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Theorem breprexp 33583
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 33584 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 12472 . . . . . 6 0 ⊆ ℝ
32a1i 11 . . . . 5 (𝜑 → ℕ0 ⊆ ℝ)
43sselda 3981 . . . 4 ((𝜑𝑆 ∈ ℕ0) → 𝑆 ∈ ℝ)
5 leid 11306 . . . 4 (𝑆 ∈ ℝ → 𝑆𝑆)
64, 5syl 17 . . 3 ((𝜑𝑆 ∈ ℕ0) → 𝑆𝑆)
7 breq1 5150 . . . . 5 (𝑡 = 0 → (𝑡𝑆 ↔ 0 ≤ 𝑆))
8 oveq2 7412 . . . . . . 7 (𝑡 = 0 → (0..^𝑡) = (0..^0))
98prodeq1d 15861 . . . . . 6 (𝑡 = 0 → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)))
10 oveq1 7411 . . . . . . . 8 (𝑡 = 0 → (𝑡 · 𝑁) = (0 · 𝑁))
1110oveq2d 7420 . . . . . . 7 (𝑡 = 0 → (0...(𝑡 · 𝑁)) = (0...(0 · 𝑁)))
12 fveq2 6888 . . . . . . . . . 10 (𝑡 = 0 → (repr‘𝑡) = (repr‘0))
1312oveqd 7421 . . . . . . . . 9 (𝑡 = 0 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘0)𝑚))
148prodeq1d 15861 . . . . . . . . . . 11 (𝑡 = 0 → ∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)))
1514oveq1d 7419 . . . . . . . . . 10 (𝑡 = 0 → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
1615adantr 482 . . . . . . . . 9 ((𝑡 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)) → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
1713, 16sumeq12dv 15648 . . . . . . . 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 482 . . . . . . 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 15648 . . . . . 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 2749 . . . . 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 5150 . . . . 5 (𝑡 = 𝑠 → (𝑡𝑆𝑠𝑆))
23 oveq2 7412 . . . . . . 7 (𝑡 = 𝑠 → (0..^𝑡) = (0..^𝑠))
2423prodeq1d 15861 . . . . . 6 (𝑡 = 𝑠 → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)))
25 oveq1 7411 . . . . . . . 8 (𝑡 = 𝑠 → (𝑡 · 𝑁) = (𝑠 · 𝑁))
2625oveq2d 7420 . . . . . . 7 (𝑡 = 𝑠 → (0...(𝑡 · 𝑁)) = (0...(𝑠 · 𝑁)))
27 fveq2 6888 . . . . . . . . . 10 (𝑡 = 𝑠 → (repr‘𝑡) = (repr‘𝑠))
2827oveqd 7421 . . . . . . . . 9 (𝑡 = 𝑠 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘𝑠)𝑚))
2923prodeq1d 15861 . . . . . . . . . . 11 (𝑡 = 𝑠 → ∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)))
3029oveq1d 7419 . . . . . . . . . 10 (𝑡 = 𝑠 → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
3130adantr 482 . . . . . . . . 9 ((𝑡 = 𝑠𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)) → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
3228, 31sumeq12dv 15648 . . . . . . . 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 482 . . . . . . 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 15648 . . . . . 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 2749 . . . . 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 5150 . . . . 5 (𝑡 = (𝑠 + 1) → (𝑡𝑆 ↔ (𝑠 + 1) ≤ 𝑆))
38 oveq2 7412 . . . . . . 7 (𝑡 = (𝑠 + 1) → (0..^𝑡) = (0..^(𝑠 + 1)))
3938prodeq1d 15861 . . . . . 6 (𝑡 = (𝑠 + 1) → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^(𝑠 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)))
40 oveq1 7411 . . . . . . . 8 (𝑡 = (𝑠 + 1) → (𝑡 · 𝑁) = ((𝑠 + 1) · 𝑁))
4140oveq2d 7420 . . . . . . 7 (𝑡 = (𝑠 + 1) → (0...(𝑡 · 𝑁)) = (0...((𝑠 + 1) · 𝑁)))
42 fveq2 6888 . . . . . . . . . 10 (𝑡 = (𝑠 + 1) → (repr‘𝑡) = (repr‘(𝑠 + 1)))
4342oveqd 7421 . . . . . . . . 9 (𝑡 = (𝑠 + 1) → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘(𝑠 + 1))𝑚))
4438prodeq1d 15861 . . . . . . . . . . 11 (𝑡 = (𝑠 + 1) → ∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿𝑎)‘(𝑐𝑎)))
4544oveq1d 7419 . . . . . . . . . 10 (𝑡 = (𝑠 + 1) → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
4645adantr 482 . . . . . . . . 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 15648 . . . . . . . 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 482 . . . . . . 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 15648 . . . . . 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 2749 . . . . 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 5150 . . . . 5 (𝑡 = 𝑆 → (𝑡𝑆𝑆𝑆))
53 oveq2 7412 . . . . . . 7 (𝑡 = 𝑆 → (0..^𝑡) = (0..^𝑆))
5453prodeq1d 15861 . . . . . 6 (𝑡 = 𝑆 → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)))
