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

Theorem circlemeth 33253
Description: The Hardy, Littlewood and Ramanujan Circle Method, in a generic form, with different weighting / smoothing functions. (Contributed by Thierry Arnoux, 13-Dec-2021.)
Hypotheses
Ref Expression
circlemeth.n (𝜑𝑁 ∈ ℕ0)
circlemeth.s (𝜑𝑆 ∈ ℕ)
circlemeth.l (𝜑𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
Assertion
Ref Expression
circlemeth (𝜑 → Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = ∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)(((𝐿𝑎)vts𝑁)‘𝑥) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥)
Distinct variable groups:   𝐿,𝑎,𝑐,𝑥   𝑁,𝑎,𝑐,𝑥   𝑆,𝑎,𝑐,𝑥   𝜑,𝑎,𝑐,𝑥

Proof of Theorem circlemeth
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 circlemeth.n . . . . . . 7 (𝜑𝑁 ∈ ℕ0)
21adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → 𝑁 ∈ ℕ0)
3 ioossre 13325 . . . . . . . . 9 (0(,)1) ⊆ ℝ
4 ax-resscn 11108 . . . . . . . . 9 ℝ ⊆ ℂ
53, 4sstri 3953 . . . . . . . 8 (0(,)1) ⊆ ℂ
65a1i 11 . . . . . . 7 (𝜑 → (0(,)1) ⊆ ℂ)
76sselda 3944 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → 𝑥 ∈ ℂ)
8 circlemeth.s . . . . . . . 8 (𝜑𝑆 ∈ ℕ)
98nnnn0d 12473 . . . . . . 7 (𝜑𝑆 ∈ ℕ0)
109adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → 𝑆 ∈ ℕ0)
11 circlemeth.l . . . . . . 7 (𝜑𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
1211adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → 𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
132, 7, 10, 12vtsprod 33252 . . . . 5 ((𝜑𝑥 ∈ (0(,)1)) → ∏𝑎 ∈ (0..^𝑆)(((𝐿𝑎)vts𝑁)‘𝑥) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))))
1413oveq1d 7372 . . . 4 ((𝜑𝑥 ∈ (0(,)1)) → (∏𝑎 ∈ (0..^𝑆)(((𝐿𝑎)vts𝑁)‘𝑥) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = (Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))))
15 fzfid 13878 . . . . 5 ((𝜑𝑥 ∈ (0(,)1)) → (0...(𝑆 · 𝑁)) ∈ Fin)
16 ax-icn 11110 . . . . . . . . 9 i ∈ ℂ
17 2cn 12228 . . . . . . . . . 10 2 ∈ ℂ
18 picn 25816 . . . . . . . . . 10 π ∈ ℂ
1917, 18mulcli 11162 . . . . . . . . 9 (2 · π) ∈ ℂ
2016, 19mulcli 11162 . . . . . . . 8 (i · (2 · π)) ∈ ℂ
2120a1i 11 . . . . . . 7 ((𝜑𝑥 ∈ (0(,)1)) → (i · (2 · π)) ∈ ℂ)
221nn0cnd 12475 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℂ)
2322negcld 11499 . . . . . . . . . 10 (𝜑 → -𝑁 ∈ ℂ)
2423ralrimivw 3147 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ (0(,)1)-𝑁 ∈ ℂ)
2524r19.21bi 3234 . . . . . . . 8 ((𝜑𝑥 ∈ (0(,)1)) → -𝑁 ∈ ℂ)
2625, 7mulcld 11175 . . . . . . 7 ((𝜑𝑥 ∈ (0(,)1)) → (-𝑁 · 𝑥) ∈ ℂ)
2721, 26mulcld 11175 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → ((i · (2 · π)) · (-𝑁 · 𝑥)) ∈ ℂ)
2827efcld 33204 . . . . 5 ((𝜑𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
29 fz1ssnn 13472 . . . . . . . 8 (1...𝑁) ⊆ ℕ
3029a1i 11 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ⊆ ℕ)
31 simpr 485 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
3231elfzelzd 13442 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
3332adantlr 713 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
3410adantr 481 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ0)
35 fzfid 13878 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
3630, 33, 34, 35reprfi 33229 . . . . . 6 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
37 fzofi 13879 . . . . . . . . 9 (0..^𝑆) ∈ Fin
3837a1i 11 . . . . . . . 8 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
391ad3antrrr 728 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℕ0)
409ad3antrrr 728 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑆 ∈ ℕ0)
4132zcnd 12608 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℂ)
4241ad2antrr 724 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑚 ∈ ℂ)
4311ad3antrrr 728 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
44 simpr 485 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
4529a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (1...𝑁) ⊆ ℕ)
4632adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑚 ∈ ℤ)
479ad2antrr 724 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑆 ∈ ℕ0)
