Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  breprexp Structured version   Visualization version   GIF version

Theorem breprexp 31904
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 31905 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 11900 . . . . . 6 0 ⊆ ℝ
32a1i 11 . . . . 5 (𝜑 → ℕ0 ⊆ ℝ)
43sselda 3966 . . . 4 ((𝜑𝑆 ∈ ℕ0) → 𝑆 ∈ ℝ)
5 leid 10735 . . . 4 (𝑆 ∈ ℝ → 𝑆𝑆)
64, 5syl 17 . . 3 ((𝜑𝑆 ∈ ℕ0) → 𝑆𝑆)
7 breq1 5068 . . . . 5 (𝑡 = 0 → (𝑡𝑆 ↔ 0 ≤ 𝑆))
8 oveq2 7163 . . . . . . 7 (𝑡 = 0 → (0..^𝑡) = (0..^0))
98prodeq1d 15274 . . . . . 6 (𝑡 = 0 → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)))
10 oveq1 7162 . . . . . . . 8 (𝑡 = 0 → (𝑡 · 𝑁) = (0 · 𝑁))
1110oveq2d 7171 . . . . . . 7 (𝑡 = 0 → (0...(𝑡 · 𝑁)) = (0...(0 · 𝑁)))
12 fveq2 6669 . . . . . . . . . 10 (𝑡 = 0 → (repr‘𝑡) = (repr‘0))
1312oveqd 7172 . . . . . . . . 9 (𝑡 = 0 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘0)𝑚))
148prodeq1d 15274 . . . . . . . . . . 11 (𝑡 = 0 → ∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)))
1514oveq1d 7170 . . . . . . . . . 10 (𝑡 = 0 → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
1615adantr 483 . . . . . . . . 9 ((𝑡 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)) → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
1713, 16sumeq12dv 15062 . . . . . . . 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 483 . . . . . . 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 15062 . . . . . 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 2837 . . . . 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 347 . . . 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 5068 . . . . 5 (𝑡 = 𝑠 → (𝑡𝑆𝑠𝑆))
23 oveq2 7163 . . . . . . 7 (𝑡 = 𝑠 → (0..^𝑡) = (0..^𝑠))
2423prodeq1d 15274 . . . . . 6 (𝑡 = 𝑠 → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)))
25 oveq1 7162 . . . . . . . 8 (𝑡 = 𝑠 → (𝑡 · 𝑁) = (𝑠 · 𝑁))
2625oveq2d 7171 . . . . . . 7 (𝑡 = 𝑠 → (0...(𝑡 · 𝑁)) = (0...(𝑠 · 𝑁)))
27 fveq2 6669 . . . . . . . . . 10 (𝑡 = 𝑠 → (repr‘𝑡) = (repr‘𝑠))
2827oveqd 7172 . . . . . . . . 9 (𝑡 = 𝑠 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘𝑠)𝑚))
2923prodeq1d 15274 . . . . . . . . . . 11 (𝑡 = 𝑠 → ∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)))
3029oveq1d 7170 . . . . . . . . . 10 (𝑡 = 𝑠 → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
3130adantr 483 . . . . . . . . 9 ((𝑡 = 𝑠𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)) → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
3228, 31sumeq12dv 15062 . . . . . . . 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 483 . . . . . . 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 15062 . . . . . 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 2837 . . . . 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 347 . . . 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 5068 . . . . 5 (𝑡 = (𝑠 + 1) → (𝑡𝑆 ↔ (𝑠 + 1) ≤ 𝑆))
38 oveq2 7163 . . . . . . 7 (𝑡 = (𝑠 + 1) → (0..^𝑡) = (0..^(𝑠 + 1)))
3938prodeq1d 15274 . . . . . 6 (𝑡 = (𝑠 + 1) → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^(𝑠 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)))
40 oveq1 7162 . . . . . . . 8 (𝑡 = (𝑠 + 1) → (𝑡 · 𝑁) = ((𝑠 + 1) · 𝑁))
4140oveq2d 7171 . . . . . . 7 (𝑡 = (𝑠 + 1) → (0...(𝑡 · 𝑁)) = (0...((𝑠 + 1) · 𝑁)))
42 fveq2 6669 . . . . . . . . . 10 (𝑡 = (𝑠 + 1) → (repr‘𝑡) = (repr‘(𝑠 + 1)))