55 oveq1 7411 . . . . . . . 8 (𝑡 = 𝑆 → (𝑡 · 𝑁) = (𝑆 · 𝑁))
5655oveq2d 7420 . . . . . . 7 (𝑡 = 𝑆 → (0...(𝑡 · 𝑁)) = (0...(𝑆 · 𝑁)))
57 fveq2 6888 . . . . . . . . . 10 (𝑡 = 𝑆 → (repr‘𝑡) = (repr‘𝑆))
5857oveqd 7421 . . . . . . . . 9 (𝑡 = 𝑆 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘𝑆)𝑚))
5953prodeq1d 15861 . . . . . . . . . . 11 (𝑡 = 𝑆 → ∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
6059oveq1d 7419 . . . . . . . . . 10 (𝑡 = 𝑆 → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
6160adantr 482 . . . . . . . . 9 ((𝑡 = 𝑆𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)) → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
6258, 61sumeq12dv 15648 . . . . . . . 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 482 . . . . . . 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 15648 . . . . . 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 2749 . . . . 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 12483 . . . . . . . 8 0 ∈ ℕ0
68 fz1ssnn 13528 . . . . . . . . . . . . 13 (1...𝑁) ⊆ ℕ
6968a1i 11 . . . . . . . . . . . 12 (𝜑 → (1...𝑁) ⊆ ℕ)
70 0zd 12566 . . . . . . . . . . . 12 (𝜑 → 0 ∈ ℤ)
71 breprexp.n . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
7269, 70, 71repr0 33561 . . . . . . . . . . 11 (𝜑 → ((1...𝑁)(repr‘0)0) = if(0 = 0, {∅}, ∅))
73 eqid 2733 . . . . . . . . . . . 12 0 = 0
7473iftruei 4534 . . . . . . . . . . 11 if(0 = 0, {∅}, ∅) = {∅}
7572, 74eqtrdi 2789 . . . . . . . . . 10 (𝜑 → ((1...𝑁)(repr‘0)0) = {∅})
76 snfi 9040 . . . . . . . . . 10 {∅} ∈ Fin
7775, 76eqeltrdi 2842 . . . . . . . . 9 (𝜑 → ((1...𝑁)(repr‘0)0) ∈ Fin)
78 fzo0 13652 . . . . . . . . . . . . . . . 16 (0..^0) = ∅
7978prodeq1i 15858 . . . . . . . . . . . . . . 15 𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ ∅ ((𝐿𝑎)‘(𝑐𝑎))
80 prod0 15883 . . . . . . . . . . . . . . 15 𝑎 ∈ ∅ ((𝐿𝑎)‘(𝑐𝑎)) = 1
8179, 80eqtri 2761 . . . . . . . . . . . . . 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 14027 . . . . . . . . . . . . . 14 (𝑍 ∈ ℂ → (𝑍↑0) = 1)
8583, 84syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑍↑0) = 1)
8682, 85oveq12d 7422 . . . . . . . . . . . 12 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = (1 · 1))
87 ax-1cn 11164 . . . . . . . . . . . . 13 1 ∈ ℂ
8887mulridi 11214 . . . . . . . . . . . 12 (1 · 1) = 1
8986, 88eqtrdi 2789 . . . . . . . . . . 11 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = 1)
9089, 87eqeltrdi 2842 . . . . . . . . . 10 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) ∈ ℂ)
9190adantr 482 . . . . . . . . 9 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘0)0)) → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) ∈ ℂ)
9277, 91fsumcl 15675 . . . . . . . 8 (𝜑 → Σ𝑐 ∈ ((1...𝑁)(repr‘0)0)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) ∈ ℂ)
93 oveq2 7412 . . . . . . . . . 10 (𝑚 = 0 → ((1...𝑁)(repr‘0)𝑚) = ((1...𝑁)(repr‘0)0))
94 simpl 484 . . . . . . . . . . . 12 ((𝑚 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)) → 𝑚 = 0)
9594oveq2d 7420 . . . . . . . . . . 11 ((𝑚 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)) → (𝑍𝑚) = (𝑍↑0))
9695oveq2d 7420 . . . . . . . . . 10 ((𝑚 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)) → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)))
9793, 96sumeq12dv 15648 . . . . . . . . 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 15688 . . . . . . . 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 588 . . . . . . 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 15643 . . . . . . 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 5306 . . . . . . . . 9 ∅ ∈ V
10278prodeq1i 15858 . . . . . . . . . . . . 13 𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) = ∏𝑎 ∈ ∅ ((𝐿𝑎)‘(∅‘𝑎))
103 prod0 15883 . . . . . . . . . . . . 13 𝑎 ∈ ∅ ((𝐿𝑎)‘(∅‘𝑎)) = 1
104102, 103eqtri 2761 . . . . . . . . . . . 12 𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) = 1
105104a1i 11 . . . . . . . . . . 11 (𝜑 → ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) = 1)