48 simpr 485 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚))
4945, 46, 47, 48reprf 33225 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
5049ffvelcdmda 7035 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ (1...𝑁))
5129, 50sselid 3942 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ ℕ)
5239, 40, 42, 43, 44, 51breprexplemb 33244 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
5352adantl3r 748 . . . . . . . 8 (((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
5438, 53fprodcl 15835 . . . . . . 7 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
5520a1i 11 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (i · (2 · π)) ∈ ℂ)
5633zcnd 12608 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℂ)
577adantr 481 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑥 ∈ ℂ)
5856, 57mulcld 11175 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 · 𝑥) ∈ ℂ)
5955, 58mulcld 11175 . . . . . . . . 9 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · (𝑚 · 𝑥)) ∈ ℂ)
6059efcld 33204 . . . . . . . 8 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘((i · (2 · π)) · (𝑚 · 𝑥))) ∈ ℂ)
6160adantr 481 . . . . . . 7 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · (𝑚 · 𝑥))) ∈ ℂ)
6254, 61mulcld 11175 . . . . . 6 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) ∈ ℂ)
6336, 62fsumcl 15618 . . . . 5 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) ∈ ℂ)
6415, 28, 63fsummulc1 15670 . . . 4 ((𝜑𝑥 ∈ (0(,)1)) → (Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))(Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))))
6528adantr 481 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
6636, 65, 62fsummulc1 15670 . . . . . 6 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)((∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))))
6765adantr 481 . . . . . . . . 9 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
6854, 61, 67mulassd 11178 . . . . . . . 8 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ((∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ((exp‘((i · (2 · π)) · (𝑚 · 𝑥))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))))))
6927adantr 481 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · (-𝑁 · 𝑥)) ∈ ℂ)
70 efadd 15976 . . . . . . . . . . . 12 ((((i · (2 · π)) · (𝑚 · 𝑥)) ∈ ℂ ∧ ((i · (2 · π)) · (-𝑁 · 𝑥)) ∈ ℂ) → (exp‘(((i · (2 · π)) · (𝑚 · 𝑥)) + ((i · (2 · π)) · (-𝑁 · 𝑥)))) = ((exp‘((i · (2 · π)) · (𝑚 · 𝑥))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))))
7159, 69, 70syl2anc 584 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘(((i · (2 · π)) · (𝑚 · 𝑥)) + ((i · (2 · π)) · (-𝑁 · 𝑥)))) = ((exp‘((i · (2 · π)) · (𝑚 · 𝑥))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))))
7226adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (-𝑁 · 𝑥) ∈ ℂ)
7355, 58, 72adddid 11179 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · ((𝑚 · 𝑥) + (-𝑁 · 𝑥))) = (((i · (2 · π)) · (𝑚 · 𝑥)) + ((i · (2 · π)) · (-𝑁 · 𝑥))))
7425adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → -𝑁 ∈ ℂ)
7556, 74, 57adddird 11180 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 + -𝑁) · 𝑥) = ((𝑚 · 𝑥) + (-𝑁 · 𝑥)))
7622ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑁 ∈ ℂ)
7756, 76negsubd 11518 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 + -𝑁) = (𝑚𝑁))
7877oveq1d 7372 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 + -𝑁) · 𝑥) = ((𝑚𝑁) · 𝑥))
7975, 78eqtr3d 2778 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 · 𝑥) + (-𝑁 · 𝑥)) = ((𝑚𝑁) · 𝑥))
8079oveq2d 7373 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · ((𝑚 · 𝑥) + (-𝑁 · 𝑥))) = ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))
8173, 80eqtr3d 2778 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (((i · (2 · π)) · (𝑚 · 𝑥)) + ((i · (2 · π)) · (-𝑁 · 𝑥))) = ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))
8281fveq2d 6846 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘(((i · (2 · π)) · (𝑚 · 𝑥)) + ((i · (2 · π)) · (-𝑁 · 𝑥)))) = (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))))