4342oveqd 7172 . . . . . . . . 9 (𝑡 = (𝑠 + 1) → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘(𝑠 + 1))𝑚))
4438prodeq1d 15274 . . . . . . . . . . 11 (𝑡 = (𝑠 + 1) → ∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿𝑎)‘(𝑐𝑎)))
4544oveq1d 7170 . . . . . . . . . 10 (𝑡 = (𝑠 + 1) → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^(𝑠 + 1))((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
4645adantr 483 . . . . . . . . 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 15062 . . . . . . . 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 483 . . . . . . 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 15062 . . . . . 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 2837 . . . . 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 347 . . . 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 5068 . . . . 5 (𝑡 = 𝑆 → (𝑡𝑆𝑆𝑆))
53 oveq2 7163 . . . . . . 7 (𝑡 = 𝑆 → (0..^𝑡) = (0..^𝑆))
5453prodeq1d 15274 . . . . . 6 (𝑡 = 𝑆 → ∏𝑎 ∈ (0..^𝑡𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)))
55 oveq1 7162 . . . . . . . 8 (𝑡 = 𝑆 → (𝑡 · 𝑁) = (𝑆 · 𝑁))
5655oveq2d 7171 . . . . . . 7 (𝑡 = 𝑆 → (0...(𝑡 · 𝑁)) = (0...(𝑆 · 𝑁)))
57 fveq2 6669 . . . . . . . . . 10 (𝑡 = 𝑆 → (repr‘𝑡) = (repr‘𝑆))
5857oveqd 7172 . . . . . . . . 9 (𝑡 = 𝑆 → ((1...𝑁)(repr‘𝑡)𝑚) = ((1...𝑁)(repr‘𝑆)𝑚))
5953prodeq1d 15274 . . . . . . . . . . 11 (𝑡 = 𝑆 → ∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
6059oveq1d 7170 . . . . . . . . . 10 (𝑡 = 𝑆 → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
6160adantr 483 . . . . . . . . 9 ((𝑡 = 𝑆𝑐 ∈ ((1...𝑁)(repr‘𝑡)𝑚)) → (∏𝑎 ∈ (0..^𝑡)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))
6258, 61sumeq12dv 15062 . . . . . . . 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 483 . . . . . . 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 15062 . . . . . 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 2837 . . . . 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 347 . . . 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 11911 . . . . . . . 8 0 ∈ ℕ0
68 fz1ssnn 12937 . . . . . . . . . . . . 13 (1...𝑁) ⊆ ℕ
6968a1i 11 . . . . . . . . . . . 12 (𝜑 → (1...𝑁) ⊆ ℕ)
70 0zd 11992 . . . . . . . . . . . 12 (𝜑 → 0 ∈ ℤ)
71 breprexp.n . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
7269, 70, 71repr0 31882 . . . . . . . . . . 11 (𝜑 → ((1...𝑁)(repr‘0)0) = if(0 = 0, {∅}, ∅))
73 eqid 2821 . . . . . . . . . . . 12 0 = 0
7473iftruei 4473 . . . . . . . . . . 11 if(0 = 0, {∅}, ∅) = {∅}
7572, 74syl6eq 2872 . . . . . . . . . 10 (𝜑 → ((1...𝑁)(repr‘0)0) = {∅})
76 snfi 8593 . . . . . . . . . 10 {∅} ∈ Fin
7775, 76eqeltrdi 2921 . . . . . . . . 9 (𝜑 → ((1...𝑁)(repr‘0)0) ∈ Fin)
78 fzo0 13060 . . . . . . . . . . . . . . . 16 (0..^0) = ∅
7978prodeq1i 15271 . . . . . . . . . . . . . . 15 𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ ∅ ((𝐿𝑎)‘(𝑐𝑎))
80 prod0 15296 . . . . . . . . . . . . . . 15 𝑎 ∈ ∅ ((𝐿𝑎)‘(𝑐𝑎)) = 1
8179, 80eqtri 2844 . . . . . . . . . . . . . 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 13432 . . . . . . . . . . . . . 14 (𝑍 ∈ ℂ → (𝑍↑0) = 1)
8583, 84syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝑍↑0) = 1)
8682, 85oveq12d 7173 . . . . . . . . . . . 12 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = (1 · 1))
87 ax-1cn 10594 . . . . . . . . . . . . 13 1 ∈ ℂ
8887mulid1i 10644 . . . . . . . . . . . 12 (1 · 1) = 1
8986, 88syl6eq 2872 . . . . . . . . . . 11 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = 1)
