7395	. . . . . . 7 ⊢ (𝑡 = 𝑠 → (0..^𝑡) = (0..^𝑠))
24	23	prodeq1d 15886	. . . . . 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 7394	. . . . . . . 8 ⊢ (𝑡 = 𝑠 → (𝑡 · 𝑁) = (𝑠 · 𝑁))
26	25	oveq2d 7403	. . . . . . 7 ⊢ (𝑡 = 𝑠 → (0...(𝑡 · 𝑁)) = (0...(𝑠 · 𝑁)))
27		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑡 = 𝑠 → (repr‘𝑡) = (repr‘𝑠))
28	27	oveqd 7404	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘𝑠)𝑚))
29	23	prodeq1d 15886	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑠 → ∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)))
30	29	oveq1d 7402	. . . . . . . . . 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 15672	. . . . . . . 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 15672	. . . . . 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 2745	. . . . 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 5110	. . . . 5 ⊢ (𝑡 = (𝑠 + 1) → (𝑡 ≤ 𝑆 ↔ (𝑠 + 1) ≤ 𝑆))
38		oveq2 7395	. . . . . . 7 ⊢ (𝑡 = (𝑠 + 1) → (0..^𝑡) = (0..^(𝑠 + 1)))
39	38	prodeq1d 15886	. . . . . 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 7394	. . . . . . . 8 ⊢ (𝑡 = (𝑠 + 1) → (𝑡 · 𝑁) = ((𝑠 + 1) · 𝑁))
41	40	oveq2d 7403	. . . . . . 7 ⊢ (𝑡 = (𝑠 + 1) → (0...(𝑡 · 𝑁)) = (0...((𝑠 + 1) · 𝑁)))
42		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑡 = (𝑠 + 1) → (repr‘𝑡) = (repr‘(𝑠 + 1)))
43	42	oveqd 7404	. . . . . . . . 9 ⊢ (𝑡 = (𝑠 + 1) → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘(𝑠 + 1))𝑚))
44	38	prodeq1d 15886	. . . . . . . . . . 11 ⊢ (𝑡 = (𝑠 + 1) → ∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿‘𝑎)‘(𝑐‘𝑎)))
45	44	oveq1d 7402	. . . . . . . . . 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 15672	. . . . . . . 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 15672	. . . . . 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 2745	. . . . 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 5110	. . . . 5 ⊢ (𝑡 = 𝑆 → (𝑡 ≤ 𝑆 ↔ 𝑆 ≤ 𝑆))
53		oveq2 7395	. . . . . . 7 ⊢ (𝑡 = 𝑆 → (0..^𝑡) = (0..^𝑆))
54	53	prodeq1d 15886	. . . . . 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 7394	. . . . . . . 8 ⊢ (𝑡 = 𝑆 → (𝑡 · 𝑁) = (𝑆 · 𝑁))
56	55	oveq2d 7403	. . . . . . 7 ⊢ (𝑡 = 𝑆 → (0...(𝑡 · 𝑁)) = (0...(𝑆 · 𝑁)))
57		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑡 = 𝑆 → (repr‘𝑡) = (repr‘𝑆))
58	57	oveqd 7404	. . . . . . . . 9 ⊢ (𝑡 = 𝑆 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘𝑆)𝑚))
59	53	prodeq1d 15886	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑆 → ∏𝑎 ∈ (0..^𝑡)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)))
60	59	oveq1d 7402	. . . . . . . . . 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 15672	. . . . . . . 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 15672	. . . . . 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 2745	. . . . 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 12457	. . . . . . . 8 ⊢ 0 ∈ ℕ0
68		fz1ssnn 13516	. . . . . . . . . . . . 13 ⊢ (1...𝑁) ⊆ ℕ
69	68	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝑁) ⊆ ℕ)
70		0zd 12541	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ ℤ)
71		breprexp.n	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
72	69, 70, 71	repr0 34602	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...𝑁)(repr‘0)0) = if(0 = 0, {∅}, ∅))
73		eqid 2729	. . . . . . . . . . . 12 ⊢ 0 = 0
74	73	iftruei 4495	. . . . . . . . . . 11 ⊢ if(0 = 0, {∅}, ∅) = {∅}
75	72, 74	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝜑 → ((1...𝑁)(repr‘0)0) = {∅})
76		snfi 9014	. . . . . . . . . 10 ⊢ {∅} ∈ Fin
77	75, 76	eqeltrdi 2836	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝑁)(repr‘0)0) ∈ Fin)
78		fzo0 13644	. . . . . . . . . . . . . . . 16 ⊢ (0..^0) = ∅
79	78	prodeq1i 15882	. . . . . . . . . . . . . . 15 ⊢ ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ ∅ ((𝐿‘𝑎)‘(𝑐‘𝑎))
80		prod0 15909	. . . . . . . . . . . . . . 15 ⊢ ∏𝑎 ∈ ∅ ((𝐿‘𝑎)‘(𝑐‘𝑎)) = 1
81	79, 80	eqtri 2752	. . . . . . . . . . . . . 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 14030	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ ℂ → (𝑍↑0) = 1)
85	83, 84	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑍↑0) = 1)