8371, 82eqtr3d 2778 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((exp‘((i · (2 · π)) · (𝑚 · 𝑥))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))))
8483oveq2d 7373 . . . . . . . . 9 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ((exp‘((i · (2 · π)) · (𝑚 · 𝑥))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))))) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))))
8584adantr 481 . . . . . . . 8 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ((exp‘((i · (2 · π)) · (𝑚 · 𝑥))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))))) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))))
8668, 85eqtrd 2776 . . . . . . 7 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ((∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))))
8786sumeq2dv 15588 . . . . . 6 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)((∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))))
8866, 87eqtrd 2776 . . . . 5 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))))
8988sumeq2dv 15588 . . . 4 ((𝜑𝑥 ∈ (0(,)1)) → Σ𝑚 ∈ (0...(𝑆 · 𝑁))(Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))))
9014, 64, 893eqtrd 2780 . . 3 ((𝜑𝑥 ∈ (0(,)1)) → (∏𝑎 ∈ (0..^𝑆)(((𝐿𝑎)vts𝑁)‘𝑥) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))))
9190itgeq2dv 25146 . 2 (𝜑 → ∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)(((𝐿𝑎)vts𝑁)‘𝑥) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥 = ∫(0(,)1)Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥)
92 ioombl 24929 . . . . 5 (0(,)1) ∈ dom vol
9392a1i 11 . . . 4 (𝜑 → (0(,)1) ∈ dom vol)
94 fzfid 13878 . . . 4 (𝜑 → (0...(𝑆 · 𝑁)) ∈ Fin)
95 sumex 15572 . . . . 5 Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ V
9695a1i 11 . . . 4 ((𝜑 ∧ (𝑥 ∈ (0(,)1) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁)))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ V)
9793adantr 481 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (0(,)1) ∈ dom vol)
9829a1i 11 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ⊆ ℕ)
999adantr 481 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ0)
100 fzfid 13878 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
10198, 32, 99, 100reprfi 33229 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
10237a1i 11 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
10352adantllr 717 . . . . . . . . 9 (((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
104102, 103fprodcl 15835 . . . . . . . 8 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
10556, 76subcld 11512 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚𝑁) ∈ ℂ)
106105, 57mulcld 11175 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚𝑁) · 𝑥) ∈ ℂ)
10755, 106mulcld 11175 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)) ∈ ℂ)
108107an32s 650 . . . . . . . . . 10 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) → ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)) ∈ ℂ)
109108adantr 481 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)) ∈ ℂ)
110109efcld 33204 . . . . . . . 8 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ ℂ)
111104, 110mulcld 11175 . . . . . . 7 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ ℂ)
112111anasss 467 . . . . . 6 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ (𝑥 ∈ (0(,)1) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚))) → (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ ℂ)
11337a1i 11 . . . . . . . 8 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
114113, 52fprodcl 15835 . . . . . . 7 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
115 fvex 6855 . . . . . . . 8 (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ V
116115a1i 11 . . . . . . 7 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ V)
117 ioossicc 13350 . . . . . . . . . 10 (0(,)1) ⊆ (0[,]1)
118117a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (0(,)1) ⊆ (0[,]1))
11992a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (0(,)1) ∈ dom vol)
120115a1i 11 . . . . . . . . 9 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0[,]1)) → (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ V)