9089, 87eqeltrdi 2921 . . . . . . . . . 10 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) ∈ ℂ)
9190adantr 483 . . . . . . . . 9 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘0)0)) → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) ∈ ℂ)
9277, 91fsumcl 15089 . . . . . . . 8 (𝜑 → Σ𝑐 ∈ ((1...𝑁)(repr‘0)0)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) ∈ ℂ)
93 oveq2 7163 . . . . . . . . . 10 (𝑚 = 0 → ((1...𝑁)(repr‘0)𝑚) = ((1...𝑁)(repr‘0)0))
94 simpl 485 . . . . . . . . . . . 12 ((𝑚 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)) → 𝑚 = 0)
9594oveq2d 7171 . . . . . . . . . . 11 ((𝑚 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)) → (𝑍𝑚) = (𝑍↑0))
9695oveq2d 7171 . . . . . . . . . 10 ((𝑚 = 0 ∧ 𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)) → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)))
9793, 96sumeq12dv 15062 . . . . . . . . 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 15100 . . . . . . . 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 589 . . . . . . 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 15057 . . . . . . 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 5210 . . . . . . . . 9 ∅ ∈ V
10278prodeq1i 15271 . . . . . . . . . . . . 13 𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) = ∏𝑎 ∈ ∅ ((𝐿𝑎)‘(∅‘𝑎))
103 prod0 15296 . . . . . . . . . . . . 13 𝑎 ∈ ∅ ((𝐿𝑎)‘(∅‘𝑎)) = 1
104102, 103eqtri 2844 . . . . . . . . . . . 12 𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) = 1
105104a1i 11 . . . . . . . . . . 11 (𝜑 → ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) = 1)
106105, 87eqeltrdi 2921 . . . . . . . . . 10 (𝜑 → ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) ∈ ℂ)
10785, 87eqeltrdi 2921 . . . . . . . . . 10 (𝜑 → (𝑍↑0) ∈ ℂ)
108106, 107mulcld 10660 . . . . . . . . 9 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)) ∈ ℂ)
109 fveq1 6668 . . . . . . . . . . . . . 14 (𝑐 = ∅ → (𝑐𝑎) = (∅‘𝑎))
110109fveq2d 6673 . . . . . . . . . . . . 13 (𝑐 = ∅ → ((𝐿𝑎)‘(𝑐𝑎)) = ((𝐿𝑎)‘(∅‘𝑎)))
111110ralrimivw 3183 . . . . . . . . . . . 12 (𝑐 = ∅ → ∀𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) = ((𝐿𝑎)‘(∅‘𝑎)))
112111prodeq2d 15275 . . . . . . . . . . 11 (𝑐 = ∅ → ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)))
113112oveq1d 7170 . . . . . . . . . 10 (𝑐 = ∅ → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)))
114113sumsn 15100 . . . . . . . . 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 589 . . . . . . . 8 (𝜑 → Σ𝑐 ∈ {∅} (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)))
116105, 85oveq12d 7173 . . . . . . . . 9 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)) = (1 · 1))
117116, 86, 893eqtr2d 2862 . . . . . . . 8 (𝜑 → (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(∅‘𝑎)) · (𝑍↑0)) = 1)
118115, 117eqtrd 2856 . . . . . . 7 (𝜑 → Σ𝑐 ∈ {∅} (∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍↑0)) = 1)
11999, 100, 1183eqtrd 2860 . . . . . 6 (𝜑 → Σ𝑚 ∈ {0}Σ𝑐 ∈ ((1...𝑁)(repr‘0)𝑚)(∏𝑎 ∈ (0..^0)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = 1)
12071nn0cnd 11956 . . . . . . . . . 10 (𝜑𝑁 ∈ ℂ)
121120mul02d 10837 . . . . . . . . 9 (𝜑 → (0 · 𝑁) = 0)
122121oveq2d 7171 . . . . . . . 8 (𝜑 → (0...(0 · 𝑁)) = (0...0))
123 fz0sn 13006 . . . . . . . 8 (0...0) = {0}
124122, 123syl6eq 2872 . . . . . . 7 (𝜑 → (0...(0 · 𝑁)) = {0})
