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Theorem breprexp 30839
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 30840 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 11334 . . . . . 6 0 ⊆ ℝ
32a1i 11 . . . . 5 (𝜑 → ℕ0 ⊆ ℝ)
43sselda 3636 . . . 4 ((𝜑𝑆 ∈ ℕ0) → 𝑆 ∈ ℝ)
5 leid 10171 . . . 4 (𝑆 ∈ ℝ → 𝑆𝑆)
64, 5syl 17 . . 3 ((𝜑𝑆 ∈ ℕ0) → 𝑆𝑆)
7 breq1 4688 . . . . 5 (𝑡 = 0 → (𝑡𝑆 ↔ 0 ≤ 𝑆))
8 oveq2 6698 . . . . . . 7 (𝑡 = 0 → (0..^𝑡) = (0..^0))
98prodeq1d 14695 . . . . . 6 (𝑡 = 0 → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)))
10 oveq1 6697 . . . . . . . 8 (𝑡 = 0 → (𝑡 · 𝑁) = (0 · 𝑁))
1110oveq2d 6706 . . . . . . 7 (𝑡 = 0 → (0...(𝑡 · 𝑁)) = (0...(0 · 𝑁)))
12 fveq2 6229 . . . . . . . . . 10 (𝑡 = 0 → (repr‘𝑡) = (repr‘0))
1312oveqd 6707 . . . . . . . . 9 (𝑡 = 0 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘0)𝑚))
148prodeq1d 14695 . . . . . . . . . . 11 (𝑡 = 0 → ∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)))
1514oveq1d 6705 . . . . . . . . . 10 (𝑡 = 0 → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
1615adantr 480 . . . . . . . . 9 ((𝑡 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)) → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
1713, 16sumeq12dv 14481 . . . . . . . 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 480 . . . . . . 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 14481 . . . . . 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 2666 . . . . 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 333 . . . 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 4688 . . . . 5 (𝑡 = 𝑠 → (𝑡𝑆𝑠𝑆))
23 oveq2 6698 . . . . . . 7 (𝑡 = 𝑠 → (0..^𝑡) = (0..^𝑠))
2423prodeq1d 14695 . . . . . 6 (𝑡 = 𝑠 → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)))
25 oveq1 6697 . . . . . . . 8 (𝑡 = 𝑠 → (𝑡 · 𝑁) = (𝑠 · 𝑁))
2625oveq2d 6706 . . . . . . 7 (𝑡 = 𝑠 → (0...(𝑡 · 𝑁)) = (0...(𝑠 · 𝑁)))
27 fveq2 6229 . . . . . . . . . 10 (𝑡 = 𝑠 → (repr‘𝑡) = (repr‘𝑠))
2827oveqd 6707 . . . . . . . . 9 (𝑡 = 𝑠 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘𝑠)𝑚))
2923prodeq1d 14695 . . . . . . . . . . 11 (𝑡 = 𝑠 → ∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)))
3029oveq1d 6705 . . . . . . . . . 10 (𝑡 = 𝑠 → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
3130adantr 480 . . . . . . . . 9 ((𝑡 = 𝑠𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)) → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
3228, 31sumeq12dv 14481 . . . . . . . 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 480 . . . . . . 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 14481 . . . . . 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 2666 . . . . 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 333 . . . 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 4688 . . . . 5 (𝑡 = (𝑠 + 1) → (𝑡𝑆 ↔ (𝑠 + 1) ≤ 𝑆))
38 oveq2 6698 . . . . . . 7 (𝑡 = (𝑠 + 1) → (0..^𝑡) = (0..^(𝑠 + 1)))
3938prodeq1d 14695 . . . . . 6 (𝑡 = (𝑠 + 1) → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^(𝑠 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)))
40 oveq1 6697 . . . . . . . 8 (𝑡 = (𝑠 + 1) → (𝑡 · 𝑁) = ((𝑠 + 1) · 𝑁))
4140oveq2d 6706 . . . . . . 7 (𝑡 = (𝑠 + 1) → (0...(𝑡 · 𝑁)) = (0...((𝑠 + 1) · 𝑁)))
42 fveq2 6229 . . . . . . . . . 10 (𝑡 = (𝑠 + 1) → (repr‘𝑡) = (repr‘(𝑠 + 1)))
4342oveqd 6707 . . . . . . . . 9 (𝑡 = (𝑠 + 1) → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘(𝑠 + 1))𝑚))
4438prodeq1d 14695 . . . . . . . . . . 11 (𝑡 = (𝑠 + 1) → ∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿𝑎)‘(𝑐𝑎)))
4544oveq1d 6705 . . . . . . . . . 10 (𝑡 = (𝑠 + 1) → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
4645adantr 480 . . . . . . . . 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 14481 . . . . . . . 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 480 . . . . . . 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 14481 . . . . . 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 2666 . . . . 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 333 . . . 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 4688 . . . . 5 (𝑡 = 𝑆 → (𝑡𝑆𝑆𝑆))
