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Theorem breprexp 32613
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 32614 for the simple case presented in the proposition of [Nathanson] p. 123. (Contributed by Thierry Arnoux, 6-Dec-2021.)
Hypotheses
Ref Expression
breprexp.n (𝜑𝑁 ∈ ℕ0)
breprexp.s (𝜑𝑆 ∈ ℕ0)
breprexp.z (𝜑𝑍 ∈ ℂ)
breprexp.h (𝜑𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
Assertion
Ref Expression
breprexp (𝜑 → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
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.
StepHypRef Expression
1 breprexp.s . 2 (𝜑𝑆 ∈ ℕ0)
2 nn0ssre 12237 . . . . . 6 0 ⊆ ℝ
32a1i 11 . . . . 5 (𝜑 → ℕ0 ⊆ ℝ)
43sselda 3921 . . . 4 ((𝜑𝑆 ∈ ℕ0) → 𝑆 ∈ ℝ)
5 leid 11071 . . . 4 (𝑆 ∈ ℝ → 𝑆𝑆)
64, 5syl 17 . . 3 ((𝜑𝑆 ∈ ℕ0) → 𝑆𝑆)
7 breq1 5077 . . . . 5 (𝑡 = 0 → (𝑡𝑆 ↔ 0 ≤ 𝑆))
8 oveq2 7283 . . . . . . 7 (𝑡 = 0 → (0..^𝑡) = (0..^0))
98prodeq1d 15631 . . . . . 6 (𝑡 = 0 → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)))
10 oveq1 7282 . . . . . . . 8 (𝑡 = 0 → (𝑡 · 𝑁) = (0 · 𝑁))
1110oveq2d 7291 . . . . . . 7 (𝑡 = 0 → (0...(𝑡 · 𝑁)) = (0...(0 · 𝑁)))
12 fveq2 6774 . . . . . . . . . 10 (𝑡 = 0 → (repr‘𝑡) = (repr‘0))
1312oveqd 7292 . . . . . . . . 9 (𝑡 = 0 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘0)𝑚))
148prodeq1d 15631 . . . . . . . . . . 11 (𝑡 = 0 → ∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)))
1514oveq1d 7290 . . . . . . . . . 10 (𝑡 = 0 → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
1615adantr 481 . . . . . . . . 9 ((𝑡 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)) → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
1713, 16sumeq12dv 15418 . . . . . . . 8 (𝑡 = 0 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
1817adantr 481 . . . . . . 7 ((𝑡 = 0 ∧ 𝑚 ∈ (0...(𝑡 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
1911, 18sumeq12dv 15418 . . . . . 6 (𝑡 = 0 → Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑚 ∈ (0...(0 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
209, 19eqeq12d 2754 . . . . 5 (𝑡 = 0 → (∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) ↔ ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(0 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚))))
217, 20imbi12d 345 . . . 4 (𝑡 = 0 → ((𝑡𝑆 → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚))) ↔ (0 ≤ 𝑆 → ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(0 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))))
22 breq1 5077 . . . . 5 (𝑡 = 𝑠 → (𝑡𝑆𝑠𝑆))
23 oveq2 7283 . . . . . . 7 (𝑡 = 𝑠 → (0..^𝑡) = (0..^𝑠))
2423prodeq1d 15631 . . . . . 6 (𝑡 = 𝑠 → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)))
25 oveq1 7282 . . . . . . . 8 (𝑡 = 𝑠 → (𝑡 · 𝑁) = (𝑠 · 𝑁))
2625oveq2d 7291 . . . . . . 7 (𝑡 = 𝑠 → (0...(𝑡 · 𝑁)) = (0...(𝑠 · 𝑁)))
27 fveq2 6774 . . . . . . . . . 10 (𝑡 = 𝑠 → (repr‘𝑡) = (repr‘𝑠))
2827oveqd 7292 . . . . . . . . 9 (𝑡 = 𝑠 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘𝑠)𝑚))
2923prodeq1d 15631 . . . . . . . . . . 11 (𝑡 = 𝑠 → ∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)))
3029oveq1d 7290 . . . . . . . . . 10 (𝑡 = 𝑠 → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
3130adantr 481 . . . . . . . . 9 ((𝑡 = 𝑠𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)) → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
3228, 31sumeq12dv 15418 . . . . . . . 8 (𝑡 = 𝑠 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
3332adantr 481 . . . . . . 7 ((𝑡 = 𝑠𝑚 ∈ (0...(𝑡 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
3426, 33sumeq12dv 15418 . . . . . 6 (𝑡 = 𝑠 → Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
3524, 34eqeq12d 2754 . . . . 5 (𝑡 = 𝑠 → (∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) ↔ ∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚))))
3622, 35imbi12d 345 . . . 4 (𝑡 = 𝑠 → ((𝑡𝑆 → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚))) ↔ (𝑠𝑆 → ∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))))