106105, 87eqeltrdi 2842 . . . . . . . . . 10 (𝜑 → ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) ∈ ℂ)
10785, 87eqeltrdi 2842 . . . . . . . . . 10 (𝜑 → (𝑍↑0) ∈ ℂ)
108106, 107mulcld 11230 . . . . . . . . 9 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)) ∈ ℂ)
109 fveq1 6887 . . . . . . . . . . . . . 14 (𝑐 = ∅ → (𝑐𝑎) = (∅‘𝑎))
110109fveq2d 6892 . . . . . . . . . . . . 13 (𝑐 = ∅ → ((𝐿𝑎)‘(𝑐𝑎)) = ((𝐿𝑎)‘(∅‘𝑎)))
111110ralrimivw 3151 . . . . . . . . . . . 12 (𝑐 = ∅ → ∀𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) = ((𝐿𝑎)‘(∅‘𝑎)))
112111prodeq2d 15862 . . . . . . . . . . 11 (𝑐 = ∅ → ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)))
113112oveq1d 7419 . . . . . . . . . 10 (𝑐 = ∅ → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)))
114113sumsn 15688 . . . . . . . . 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 588 . . . . . . . 8 (𝜑 → Σ𝑐 ∈ {∅} (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)))
116105, 85oveq12d 7422 . . . . . . . . 9 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)) = (1 · 1))
117116, 86, 893eqtr2d 2779 . . . . . . . 8 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)) = 1)
118115, 117eqtrd 2773 . . . . . . 7 (𝜑 → Σ𝑐 ∈ {∅} (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = 1)
11999, 100, 1183eqtrd 2777 . . . . . 6 (𝜑 → Σ𝑚 ∈ {0}Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = 1)
12071nn0cnd 12530 . . . . . . . . . 10 (𝜑𝑁 ∈ ℂ)
121120mul02d 11408 . . . . . . . . 9 (𝜑 → (0 · 𝑁) = 0)
122121oveq2d 7420 . . . . . . . 8 (𝜑 → (0...(0 · 𝑁)) = (0...0))
123 fz0sn 13597 . . . . . . . 8 (0...0) = {0}
124122, 123eqtrdi 2789 . . . . . . 7 (𝜑 → (0...(0 · 𝑁)) = {0})
125124sumeq1d 15643 . . . . . 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 15858 . . . . . . . 8 𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ ∅ Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏))
127 prod0 15883 . . . . . . . 8 𝑎 ∈ ∅ Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = 1
128126, 127eqtri 2761 . . . . . . 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 2784 . . . . 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 766 . . . . . 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 768 . . . . . . 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 7412 . . . . . . . . . . . 12 (𝑚 = 𝑛 → ((1...𝑁)(repr‘𝑠)𝑚) = ((1...𝑁)(repr‘𝑠)𝑛))
135 oveq2 7412 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (𝑍𝑚) = (𝑍𝑛))
136135oveq2d 7420 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛)))
137136adantr 482 . . . . . . . . . . . 12 ((𝑚 = 𝑛𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)) → (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛)))
138134, 137sumeq12dv 15648 . . . . . . . . . . 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 15638 . . . . . . . . . 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 2746 . . . . . . . . 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 484 . . . . . . . . . . . . . . . 16 ((𝑎 = 𝑖𝑏 ∈ (1...𝑁)) → 𝑎 = 𝑖)
142141fveq2d 6892 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑖𝑏 ∈ (1...𝑁)) → (𝐿𝑎) = (𝐿𝑖))
143142fveq1d 6890 . . . . . . . . . . . . . 14 ((𝑎 = 𝑖𝑏 ∈ (1...𝑁)) → ((𝐿𝑎)‘𝑏) = ((𝐿𝑖)‘𝑏))
144143oveq1d 7419 . . . . . . . . . . . . 13 ((𝑎 = 𝑖𝑏 ∈ (1...𝑁)) → (((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = (((𝐿𝑖)‘𝑏) · (𝑍𝑏)))
145144sumeq2dv 15645 . . . . . . . . . . . 12 (𝑎 = 𝑖 → Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑏 ∈ (1...𝑁)(((𝐿𝑖)‘𝑏) · (𝑍𝑏)))
146145cbvprodv 15856 . . . . . . . . . . 11 𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑖 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑖)‘𝑏) · (𝑍𝑏))
147 fveq2 6888 . . . . . . . . . . . . . . 15 (𝑏 = 𝑗 → ((𝐿𝑖)‘𝑏) = ((𝐿𝑖)‘𝑗))
148 oveq2 7412 . . . . . . . . . . . . . . 15 (𝑏 = 𝑗 → (𝑍𝑏) = (𝑍𝑗))
149147, 148oveq12d 7422 . . . . . . . . . . . . . 14 (𝑏 = 𝑗 → (((𝐿𝑖)‘𝑏) · (𝑍𝑏)) = (((𝐿𝑖)‘𝑗) · (𝑍𝑗)))