121 0red 11158 . . . . . . . . . 10 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 0 ∈ ℝ)
122 1red 11156 . . . . . . . . . 10 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 1 ∈ ℝ)
12322adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑁 ∈ ℂ)
12441, 123subcld 11512 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚𝑁) ∈ ℂ)
125 unitsscn 13417 . . . . . . . . . . . . . 14 (0[,]1) ⊆ ℂ
126125a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (0[,]1) ⊆ ℂ)
127 ssidd 3967 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ℂ ⊆ ℂ)
128 cncfmptc 24275 . . . . . . . . . . . . 13 (((𝑚𝑁) ∈ ℂ ∧ (0[,]1) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]1) ↦ (𝑚𝑁)) ∈ ((0[,]1)–cn→ℂ))
129124, 126, 127, 128syl3anc 1371 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (𝑚𝑁)) ∈ ((0[,]1)–cn→ℂ))
130 cncfmptid 24276 . . . . . . . . . . . . 13 (((0[,]1) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ ((0[,]1)–cn→ℂ))
131126, 127, 130syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ ((0[,]1)–cn→ℂ))
132129, 131mulcncf 24810 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ ((𝑚𝑁) · 𝑥)) ∈ ((0[,]1)–cn→ℂ))
133132efmul2picn 33209 . . . . . . . . . 10 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ ((0[,]1)–cn→ℂ))
134 cniccibl 25205 . . . . . . . . . 10 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ ((0[,]1)–cn→ℂ)) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ 𝐿1)
135121, 122, 133, 134syl3anc 1371 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ 𝐿1)
136118, 119, 120, 135iblss 25169 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0(,)1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ 𝐿1)
137136adantr 481 . . . . . . 7 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (𝑥 ∈ (0(,)1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ 𝐿1)
138114, 116, 137iblmulc2 25195 . . . . . 6 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (𝑥 ∈ (0(,)1) ↦ (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))))) ∈ 𝐿1)
13997, 101, 112, 138itgfsum 25191 . . . . 5 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑥 ∈ (0(,)1) ↦ Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))))) ∈ 𝐿1 ∧ ∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥 = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥))
140139simpld 495 . . . 4 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0(,)1) ↦ Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))))) ∈ 𝐿1)
14193, 94, 96, 140itgfsum 25191 . . 3 (𝜑 → ((𝑥 ∈ (0(,)1) ↦ Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))))) ∈ 𝐿1 ∧ ∫(0(,)1)Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥 = Σ𝑚 ∈ (0...(𝑆 · 𝑁))∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥))
142141simprd 496 . 2 (𝜑 → ∫(0(,)1)Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥 = Σ𝑚 ∈ (0...(𝑆 · 𝑁))∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥)
143 oveq2 7365 . . . . . . 7 (if((𝑚𝑁) = 0, 1, 0) = 1 → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · if((𝑚𝑁) = 0, 1, 0)) = (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · 1))
144 oveq2 7365 . . . . . . 7 (if((𝑚𝑁) = 0, 1, 0) = 0 → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · if((𝑚𝑁) = 0, 1, 0)) = (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · 0))
145101, 114fsumcl 15618 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
146145mulid1d 11172 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · 1) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
147145mul01d 11354 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · 0) = 0)
148143, 144, 146, 147ifeq3da 31468 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → if((𝑚𝑁) = 0, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)), 0) = (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · if((𝑚𝑁) = 0, 1, 0)))
149 velsn 4602 . . . . . . . 8 (𝑚 ∈ {𝑁} ↔ 𝑚 = 𝑁)
15041, 123subeq0ad 11522 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚𝑁) = 0 ↔ 𝑚 = 𝑁))