37 breq1 5077 . . . . 5 (𝑡 = (𝑠 + 1) → (𝑡𝑆 ↔ (𝑠 + 1) ≤ 𝑆))
38 oveq2 7283 . . . . . . 7 (𝑡 = (𝑠 + 1) → (0..^𝑡) = (0..^(𝑠 + 1)))
3938prodeq1d 15631 . . . . . 6 (𝑡 = (𝑠 + 1) → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^(𝑠 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)))
40 oveq1 7282 . . . . . . . 8 (𝑡 = (𝑠 + 1) → (𝑡 · 𝑁) = ((𝑠 + 1) · 𝑁))
4140oveq2d 7291 . . . . . . 7 (𝑡 = (𝑠 + 1) → (0...(𝑡 · 𝑁)) = (0...((𝑠 + 1) · 𝑁)))
42 fveq2 6774 . . . . . . . . . 10 (𝑡 = (𝑠 + 1) → (repr‘𝑡) = (repr‘(𝑠 + 1)))
4342oveqd 7292 . . . . . . . . 9 (𝑡 = (𝑠 + 1) → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘(𝑠 + 1))𝑚))
4438prodeq1d 15631 . . . . . . . . . . 11 (𝑡 = (𝑠 + 1) → ∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿𝑎)‘(𝑐𝑎)))
4544oveq1d 7290 . . . . . . . . . 10 (𝑡 = (𝑠 + 1) → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
4645adantr 481 . . . . . . . . 9 ((𝑡 = (𝑠 + 1) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)) → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
4743, 46sumeq12dv 15418 . . . . . . . 8 (𝑡 = (𝑠 + 1) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘(𝑠 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
4847adantr 481 . . . . . . 7 ((𝑡 = (𝑠 + 1) ∧ 𝑚 ∈ (0...(𝑡 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘(𝑠 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
4941, 48sumeq12dv 15418 . . . . . 6 (𝑡 = (𝑠 + 1) → Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑚 ∈ (0...((𝑠 + 1) · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘(𝑠 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
5039, 49eqeq12d 2754 . . . . 5 (𝑡 = (𝑠 + 1) → (∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) ↔ ∏𝑎 ∈ (0..^(𝑠 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...((𝑠 + 1) · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘(𝑠 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚))))
5137, 50imbi12d 345 . . . 4 (𝑡 = (𝑠 + 1) → ((𝑡𝑆 → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚))) ↔ ((𝑠 + 1) ≤ 𝑆 → ∏𝑎 ∈ (0..^(𝑠 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...((𝑠 + 1) · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘(𝑠 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))))
52 breq1 5077 . . . . 5 (𝑡 = 𝑆 → (𝑡𝑆𝑆𝑆))
53 oveq2 7283 . . . . . . 7 (𝑡 = 𝑆 → (0..^𝑡) = (0..^𝑆))
5453prodeq1d 15631 . . . . . 6 (𝑡 = 𝑆 → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)))
55 oveq1 7282 . . . . . . . 8 (𝑡 = 𝑆 → (𝑡 · 𝑁) = (𝑆 · 𝑁))
5655oveq2d 7291 . . . . . . 7 (𝑡 = 𝑆 → (0...(𝑡 · 𝑁)) = (0...(𝑆 · 𝑁)))
57 fveq2 6774 . . . . . . . . . 10 (𝑡 = 𝑆 → (repr‘𝑡) = (repr‘𝑆))
5857oveqd 7292 . . . . . . . . 9 (𝑡 = 𝑆 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘𝑆)𝑚))
5953prodeq1d 15631 . . . . . . . . . . 11 (𝑡 = 𝑆 → ∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
6059oveq1d 7290 . . . . . . . . . 10 (𝑡 = 𝑆 → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
6160adantr 481 . . . . . . . . 9 ((𝑡 = 𝑆𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)) → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
6258, 61sumeq12dv 15418 . . . . . . . 8 (𝑡 = 𝑆 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
6362adantr 481 . . . . . . 7 ((𝑡 = 𝑆𝑚 ∈ (0...(𝑡 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
6456, 63sumeq12dv 15418 . . . . . 6 (𝑡 = 𝑆 → Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
6554, 64eqeq12d 2754 . . . . 5 (𝑡 = 𝑆 → (∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) ↔ ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚))))
6652, 65imbi12d 345 . . . 4 (𝑡 = 𝑆 → ((𝑡𝑆 → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑡 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)(∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚))) ↔ (𝑆𝑆 → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))))
67 0nn0 12248 . . . . . . . 8 0 ∈ ℕ0
68 fz1ssnn 13287 . . . . . . . . . . . . 13 (1...𝑁) ⊆ ℕ
6968a1i 11 . . . . . . . . . . . 12 (𝜑 → (1...𝑁) ⊆ ℕ)