53 oveq2 6698 . . . . . . 7 (𝑡 = 𝑆 → (0..^𝑡) = (0..^𝑆))
5453prodeq1d 14695 . . . . . 6 (𝑡 = 𝑆 → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)))
55 oveq1 6697 . . . . . . . 8 (𝑡 = 𝑆 → (𝑡 · 𝑁) = (𝑆 · 𝑁))
5655oveq2d 6706 . . . . . . 7 (𝑡 = 𝑆 → (0...(𝑡 · 𝑁)) = (0...(𝑆 · 𝑁)))
57 fveq2 6229 . . . . . . . . . 10 (𝑡 = 𝑆 → (repr‘𝑡) = (repr‘𝑆))
5857oveqd 6707 . . . . . . . . 9 (𝑡 = 𝑆 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘𝑆)𝑚))
5953prodeq1d 14695 . . . . . . . . . . 11 (𝑡 = 𝑆 → ∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
6059oveq1d 6705 . . . . . . . . . 10 (𝑡 = 𝑆 → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
6160adantr 480 . . . . . . . . 9 ((𝑡 = 𝑆𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)) → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
6258, 61sumeq12dv 14481 . . . . . . . 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 480 . . . . . . 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 14481 . . . . . 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 2666 . . . . 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 333 . . . 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 11345 . . . . . . . 8 0 ∈ ℕ0
68 fz1ssnn 12410 . . . . . . . . . . . . 13 (1...𝑁) ⊆ ℕ
6968a1i 11 . . . . . . . . . . . 12 (𝜑 → (1...𝑁) ⊆ ℕ)
70 0zd 11427 . . . . . . . . . . . 12 (𝜑 → 0 ∈ ℤ)
71 breprexp.n . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
7269, 70, 71repr0 30817 . . . . . . . . . . 11 (𝜑 → ((1...𝑁)(repr‘0)0) = if(0 = 0, {∅}, ∅))
73 eqid 2651 . . . . . . . . . . . 12 0 = 0
7473iftruei 4126 . . . . . . . . . . 11 if(0 = 0, {∅}, ∅) = {∅}
7572, 74syl6eq 2701 . . . . . . . . . 10 (𝜑 → ((1...𝑁)(repr‘0)0) = {∅})
76 snfi 8079 . . . . . . . . . 10 {∅} ∈ Fin
7775, 76syl6eqel 2738 . . . . . . . . 9 (𝜑 → ((1...𝑁)(repr‘0)0) ∈ Fin)
78 fzo0 12531 . . . . . . . . . . . . . . . 16 (0..^0) = ∅
7978prodeq1i 14692 . . . . . . . . . . . . . . 15 𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ ∅ ((𝐿𝑎)‘(𝑐𝑎))
80 prod0 14717 . . . . . . . . . . . . . . 15 𝑎 ∈ ∅ ((𝐿𝑎)‘(𝑐𝑎)) = 1
8179, 80eqtri 2673 . . . . . . . . . . . . . 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 12904 . . . . . . . . . . . . . 14 (𝑍 ∈ ℂ → (𝑍↑0) = 1)
8583, 84syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑍↑0) = 1)
8682, 85oveq12d 6708 . . . . . . . . . . . 12 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = (1 · 1))
87 ax-1cn 10032 . . . . . . . . . . . . 13 1 ∈ ℂ
8887mulid1i 10080 . . . . . . . . . . . 12 (1 · 1) = 1
8986, 88syl6eq 2701 . . . . . . . . . . 11 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = 1)
9089, 87syl6eqel 2738 . . . . . . . . . 10 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) ∈ ℂ)
9190adantr 480 . . . . . . . . 9 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘0)0)) → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) ∈ ℂ)
9277, 91fsumcl 14508 . . . . . . . 8 (𝜑 → Σ𝑐 ∈ ((1...𝑁)(repr‘0)0)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) ∈ ℂ)
93 oveq2 6698 . . . . . . . . . 10 (𝑚 = 0 → ((1...𝑁)(repr‘0)𝑚) = ((1...𝑁)(repr‘0)0))
94 simpl 472 . . . . . . . . . . . 12 ((𝑚 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)) → 𝑚 = 0)
9594oveq2d 6706 . . . . . . . . . . 11 ((𝑚 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)) → (𝑍𝑚) = (𝑍↑0))
9695oveq2d 6706 . . . . . . . . . 10 ((𝑚 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)) → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)))
9793, 96sumeq12dv 14481 . . . . . . . . 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 14519 . . . . . . . 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 696 . . . . . . 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 14475 . . . . . . 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 4823 . . . . . . . . 9 ∅ ∈ V
10278prodeq1i 14692 . . . . . . . . . . . . 13 𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) = ∏𝑎 ∈ ∅ ((𝐿𝑎)‘(∅‘𝑎))
103 prod0 14717 . . . . . . . . . . . . 13 𝑎 ∈ ∅ ((𝐿𝑎)‘(∅‘𝑎)) = 1
104102, 103eqtri 2673 . . . . . . . . . . . 12 𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) = 1