151149, 150bitr4id 289 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 ∈ {𝑁} ↔ (𝑚𝑁) = 0))
152151ifbid 4509 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → if(𝑚 ∈ {𝑁}, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)), 0) = if((𝑚𝑁) = 0, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)), 0))
1531nn0zd 12525 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
154153ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑁 ∈ ℤ)
15546, 154zsubcld 12612 . . . . . . . . . 10 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (𝑚𝑁) ∈ ℤ)
156 itgexpif 33219 . . . . . . . . . 10 ((𝑚𝑁) ∈ ℤ → ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥 = if((𝑚𝑁) = 0, 1, 0))
157155, 156syl 17 . . . . . . . . 9 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥 = if((𝑚𝑁) = 0, 1, 0))
158157oveq2d 7373 . . . . . . . 8 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · if((𝑚𝑁) = 0, 1, 0)))
159158sumeq2dv 15588 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · if((𝑚𝑁) = 0, 1, 0)))
160 1cnd 11150 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 1 ∈ ℂ)
161 0cnd 11148 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 0 ∈ ℂ)
162160, 161ifcld 4532 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → if((𝑚𝑁) = 0, 1, 0) ∈ ℂ)
163101, 162, 114fsummulc1 15670 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · if((𝑚𝑁) = 0, 1, 0)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · if((𝑚𝑁) = 0, 1, 0)))
164159, 163eqtr4d 2779 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥) = (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · if((𝑚𝑁) = 0, 1, 0)))
165148, 152, 1643eqtr4rd 2787 . . . . 5 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥) = if(𝑚 ∈ {𝑁}, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)), 0))
166165sumeq2dv 15588 . . . 4 (𝜑 → Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))if(𝑚 ∈ {𝑁}, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)), 0))
167 0zd 12511 . . . . . . 7 (𝜑 → 0 ∈ ℤ)
1689nn0zd 12525 . . . . . . . 8 (𝜑𝑆 ∈ ℤ)
169168, 153zmulcld 12613 . . . . . . 7 (𝜑 → (𝑆 · 𝑁) ∈ ℤ)
1701nn0ge0d 12476 . . . . . . 7 (𝜑 → 0 ≤ 𝑁)
171 nnmulge 31655 . . . . . . . 8 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (𝑆 · 𝑁))
1728, 1, 171syl2anc 584 . . . . . . 7 (𝜑𝑁 ≤ (𝑆 · 𝑁))
173167, 169, 153, 170, 172elfzd 13432 . . . . . 6 (𝜑𝑁 ∈ (0...(𝑆 · 𝑁)))
174173snssd 4769 . . . . 5 (𝜑 → {𝑁} ⊆ (0...(𝑆 · 𝑁)))
175174sselda 3944 . . . . . . 7 ((𝜑𝑚 ∈ {𝑁}) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
176175, 145syldan 591 . . . . . 6 ((𝜑𝑚 ∈ {𝑁}) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
177176ralrimiva 3143 . . . . 5 (𝜑 → ∀𝑚 ∈ {𝑁𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
17894olcd 872 . . . . 5 (𝜑 → ((0...(𝑆 · 𝑁)) ⊆ (ℤ‘0) ∨ (0...(𝑆 · 𝑁)) ∈ Fin))
179 sumss2 15611 . . . . 5 ((({𝑁} ⊆ (0...(𝑆 · 𝑁)) ∧ ∀𝑚 ∈ {𝑁𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ) ∧ ((0...(𝑆 · 𝑁)) ⊆ (ℤ‘0) ∨ (0...(𝑆 · 𝑁)) ∈ Fin)) → Σ𝑚 ∈ {𝑁𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))if(𝑚 ∈ {𝑁}, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)), 0))
180174, 177, 178, 179syl21anc 836 . . . 4 (𝜑 → Σ𝑚 ∈ {𝑁𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))if(𝑚 ∈ {𝑁}, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)), 0))
18129a1i 11 . . . . . . 7 (𝜑 → (1...𝑁) ⊆ ℕ)
182 fzfid 13878 . . . . . . 7 (𝜑 → (1...𝑁) ∈ Fin)
183181, 153, 9, 182reprfi 33229 . . . . . 6 (𝜑 → ((1...𝑁)(repr‘𝑆)𝑁) ∈ Fin)
18437a1i 11 . . . . . . 7 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → (0..^𝑆) ∈ Fin)
1851ad2antrr 724 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℕ0)
1869ad2antrr 724 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑆 ∈ ℕ0)
18722ad2antrr 724 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℂ)
18811ad2antrr 724 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
189 simpr 485 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
19029a1i 11 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → (1...𝑁) ⊆ ℕ)
191153adantr 481 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑁 ∈ ℤ)