70 0zd 12331 . . . . . . . . . . . 12 (𝜑 → 0 ∈ ℤ)
71 breprexp.n . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
7269, 70, 71repr0 32591 . . . . . . . . . . 11 (𝜑 → ((1...𝑁)(repr‘0)0) = if(0 = 0, {∅}, ∅))
73 eqid 2738 . . . . . . . . . . . 12 0 = 0
7473iftruei 4466 . . . . . . . . . . 11 if(0 = 0, {∅}, ∅) = {∅}
7572, 74eqtrdi 2794 . . . . . . . . . 10 (𝜑 → ((1...𝑁)(repr‘0)0) = {∅})
76 snfi 8834 . . . . . . . . . 10 {∅} ∈ Fin
7775, 76eqeltrdi 2847 . . . . . . . . 9 (𝜑 → ((1...𝑁)(repr‘0)0) ∈ Fin)
78 fzo0 13411 . . . . . . . . . . . . . . . 16 (0..^0) = ∅
7978prodeq1i 15628 . . . . . . . . . . . . . . 15 𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ ∅ ((𝐿𝑎)‘(𝑐𝑎))
80 prod0 15653 . . . . . . . . . . . . . . 15 𝑎 ∈ ∅ ((𝐿𝑎)‘(𝑐𝑎)) = 1
8179, 80eqtri 2766 . . . . . . . . . . . . . 14 𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) = 1
8281a1i 11 . . . . . . . . . . . . 13 (𝜑 → ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) = 1)
83 breprexp.z . . . . . . . . . . . . . 14 (𝜑𝑍 ∈ ℂ)
84 exp0 13786 . . . . . . . . . . . . . 14 (𝑍 ∈ ℂ → (𝑍↑0) = 1)
8583, 84syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑍↑0) = 1)
8682, 85oveq12d 7293 . . . . . . . . . . . 12 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = (1 · 1))
87 ax-1cn 10929 . . . . . . . . . . . . 13 1 ∈ ℂ
8887mulid1i 10979 . . . . . . . . . . . 12 (1 · 1) = 1
8986, 88eqtrdi 2794 . . . . . . . . . . 11 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = 1)
9089, 87eqeltrdi 2847 . . . . . . . . . 10 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) ∈ ℂ)
9190adantr 481 . . . . . . . . 9 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘0)0)) → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) ∈ ℂ)
9277, 91fsumcl 15445 . . . . . . . 8 (𝜑 → Σ𝑐 ∈ ((1...𝑁)(repr‘0)0)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) ∈ ℂ)
93 oveq2 7283 . . . . . . . . . 10 (𝑚 = 0 → ((1...𝑁)(repr‘0)𝑚) = ((1...𝑁)(repr‘0)0))
94 simpl 483 . . . . . . . . . . . 12 ((𝑚 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)) → 𝑚 = 0)
9594oveq2d 7291 . . . . . . . . . . 11 ((𝑚 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)) → (𝑍𝑚) = (𝑍↑0))
9695oveq2d 7291 . . . . . . . . . 10 ((𝑚 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)) → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)))
9793, 96sumeq12dv 15418 . . . . . . . . 9 (𝑚 = 0 → Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘0)0)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)))
9897sumsn 15458 . . . . . . . 8 ((0 ∈ ℕ0 ∧ Σ𝑐 ∈ ((1...𝑁)(repr‘0)0)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) ∈ ℂ) → Σ𝑚 ∈ {0}Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘0)0)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)))
9967, 92, 98sylancr 587 . . . . . . 7 (𝜑 → Σ𝑚 ∈ {0}Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘0)0)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)))
10075sumeq1d 15413 . . . . . . 7 (𝜑 → Σ𝑐 ∈ ((1...𝑁)(repr‘0)0)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = Σ𝑐 ∈ {∅} (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)))
101 0ex 5231 . . . . . . . . 9 ∅ ∈ V
10278prodeq1i 15628 . . . . . . . . . . . . 13 𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) = ∏𝑎 ∈ ∅ ((𝐿𝑎)‘(∅‘𝑎))
103 prod0 15653 . . . . . . . . . . . . 13 𝑎 ∈ ∅ ((𝐿𝑎)‘(∅‘𝑎)) = 1
104102, 103eqtri 2766 . . . . . . . . . . . 12 𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) = 1
105104a1i 11 . . . . . . . . . . 11 (𝜑 → ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) = 1)
106105, 87eqeltrdi 2847 . . . . . . . . . 10 (𝜑 → ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) ∈ ℂ)
10785, 87eqeltrdi 2847 . . . . . . . . . 10 (𝜑 → (𝑍↑0) ∈ ℂ)
108106, 107mulcld 10995 . . . . . . . . 9 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)) ∈ ℂ)
109 fveq1 6773 . . . . . . . . . . . . . 14 (𝑐 = ∅ → (𝑐𝑎) = (∅‘𝑎))
110109fveq2d 6778 . . . . . . . . . . . . 13 (𝑐 = ∅ → ((𝐿𝑎)‘(𝑐𝑎)) = ((𝐿𝑎)‘(∅‘𝑎)))