105104a1i 11 . . . . . . . . . . 11 (𝜑 → ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) = 1)
106105, 87syl6eqel 2738 . . . . . . . . . 10 (𝜑 → ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) ∈ ℂ)
10785, 87syl6eqel 2738 . . . . . . . . . 10 (𝜑 → (𝑍↑0) ∈ ℂ)
108106, 107mulcld 10098 . . . . . . . . 9 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)) ∈ ℂ)
109 fveq1 6228 . . . . . . . . . . . . . 14 (𝑐 = ∅ → (𝑐𝑎) = (∅‘𝑎))
110109fveq2d 6233 . . . . . . . . . . . . 13 (𝑐 = ∅ → ((𝐿𝑎)‘(𝑐𝑎)) = ((𝐿𝑎)‘(∅‘𝑎)))
111110ralrimivw 2996 . . . . . . . . . . . 12 (𝑐 = ∅ → ∀𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) = ((𝐿𝑎)‘(∅‘𝑎)))
112111prodeq2d 14696 . . . . . . . . . . 11 (𝑐 = ∅ → ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)))
113112oveq1d 6705 . . . . . . . . . 10 (𝑐 = ∅ → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)))
114113sumsn 14519 . . . . . . . . 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 696 . . . . . . . 8 (𝜑 → Σ𝑐 ∈ {∅} (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)))
116105, 85oveq12d 6708 . . . . . . . . 9 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)) = (1 · 1))
117116, 86, 893eqtr2d 2691 . . . . . . . 8 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)) = 1)
118115, 117eqtrd 2685 . . . . . . 7 (𝜑 → Σ𝑐 ∈ {∅} (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = 1)
11999, 100, 1183eqtrd 2689 . . . . . 6 (𝜑 → Σ𝑚 ∈ {0}Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = 1)
12071nn0cnd 11391 . . . . . . . . . 10 (𝜑𝑁 ∈ ℂ)
121120mul02d 10272 . . . . . . . . 9 (𝜑 → (0 · 𝑁) = 0)
122121oveq2d 6706 . . . . . . . 8 (𝜑 → (0...(0 · 𝑁)) = (0...0))
123 fz0sn 12478 . . . . . . . 8 (0...0) = {0}
124122, 123syl6eq 2701 . . . . . . 7 (𝜑 → (0...(0 · 𝑁)) = {0})
125124sumeq1d 14475 . . . . . 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 14692 . . . . . . . 8 𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ ∅ Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏))
127 prod0 14717 . . . . . . . 8 𝑎 ∈ ∅ Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = 1
128126, 127eqtri 2673 . . . . . . 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 2696 . . . . 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 805 . . . . . 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 807 . . . . . . 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 6698 . . . . . . . . . . . 12 (𝑚 = 𝑛 → ((1...𝑁)(repr‘𝑠)𝑚) = ((1...𝑁)(repr‘𝑠)𝑛))
135 oveq2 6698 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (𝑍𝑚) = (𝑍𝑛))
136135oveq2d 6706 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛)))
137136adantr 480 . . . . . . . . . . . 12 ((𝑚 = 𝑛𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)) → (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛)))
138134, 137sumeq12dv 14481 . . . . . . . . . . 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 14470 . . . . . . . . . 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 2663 . . . . . . . . 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 472 . . . . . . . . . . . . . . . 16 ((𝑎 = 𝑖𝑏 ∈ (1...𝑁)) → 𝑎 = 𝑖)
142141fveq2d 6233 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑖𝑏 ∈ (1...𝑁)) → (𝐿𝑎) = (𝐿𝑖))
143142fveq1d 6231 . . . . . . . . . . . . . 14 ((𝑎 = 𝑖𝑏 ∈ (1...𝑁)) → ((𝐿𝑎)‘𝑏) = ((𝐿𝑖)‘𝑏))
144143oveq1d 6705 . . . . . . . . . . . . 13 ((𝑎 = 𝑖𝑏 ∈ (1...𝑁)) → (((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = (((𝐿𝑖)‘𝑏) · (𝑍𝑏)))
145144sumeq2dv 14477 . . . . . . . . . . . 12 (𝑎 = 𝑖 → Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑏 ∈ (1...𝑁)(((𝐿𝑖)‘𝑏) · (𝑍𝑏)))
146145cbvprodv 14690 . . . . . . . . . . 11 𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑖 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑖)‘𝑏) · (𝑍𝑏))
147 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑏 = 𝑗 → ((𝐿𝑖)‘𝑏) = ((𝐿𝑖)‘𝑗))
148 oveq2 6698 . . . . . . . . . . . . . . 15 (𝑏 = 𝑗 → (𝑍𝑏) = (𝑍𝑗))
149147, 148oveq12d 6708 . . . . . . . . . . . . . 14 (𝑏 = 𝑗 → (((𝐿𝑖)‘𝑏) · (𝑍𝑏)) = (((𝐿𝑖)‘𝑗) · (𝑍𝑗)))