125124sumeq1d 15057 . . . . . 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 15271 . . . . . . . 8 𝑎 ∈ (0..^0)Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑎 ∈ ∅ Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏))
127 prod0 15296 . . . . . . . 8 𝑎 ∈ ∅ Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = 1
128126, 127eqtri 2844 . . . . . . 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 2867 . . . . 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 765 . . . . . 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 767 . . . . . . 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 7163 . . . . . . . . . . . 12 (𝑚 = 𝑛 → ((1...𝑁)(repr‘𝑠)𝑚) = ((1...𝑁)(repr‘𝑠)𝑛))
135 oveq2 7163 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (𝑍𝑚) = (𝑍𝑛))
136135oveq2d 7171 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛)))
137136adantr 483 . . . . . . . . . . . 12 ((𝑚 = 𝑛𝑐 ∈ ((1...𝑁)(repr‘𝑠)𝑚)) → (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)) = (∏𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑛)))
138134, 137sumeq12dv 15062 . . . . . . . . . . 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 15052 . . . . . . . . . 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 2834 . . . . . . . . 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 485 . . . . . . . . . . . . . . . 16 ((𝑎 = 𝑖𝑏 ∈ (1...𝑁)) → 𝑎 = 𝑖)
142141fveq2d 6673 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑖𝑏 ∈ (1...𝑁)) → (𝐿𝑎) = (𝐿𝑖))
143142fveq1d 6671 . . . . . . . . . . . . . 14 ((𝑎 = 𝑖𝑏 ∈ (1...𝑁)) → ((𝐿𝑎)‘𝑏) = ((𝐿𝑖)‘𝑏))
144143oveq1d 7170 . . . . . . . . . . . . 13 ((𝑎 = 𝑖𝑏 ∈ (1...𝑁)) → (((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = (((𝐿𝑖)‘𝑏) · (𝑍𝑏)))
145144sumeq2dv 15059 . . . . . . . . . . . 12 (𝑎 = 𝑖 → Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑏 ∈ (1...𝑁)(((𝐿𝑖)‘𝑏) · (𝑍𝑏)))
146145cbvprodv 15269 . . . . . . . . . . 11 𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑖 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑖)‘𝑏) · (𝑍𝑏))
147 fveq2 6669 . . . . . . . . . . . . . . 15 (𝑏 = 𝑗 → ((𝐿𝑖)‘𝑏) = ((𝐿𝑖)‘𝑗))
148 oveq2 7163 . . . . . . . . . . . . . . 15 (𝑏 = 𝑗 → (𝑍𝑏) = (𝑍𝑗))
149147, 148oveq12d 7173 . . . . . . . . . . . . . 14 (𝑏 = 𝑗 → (((𝐿𝑖)‘𝑏) · (𝑍𝑏)) = (((𝐿𝑖)‘𝑗) · (𝑍𝑗)))
150149cbvsumv 15052 . . . . . . . . . . . . 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 15272 . . . . . . . . . . 11 𝑖 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑖)‘𝑏) · (𝑍𝑏)) = ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗))
153146, 152eqtri 2844 . . . . . . . . . 10 𝑎 ∈ (0..^𝑠𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = ∏𝑖 ∈ (0..^𝑠𝑗 ∈ (1...𝑁)(((𝐿𝑖)‘𝑗) · (𝑍𝑗))
154 fveq2 6669 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑖 → (𝐿𝑎) = (𝐿𝑖))
155 fveq2 6669 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑖 → (𝑐𝑎) = (𝑐𝑖))
156154, 155fveq12d 6676 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑖 → ((𝐿𝑎)‘(𝑐𝑎)) = ((𝐿𝑖)‘(𝑐𝑖)))
157156cbvprodv 15269 . . . . . . . . . . . . . . . 16 𝑎 ∈ (0..^𝑠)((𝐿𝑎)‘(𝑐𝑎)) = ∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑐𝑖))
158157oveq1i 7165 . . . . . . . . . . . . . . 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 15055 . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . 18 ((𝑐 = 𝑘𝑖 ∈ (0..^𝑠)) → 𝑐 = 𝑘)
162161fveq1d 6671 . . . . . . . . . . . . . . . . 17 ((𝑐 = 𝑘𝑖 ∈ (0..^𝑠)) → (𝑐𝑖) = (𝑘𝑖))
163162fveq2d 6673 . . . . . . . . . . . . . . . 16 ((𝑐 = 𝑘𝑖 ∈ (0..^𝑠)) → ((𝐿𝑖)‘(𝑐𝑖)) = ((𝐿𝑖)‘(𝑘𝑖)))