111110ralrimivw 3104 . . . . . . . . . . . 12 (𝑐 = ∅ → ∀𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) = ((𝐿𝑎)‘(∅‘𝑎)))
112111prodeq2d 15632 . . . . . . . . . . 11 (𝑐 = ∅ → ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)))
113112oveq1d 7290 . . . . . . . . . 10 (𝑐 = ∅ → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)))
114113sumsn 15458 . . . . . . . . 9 ((∅ ∈ V ∧ (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)) ∈ ℂ) → Σ𝑐 ∈ {∅} (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)))
115101, 108, 114sylancr 587 . . . . . . . 8 (𝜑 → Σ𝑐 ∈ {∅} (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)))
116105, 85oveq12d 7293 . . . . . . . . 9 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)) = (1 · 1))
117116, 86, 893eqtr2d 2784 . . . . . . . 8 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)) = 1)
118115, 117eqtrd 2778 . . . . . . 7 (𝜑 → Σ𝑐 ∈ {∅} (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = 1)
11999, 100, 1183eqtrd 2782 . . . . . 6 (𝜑 → Σ𝑚 ∈ {0}Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = 1)
12071nn0cnd 12295 . . . . . . . . . 10 (𝜑𝑁 ∈ ℂ)
121120mul02d 11173 . . . . . . . . 9 (𝜑 → (0 · 𝑁) = 0)
122121oveq2d 7291 . . . . . . . 8 (𝜑 → (0...(0 · 𝑁)) = (0...0))
123 fz0sn 13356 . . . . . . . 8 (0...0) = {0}
124122, 123eqtrdi 2794 . . . . . . 7 (𝜑 → (0...(0 · 𝑁)) = {0})
125124sumeq1d 15413 . . . . . 6 (𝜑 → Σ𝑚 ∈ (0...(0 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑚 ∈ {0}Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
12678prodeq1i 15628 . . . . . . . 8 𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ ∅ Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏))
127 prod0 15653 . . . . . . . 8 𝑎 ∈ ∅ Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = 1
128126, 127eqtri 2766 . . . . . . 7 𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = 1
129128a1i 11 . . . . . 6 (𝜑 → ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = 1)
130119, 125, 1293eqtr4rd 2789 . . . . 5 (𝜑 → ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(0 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
131130a1d 25 . . . 4 (𝜑 → (0 ≤ 𝑆 → ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(0 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚))))
132 simpll 764 . . . . . 6 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))) ∧ (𝑠 + 1) ≤ 𝑆) → (𝜑𝑠 ∈ ℕ0))
133 simplr 766 . . . . . . 7 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))) ∧ (𝑠 + 1) ≤ 𝑆) → (𝑠𝑆 → ∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚))))
134 oveq2 7283 . . . . . . . . . . . 12 (𝑚 = 𝑛 → ((1...𝑁)(repr‘𝑠)𝑚) = ((1...𝑁)(repr‘𝑠)𝑛))
135 oveq2 7283 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (𝑍𝑚) = (𝑍𝑛))
136135oveq2d 7291 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛)))
137136adantr 481 . . . . . . . . . . . 12 ((𝑚 = 𝑛𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)) → (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛)))
138134, 137sumeq12dv 15418 . . . . . . . . . . 11 (𝑚 = 𝑛 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛)))
139138cbvsumv 15408 . . . . . . . . . 10 Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛))
140139eqeq2i 2751 . . . . . . . . 9 (∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) ↔ ∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛)))
141 simpl 483 . . . . . . . . . . . . . . . 16 ((𝑎 = 𝑖𝑏 ∈ (1...𝑁)) → 𝑎 = 𝑖)
142141fveq2d 6778 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑖𝑏 ∈ (1...𝑁)) → (𝐿𝑎) = (𝐿𝑖))
143142fveq1d 6776 . . . . . . . . . . . . . 14 ((𝑎 = 𝑖𝑏 ∈ (1...𝑁)) → ((𝐿𝑎)‘𝑏) = ((𝐿𝑖)‘𝑏))
144143oveq1d 7290 . . . . . . . . . . . . 13 ((𝑎 = 𝑖𝑏 ∈ (1...𝑁)) → (((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = (((𝐿𝑖)‘𝑏) · (𝑍𝑏)))
145144sumeq2dv 15415 . . . . . . . . . . . 12 (𝑎 = 𝑖 → Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑏 ∈ (1...𝑁)(((𝐿𝑖)‘𝑏) · (𝑍𝑏)))