86	82, 85	oveq12d 7405	. . . . . . . . . . . 12 ⊢ (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) = (1 · 1))
87		ax-1cn 11126	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
88	87	mulridi 11178	. . . . . . . . . . . 12 ⊢ (1 · 1) = 1
89	86, 88	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) = 1)
90	89, 87	eqeltrdi 2836	. . . . . . . . . 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 15699	. . . . . . . 8 ⊢ (𝜑 → Σ𝑐 ∈ ((1...𝑁)(repr‘0)0)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) ∈ ℂ)
93		oveq2 7395	. . . . . . . . . 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 7403	. . . . . . . . . . 11 ⊢ ((𝑚 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)) → (𝑍↑𝑚) = (𝑍↑0))
96	95	oveq2d 7403	. . . . . . . . . 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 15672	. . . . . . . . 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 15712	. . . . . . . 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 15666	. . . . . . 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 5262	. . . . . . . . 9 ⊢ ∅ ∈ V
102	78	prodeq1i 15882	. . . . . . . . . . . . 13 ⊢ ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) = ∏𝑎 ∈ ∅ ((𝐿‘𝑎)‘(∅‘𝑎))
103		prod0 15909	. . . . . . . . . . . . 13 ⊢ ∏𝑎 ∈ ∅ ((𝐿‘𝑎)‘(∅‘𝑎)) = 1
104	102, 103	eqtri 2752	. . . . . . . . . . . 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 2836	. . . . . . . . . 10 ⊢ (𝜑 → ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) ∈ ℂ)
107	85, 87	eqeltrdi 2836	. . . . . . . . . 10 ⊢ (𝜑 → (𝑍↑0) ∈ ℂ)
108	106, 107	mulcld 11194	. . . . . . . . 9 ⊢ (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) · (𝑍↑0)) ∈ ℂ)
109		fveq1 6857	. . . . . . . . . . . . . 14 ⊢ (𝑐 = ∅ → (𝑐‘𝑎) = (∅‘𝑎))
110	109	fveq2d 6862	. . . . . . . . . . . . 13 ⊢ (𝑐 = ∅ → ((𝐿‘𝑎)‘(𝑐‘𝑎)) = ((𝐿‘𝑎)‘(∅‘𝑎)))
111	110	ralrimivw 3129	. . . . . . . . . . . 12 ⊢ (𝑐 = ∅ → ∀𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ((𝐿‘𝑎)‘(∅‘𝑎)))
112	111	prodeq2d 15887	. . . . . . . . . . 11 ⊢ (𝑐 = ∅ → ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)))
113	112	oveq1d 7402	. . . . . . . . . 10 ⊢ (𝑐 = ∅ → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) = (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) · (𝑍↑0)))
114	113	sumsn 15712	. . . . . . . . 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 7405	. . . . . . . . 9 ⊢ (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) · (𝑍↑0)) = (1 · 1))
117	116, 86, 89	3eqtr2d 2770	. . . . . . . 8 ⊢ (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(∅‘𝑎)) · (𝑍↑0)) = 1)
118	115, 117	eqtrd 2764	. . . . . . 7 ⊢ (𝜑 → Σ𝑐 ∈ {∅} (∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑0)) = 1)
119	99, 100, 118	3eqtrd 2768	. . . . . 6 ⊢ (𝜑 → Σ𝑚 ∈ {0}Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚)) = 1)
120	71	nn0cnd 12505	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ)
121	120	mul02d 11372	. . . . . . . . 9 ⊢ (𝜑 → (0 · 𝑁) = 0)
122	121	oveq2d 7403	. . . . . . . 8 ⊢ (𝜑 → (0...(0 · 𝑁)) = (0...0))
123		fz0sn 13588	. . . . . . . 8 ⊢ (0...0) = {0}
124	122, 123	eqtrdi 2780	. . . . . . 7 ⊢ (𝜑 → (0...(0 · 𝑁)) = {0})
125	124	sumeq1d 15666	. . . . . 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 15882	. . . . . . . 8 ⊢ ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = ∏𝑎 ∈ ∅ Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏))
127		prod0 15909	. . . . . . . 8 ⊢ ∏𝑎 ∈ ∅ Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = 1
128	126, 127	eqtri 2752	. . . . . . 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 2775	. . . . 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 7395	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((1...𝑁)(repr‘𝑠)𝑚) = ((1...𝑁)(repr‘𝑠)𝑛))
135		oveq2 7395	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑍↑𝑚) = (𝑍↑𝑛))
136	135	oveq2d 7403	. . . . . . . . . . . . 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 15672	. . . . . . . . . . 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 15662	. . . . . . . . . 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 2742	. . . . . . . . 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 6862	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑖 ∧ 𝑏 ∈ (1...𝑁)) → (𝐿‘𝑎) = (𝐿‘𝑖))