150149cbvsumv 14470 . . . . . . . . . . . . 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 14693 . . . . . . . . . . 11 𝑖 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑖)‘𝑏) · (𝑍𝑏)) = ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗))
153146, 152eqtri 2673 . . . . . . . . . 10 𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗))
154 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑖 → (𝐿𝑎) = (𝐿𝑖))
155 fveq2 6229 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑖 → (𝑐𝑎) = (𝑐𝑖))
156154, 155fveq12d 6235 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑖 → ((𝐿𝑎)‘(𝑐𝑎)) = ((𝐿𝑖)‘(𝑐𝑖)))
157156cbvprodv 14690 . . . . . . . . . . . . . . . 16 𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑐𝑖))
158157oveq1i 6700 . . . . . . . . . . . . . . 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 14473 . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝑘𝑖 ∈ (0..^𝑠)) → 𝑐 = 𝑘)
162161fveq1d 6231 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝑘𝑖 ∈ (0..^𝑠)) → (𝑐𝑖) = (𝑘𝑖))
163162fveq2d 6233 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝑘𝑖 ∈ (0..^𝑠)) → ((𝐿𝑖)‘(𝑐𝑖)) = ((𝐿𝑖)‘(𝑘𝑖)))
164163prodeq2dv 14697 . . . . . . . . . . . . . . 15 (𝑐 = 𝑘 → ∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑐𝑖)) = ∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)))
165164oveq1d 6705 . . . . . . . . . . . . . 14 (𝑐 = 𝑘 → (∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑐𝑖)) · (𝑍𝑛)) = (∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))
166165cbvsumv 14470 . . . . . . . . . . . . 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 2673 . . . . . . . . . . . 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 14473 . . . . . . . . . 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 2665 . . . . . . . . 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 264 . . . . . . . 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 325 . . . . . . 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 208 . . . . . 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 476 . . . . . 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 766 . . . . . . 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 766 . . . . . . 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 766 . . . . . . 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 766 . . . . . . 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 815 . . . . . . 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 476 . . . . . . 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, 180sseldi 3634 . . . . . . . . 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 10093 . . . . . . . . . 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 10107 . . . . . . . . 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, 176sseldi 3634 . . . . . . . . 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 10992 . . . . . . . . . 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 10223 . . . . . . . . 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 10232 . . . . . . . 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 807 . . . . . . . . 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 224 . . . . . . . 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 30838 . . . . . 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 1365 . . . . 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 449 . . . 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 11512 . . 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 703 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 383   = wceq 1523  wcel 2030  Vcvv 3231  wss 3607  c0 3948  ifcif 4119  {csn 4210   class class class wbr 4685  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  Fincfn 7997  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  cle 10113  cn 11058  0cn0 11330  ...cfz 12364  ..^cfzo 12504  cexp 12900  Σcsu 14460  cprod 14679  reprcrepr 30814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ico 12219  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-prod 14680  df-repr 30815
This theorem is referenced by:  breprexpnat  30840  vtsprod  30845
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