150149cbvsumv 15638 . . . . . . . . . . . . 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 15859 . . . . . . . . . . 11 𝑖 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑖)‘𝑏) · (𝑍𝑏)) = ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗))
153146, 152eqtri 2761 . . . . . . . . . 10 𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗))
154 fveq2 6888 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑖 → (𝐿𝑎) = (𝐿𝑖))
155 fveq2 6888 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑖 → (𝑐𝑎) = (𝑐𝑖))
156154, 155fveq12d 6895 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑖 → ((𝐿𝑎)‘(𝑐𝑎)) = ((𝐿𝑖)‘(𝑐𝑖)))
157156cbvprodv 15856 . . . . . . . . . . . . . . . 16 𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑐𝑖))
158157oveq1i 7414 . . . . . . . . . . . . . . 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 15641 . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝑘𝑖 ∈ (0..^𝑠)) → 𝑐 = 𝑘)
162161fveq1d 6890 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝑘𝑖 ∈ (0..^𝑠)) → (𝑐𝑖) = (𝑘𝑖))
163162fveq2d 6892 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝑘𝑖 ∈ (0..^𝑠)) → ((𝐿𝑖)‘(𝑐𝑖)) = ((𝐿𝑖)‘(𝑘𝑖)))
164163prodeq2dv 15863 . . . . . . . . . . . . . . 15 (𝑐 = 𝑘 → ∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑐𝑖)) = ∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)))
165164oveq1d 7419 . . . . . . . . . . . . . 14 (𝑐 = 𝑘 → (∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑐𝑖)) · (𝑍𝑛)) = (∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))
166165cbvsumv 15638 . . . . . . . . . . . . 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 2761 . . . . . . . . . . . 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 15641 . . . . . . . . . 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 2751 . . . . . . . . 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 275 . . . . . . . 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 486 . . . . . 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 729 . . . . . . 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 729 . . . . . . 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 729 . . . . . . 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 729 . . . . . . 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 775 . . . . . . 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 486 . . . . . . 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 3979 . . . . . . . . 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 11211 . . . . . . . . . 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 11239 . . . . . . . . 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 3979 . . . . . . . . 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 12140 . . . . . . . . . 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 11358 . . . . . . . . 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 11367 . . . . . . . 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 768 . . . . . . . . 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 33582 . . . . . 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 837 . . . . 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 414 . . . 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 12655 . . 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 686 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 397   = wceq 1542  wcel 2107  Vcvv 3475  wss 3947  c0 4321  ifcif 4527  {csn 4627   class class class wbr 5147  wf 6536  cfv 6540  (class class class)co 7404  m cmap 8816  Fincfn 8935  cc 11104  cr 11105  0cc0 11106  1c1 11107   + caddc 11109   · cmul 11111  cle 11245  cn 12208  0cn0 12468  ...cfz 13480  ..^cfzo 13623  cexp 14023  Σcsu 15628  cprod 15845  reprcrepr 33558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-disj 5113  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-ico 13326  df-fz 13481  df-fzo 13624  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-sum 15629  df-prod 15846  df-repr 33559
This theorem is referenced by:  breprexpnat  33584  vtsprod  33589
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