143	142	fveq1d 6860	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑖 ∧ 𝑏 ∈ (1...𝑁)) → ((𝐿‘𝑎)‘𝑏) = ((𝐿‘𝑖)‘𝑏))
144	143	oveq1d 7402	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑖 ∧ 𝑏 ∈ (1...𝑁)) → (((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = (((𝐿‘𝑖)‘𝑏) · (𝑍↑𝑏)))
145	144	sumeq2dv 15668	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑖 → Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑖)‘𝑏) · (𝑍↑𝑏)))
146	145	cbvprodv 15880	. . . . . . . . . . 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 6858	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑗 → ((𝐿‘𝑖)‘𝑏) = ((𝐿‘𝑖)‘𝑗))
148		oveq2 7395	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑗 → (𝑍↑𝑏) = (𝑍↑𝑗))
149	147, 148	oveq12d 7405	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑗 → (((𝐿‘𝑖)‘𝑏) · (𝑍↑𝑏)) = (((𝐿‘𝑖)‘𝑗) · (𝑍↑𝑗)))
150	149	cbvsumv 15662	. . . . . . . . . . . . 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 15884	. . . . . . . . . . 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 2752	. . . . . . . . . 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 6858	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑖 → (𝐿‘𝑎) = (𝐿‘𝑖))
155		fveq2 6858	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑖 → (𝑐‘𝑎) = (𝑐‘𝑖))
156	154, 155	fveq12d 6865	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑖 → ((𝐿‘𝑎)‘(𝑐‘𝑎)) = ((𝐿‘𝑖)‘(𝑐‘𝑖)))
157	156	cbvprodv 15880	. . . . . . . . . . . . . . . 16 ⊢ ∏𝑎 ∈ (0..^𝑠)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑐‘𝑖))
158	157	oveq1i 7397	. . . . . . . . . . . . . . 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 15664	. . . . . . . . . . . . 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 6860	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 = 𝑘 ∧ 𝑖 ∈ (0..^𝑠)) → (𝑐‘𝑖) = (𝑘‘𝑖))
163	162	fveq2d 6862	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 = 𝑘 ∧ 𝑖 ∈ (0..^𝑠)) → ((𝐿‘𝑖)‘(𝑐‘𝑖)) = ((𝐿‘𝑖)‘(𝑘‘𝑖)))
164	163	prodeq2dv 15888	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑘 → ∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑐‘𝑖)) = ∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)))
165	164	oveq1d 7402	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑘 → (∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑐‘𝑖)) · (𝑍↑𝑛)) = (∏𝑖 ∈ (0..^𝑠)((𝐿‘𝑖)‘(𝑘‘𝑖)) · (𝑍↑𝑛)))
166	165	cbvsumv 15662	. . . . . . . . . . . . 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 2752	. . . . . . . . . . . 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 15664	. . . . . . . . . 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 2747	. . . . . . . . 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 3944	. . . . . . . . 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 11175	. . . . . . . . . 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 11203	. . . . . . . . 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 3944	. . . . . . . . 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 12113	. . . . . . . . . 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 11322	. . . . . . . . 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 11331	. . . . . . . 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 34623	. . . . . 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 12631	. . 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 1540   ∈ wcel 2109  Vcvv 3447   ⊆ wss 3914  ∅c0 4296  ifcif 4488  {csn 4589   class class class wbr 5107  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ↑_m cmap 8799  Fincfn 8918  ℂcc 11066  ℝcr 11067  0cc0 11068  1c1 11069   + caddc 11071   · cmul 11073   ≤ cle 11209  ℕcn 12186  ℕ₀cn0 12442  ...cfz 13468  ..^cfzo 13615  ↑cexp 14026  Σcsu 15652  ∏cprod 15869  reprcrepr 34599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-disj 5075  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-pm 8802  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-n0 12443  df-z 12530  df-uz 12794  df-rp 12952  df-ico 13312  df-fz 13469  df-fzo 13616  df-seq 13967  df-exp 14027  df-hash 14296  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-abs 15202  df-clim 15454  df-sum 15653  df-prod 15870  df-repr 34600

This theorem is referenced by:  breprexpnat  34625  vtsprod  34630