164163prodeq2dv 15276 . . . . . . . . . . . . . . 15 (𝑐 = 𝑘 → ∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑐𝑖)) = ∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)))
165164oveq1d 7170 . . . . . . . . . . . . . 14 (𝑐 = 𝑘 → (∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑐𝑖)) · (𝑍𝑛)) = (∏𝑖 ∈ (0..^𝑠)((𝐿𝑖)‘(𝑘𝑖)) · (𝑍𝑛)))
166165cbvsumv 15052 . . . . . . . . . . . . 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 2844 . . . . . . . . . . . 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 15055 . . . . . . . . . 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 2836 . . . . . . . . 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 277 . . . . . . . 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 338 . . . . . . 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 220 . . . . . 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 487 . . . . . 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 728 . . . . . . 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 728 . . . . . . 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 728 . . . . . . 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 728 . . . . . . 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 774 . . . . . . 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 487 . . . . . . 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 3964 . . . . . . . . 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 10641 . . . . . . . . . 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 10669 . . . . . . . . 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 3964 . . . . . . . . 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 11569 . . . . . . . . . 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 10787 . . . . . . . . 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 10796 . . . . . . . 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 767 . . . . . . . . 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 236 . . . . . . . 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 31903 . . . . . 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 415 . . . 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 12078 . . 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 685 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 398   = wceq 1533  wcel 2110  Vcvv 3494  wss 3935  c0 4290  ifcif 4466  {csn 4566   class class class wbr 5065  wf 6350  cfv 6354  (class class class)co 7155  m cmap 8405  Fincfn 8508  cc 10534  cr 10535  0cc0 10536  1c1 10537   + caddc 10539   · cmul 10541  cle 10675  cn 11637  0cn0 11896  ...cfz 12891  ..^cfzo 13032  cexp 13428  Σcsu 15041  cprod 15258  reprcrepr 31879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-inf2 9103  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-disj 5031  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-2o 8102  df-oadd 8105  df-er 8288  df-map 8407  df-pm 8408  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-sup 8905  df-oi 8973  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-n0 11897  df-z 11981  df-uz 12243  df-rp 12389  df-ico 12743  df-fz 12892  df-fzo 13033  df-seq 13369  df-exp 13429  df-hash 13690  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-clim 14844  df-sum 15042  df-prod 15259  df-repr 31880
This theorem is referenced by:  breprexpnat  31905  vtsprod  31910
  Copyright terms: Public domain W3C validator