1929adantr 481 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑆 ∈ ℕ0)
193 simpr 485 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁))
194190, 191, 192, 193reprf 33225 . . . . . . . . . 10 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
195194ffvelcdmda 7035 . . . . . . . . 9 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ (1...𝑁))
19629, 195sselid 3942 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ ℕ)
197185, 186, 187, 188, 189, 196breprexplemb 33244 . . . . . . 7 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
198184, 197fprodcl 15835 . . . . . 6 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
199183, 198fsumcl 15618 . . . . 5 (𝜑 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
200 oveq2 7365 . . . . . . 7 (𝑚 = 𝑁 → ((1...𝑁)(repr‘𝑆)𝑚) = ((1...𝑁)(repr‘𝑆)𝑁))
201200sumeq1d 15586 . . . . . 6 (𝑚 = 𝑁 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
202201sumsn 15631 . . . . 5 ((𝑁 ∈ ℕ0 ∧ Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ) → Σ𝑚 ∈ {𝑁𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
2031, 199, 202syl2anc 584 . . . 4 (𝜑 → Σ𝑚 ∈ {𝑁𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
204166, 180, 2033eqtr2d 2782 . . 3 (𝜑 → Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
205139simprd 496 . . . . 5 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥 = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥)
206110an32s 650 . . . . . . 7 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ ℂ)
207114, 206, 137itgmulc2 25198 . . . . . 6 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥) = ∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥)
208207sumeq2dv 15588 . . . . 5 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥)
209205, 208eqtr4d 2779 . . . 4 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥 = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥))
210209sumeq2dv 15588 . . 3 (𝜑 → Σ𝑚 ∈ (0...(𝑆 · 𝑁))∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥 = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥))
2111, 9reprfz1 33237 . . . 4 (𝜑 → (ℕ(repr‘𝑆)𝑁) = ((1...𝑁)(repr‘𝑆)𝑁))
212211sumeq1d 15586 . . 3 (𝜑 → Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
213204, 210, 2123eqtr4d 2786 . 2 (𝜑 → Σ𝑚 ∈ (0...(𝑆 · 𝑁))∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) d𝑥 = Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
21491, 142, 2133eqtrrd 2781 1 (𝜑 → Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = ∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)(((𝐿𝑎)vts𝑁)‘𝑥) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 845   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  wss 3910  ifcif 4486  {csn 4586   class class class wbr 5105  cmpt 5188  dom cdm 5633  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  Fincfn 8883  cc 11049  cr 11050  0cc0 11051  1c1 11052  ici 11053   + caddc 11054   · cmul 11056  cle 11190  cmin 11385  -cneg 11386  cn 12153  2c2 12208  0cn0 12413  cz 12499  cuz 12763  (,)cioo 13264  [,]cicc 13267  ...cfz 13424  ..^cfzo 13567  Σcsu 15570  cprod 15788  expce 15944  πcpi 15949  cnccncf 24239  volcvol 24827  𝐿1cibl 24981  citg 24982  reprcrepr 33221  vtscvts 33248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cc 10371  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-symdif 4202  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-omul 8417  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-fac 14174  df-bc 14203  df-hash 14231  df-shft 14952  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-sum 15571  df-prod 15789  df-ef 15950  df-sin 15952  df-cos 15953  df-pi 15955  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-haus 22666  df-cmp 22738  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241  df-ovol 24828  df-vol 24829  df-mbf 24983  df-itg1 24984  df-itg2 24985  df-ibl 24986  df-itg 24987  df-0p 25034  df-limc 25230  df-dv 25231  df-repr 33222  df-vts 33249
This theorem is referenced by:  circlemethnat  33254  circlevma  33255  circlemethhgt  33256
  Copyright terms: Public domain W3C validator