146145cbvprodv 15626 . . . . . . . . . . 11 𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑖 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑖)‘𝑏) · (𝑍𝑏))
147 fveq2 6774 . . . . . . . . . . . . . . 15 (𝑏 = 𝑗 → ((𝐿𝑖)‘𝑏) = ((𝐿𝑖)‘𝑗))
148 oveq2 7283 . . . . . . . . . . . . . . 15 (𝑏 = 𝑗 → (𝑍𝑏) = (𝑍𝑗))
149147, 148oveq12d 7293 . . . . . . . . . . . . . 14 (𝑏 = 𝑗 → (((𝐿𝑖)‘𝑏) · (𝑍𝑏)) = (((𝐿𝑖)‘𝑗) · (𝑍𝑗)))
150149cbvsumv 15408 . . . . . . . . . . . . 13 Σ𝑏 ∈ (1...𝑁)(((𝐿𝑖)‘𝑏) · (𝑍𝑏)) = Σ𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗))
151150a1i 11 . . . . . . . . . . . 12 (𝑖 ∈ (0..^𝑠) → Σ𝑏 ∈ (1...𝑁)(((𝐿𝑖)‘𝑏) · (𝑍𝑏)) = Σ𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)))
152151prodeq2i 15629 . . . . . . . . . . 11 𝑖 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑖)‘𝑏) · (𝑍𝑏)) = ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗))
153146, 152eqtri 2766 . . . . . . . . . 10 𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗))
154 fveq2 6774 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑖 → (𝐿𝑎) = (𝐿𝑖))
155 fveq2 6774 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑖 → (𝑐𝑎) = (𝑐𝑖))
156154, 155fveq12d 6781 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑖 → ((𝐿𝑎)‘(𝑐𝑎)) = ((𝐿𝑖)‘(𝑐𝑖)))
157156cbvprodv 15626 . . . . . . . . . . . . . . . 16 𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑐𝑖))
158157oveq1i 7285 . . . . . . . . . . . . . . 15 (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛)) = (∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑐𝑖)) · (𝑍𝑛))
159158a1i 11 . . . . . . . . . . . . . 14 (𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛) → (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛)) = (∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑐𝑖)) · (𝑍𝑛)))
160159sumeq2i 15411 . . . . . . . . . . . . 13 Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑐𝑖)) · (𝑍𝑛))
161 simpl 483 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝑘𝑖 ∈ (0..^𝑠)) → 𝑐 = 𝑘)
162161fveq1d 6776 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝑘𝑖 ∈ (0..^𝑠)) → (𝑐𝑖) = (𝑘𝑖))
163162fveq2d 6778 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝑘𝑖 ∈ (0..^𝑠)) → ((𝐿𝑖)‘(𝑐𝑖)) = ((𝐿𝑖)‘(𝑘𝑖)))
164163prodeq2dv 15633 . . . . . . . . . . . . . . 15 (𝑐 = 𝑘 → ∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑐𝑖)) = ∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)))
165164oveq1d 7290 . . . . . . . . . . . . . 14 (𝑐 = 𝑘 → (∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑐𝑖)) · (𝑍𝑛)) = (∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))
166165cbvsumv 15408 . . . . . . . . . . . . 13 Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑐𝑖)) · (𝑍𝑛)) = Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛))
167160, 166eqtri 2766 . . . . . . . . . . . 12 Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛)) = Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛))
168167a1i 11 . . . . . . . . . . 11 (𝑛 ∈ (0...(𝑠 · 𝑁)) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛)) = Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))
169168sumeq2i 15411 . . . . . . . . . 10 Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛))
170153, 169eqeq12i 2756 . . . . . . . . 9 (∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛)) ↔ ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))
171140, 170bitri 274 . . . . . . . 8 (∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) ↔ ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))
172171imbi2i 336 . . . . . . 7 ((𝑠𝑆 → ∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚))) ↔ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛))))
173133, 172sylib 217 . . . . . 6 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))) ∧ (𝑠 + 1) ≤ 𝑆) → (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛))))
174 simpr 485 . . . . . 6 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))) ∧ (𝑠 + 1) ≤ 𝑆) → (𝑠 + 1) ≤ 𝑆)
17571ad3antrrr 727 . . . . . . 7 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑁 ∈ ℕ0)
1761ad3antrrr 727 . . . . . . 7 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑆 ∈ ℕ0)
17783ad3antrrr 727 . . . . . . 7 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑍 ∈ ℂ)
178 breprexp.h . . . . . . . 8 (𝜑𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
179178ad3antrrr 727 . . . . . . 7 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
180 simpllr 773 . . . . . . 7 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑠 ∈ ℕ0)
181 simpr 485 . . . . . . 7 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → (𝑠 + 1) ≤ 𝑆)
1822, 180sselid 3919 . . . . . . . . 9 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑠 ∈ ℝ)
183 1red 10976 . . . . . . . . . 10 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 1 ∈ ℝ)
184182, 183readdcld 11004 . . . . . . . . 9 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → (𝑠 + 1) ∈ ℝ)
1852, 176sselid 3919 . . . . . . . . 9 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑆 ∈ ℝ)
186182ltp1d 11905 . . . . . . . . . 10 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑠 < (𝑠 + 1))
187182, 184, 186ltled 11123 . . . . . . . . 9 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑠 ≤ (𝑠 + 1))
188182, 184, 185, 187, 181letrd 11132 . . . . . . . 8 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → 𝑠𝑆)
189 simplr 766 . . . . . . . . 9 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛))))
190189, 172sylibr 233 . . . . . . . 8 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → (𝑠𝑆 → ∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚))))
191188, 190mpd 15 . . . . . . 7 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → ∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
192175, 176, 177, 179, 180, 181, 191breprexplemc 32612 . . . . . 6 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗)) = Σ𝑛 ∈ (0...(𝑠 · 𝑁))Σ𝑘 ∈ ((1...𝑁)(repr‘𝑠)𝑛)(∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))) ∧ (𝑠 + 1) ≤ 𝑆) → ∏𝑎 ∈ (0..^(𝑠 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...((𝑠 + 1) · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘(𝑠 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
193132, 173, 174, 192syl21anc 835 . . . . 5 ((((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))) ∧ (𝑠 + 1) ≤ 𝑆) → ∏𝑎 ∈ (0..^(𝑠 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...((𝑠 + 1) · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘(𝑠 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
194193ex 413 . . . 4 (((𝜑𝑠 ∈ ℕ0) ∧ (𝑠𝑆 → ∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑠 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)(∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))) → ((𝑠 + 1) ≤ 𝑆 → ∏𝑎 ∈ (0..^(𝑠 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...((𝑠 + 1) · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘(𝑠 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚))))
19521, 36, 51, 66, 131, 194nn0indd 12417 . . 3 ((𝜑𝑆 ∈ ℕ0) → (𝑆𝑆 → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚))))
1966, 195mpd 15 . 2 ((𝜑𝑆 ∈ ℕ0) → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
1971, 196mpdan 684 1 (𝜑 → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  Vcvv 3432  wss 3887  c0 4256  ifcif 4459  {csn 4561   class class class wbr 5074  wf 6429  cfv 6433  (class class class)co 7275  m cmap 8615  Fincfn 8733  cc 10869  cr 10870  0cc0 10871  1c1 10872   + caddc 10874   · cmul 10876  cle 11010  cn 11973  0cn0 12233  ...cfz 13239  ..^cfzo 13382  cexp 13782  Σcsu 15397  cprod 15615  reprcrepr 32588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-disj 5040  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-ico 13085  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-sum 15398  df-prod 15616  df-repr 32589
This theorem is referenced by:  breprexpnat  32614  vtsprod  32619
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