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 34633
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 480 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → 𝑁 ∈ ℕ0)
3 ioossre 13444 . . . . . . . . 9 (0(,)1) ⊆ ℝ
4 ax-resscn 11209 . . . . . . . . 9 ℝ ⊆ ℂ
53, 4sstri 4004 . . . . . . . 8 (0(,)1) ⊆ ℂ
65a1i 11 . . . . . . 7 (𝜑 → (0(,)1) ⊆ ℂ)
76sselda 3994 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → 𝑥 ∈ ℂ)
8 circlemeth.s . . . . . . . 8 (𝜑𝑆 ∈ ℕ)
98nnnn0d 12584 . . . . . . 7 (𝜑𝑆 ∈ ℕ0)
109adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → 𝑆 ∈ ℕ0)
11 circlemeth.l . . . . . . 7 (𝜑𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
1211adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → 𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
132, 7, 10, 12vtsprod 34632 . . . . 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 7445 . . . 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 14010 . . . . 5 ((𝜑𝑥 ∈ (0(,)1)) → (0...(𝑆 · 𝑁)) ∈ Fin)
16 ax-icn 11211 . . . . . . . . 9 i ∈ ℂ
17 2cn 12338 . . . . . . . . . 10 2 ∈ ℂ
18 picn 26515 . . . . . . . . . 10 π ∈ ℂ
1917, 18mulcli 11265 . . . . . . . . 9 (2 · π) ∈ ℂ
2016, 19mulcli 11265 . . . . . . . 8 (i · (2 · π)) ∈ ℂ
2120a1i 11 . . . . . . 7 ((𝜑𝑥 ∈ (0(,)1)) → (i · (2 · π)) ∈ ℂ)
221nn0cnd 12586 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℂ)
2322negcld 11604 . . . . . . . . . 10 (𝜑 → -𝑁 ∈ ℂ)
2423ralrimivw 3147 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ (0(,)1)-𝑁 ∈ ℂ)
2524r19.21bi 3248 . . . . . . . 8 ((𝜑𝑥 ∈ (0(,)1)) → -𝑁 ∈ ℂ)
2625, 7mulcld 11278 . . . . . . 7 ((𝜑𝑥 ∈ (0(,)1)) → (-𝑁 · 𝑥) ∈ ℂ)
2721, 26mulcld 11278 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → ((i · (2 · π)) · (-𝑁 · 𝑥)) ∈ ℂ)
2827efcld 16115 . . . . 5 ((𝜑𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
29 fz1ssnn 13591 . . . . . . . 8 (1...𝑁) ⊆ ℕ
3029a1i 11 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ⊆ ℕ)
31 simpr 484 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
3231elfzelzd 13561 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
3332adantlr 715 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
3410adantr 480 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ0)
35 fzfid 14010 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
3630, 33, 34, 35reprfi 34609 . . . . . 6 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
37 fzofi 14011 . . . . . . . . 9 (0..^𝑆) ∈ Fin
3837a1i 11 . . . . . . . 8 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
391ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℕ0)
409ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑆 ∈ ℕ0)
4132zcnd 12720 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℂ)
4241ad2antrr 726 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑚 ∈ ℂ)
4311ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
44 simpr 484 . . . . . . . . . 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 480 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑚 ∈ ℤ)
479ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑆 ∈ ℕ0)
48 simpr 484 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚))
4945, 46, 47, 48reprf 34605 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
5049ffvelcdmda 7103 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ (1...𝑁))
5129, 50sselid 3992 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ ℕ)
5239, 40, 42, 43, 44, 51breprexplemb 34624 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
5352adantl3r 750 . . . . . . . 8 (((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
5438, 53fprodcl 15984 . . . . . . 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 12720 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℂ)
577adantr 480 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑥 ∈ ℂ)
5856, 57mulcld 11278 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 · 𝑥) ∈ ℂ)
5955, 58mulcld 11278 . . . . . . . . 9 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · (𝑚 · 𝑥)) ∈ ℂ)
6059efcld 16115 . . . . . . . 8 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘((i · (2 · π)) · (𝑚 · 𝑥))) ∈ ℂ)
6160adantr 480 . . . . . . 7 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · (𝑚 · 𝑥))) ∈ ℂ)
6254, 61mulcld 11278 . . . . . 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 15765 . . . . 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 15817 . . . 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 480 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
6636, 65, 62fsummulc1 15817 . . . . . 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 480 . . . . . . . . 9 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
6854, 61, 67mulassd 11281 . . . . . . . 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 480 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · (-𝑁 · 𝑥)) ∈ ℂ)
70 efadd 16126 . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (-𝑁 · 𝑥) ∈ ℂ)
7355, 58, 72adddid 11282 . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → -𝑁 ∈ ℂ)
7556, 74, 57adddird 11283 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 + -𝑁) · 𝑥) = ((𝑚 · 𝑥) + (-𝑁 · 𝑥)))
7622ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑁 ∈ ℂ)
7756, 76negsubd 11623 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 + -𝑁) = (𝑚𝑁))
7877oveq1d 7445 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 + -𝑁) · 𝑥) = ((𝑚𝑁) · 𝑥))
7975, 78eqtr3d 2776 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 · 𝑥) + (-𝑁 · 𝑥)) = ((𝑚𝑁) · 𝑥))
8079oveq2d 7446 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · ((𝑚 · 𝑥) + (-𝑁 · 𝑥))) = ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))
8173, 80eqtr3d 2776 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (((i · (2 · π)) · (𝑚 · 𝑥)) + ((i · (2 · π)) · (-𝑁 · 𝑥))) = ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))
8281fveq2d 6910 . . . . . . . . . . 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 2776 . . . . . . . . . 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 7446 . . . . . . . . 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 480 . . . . . . . 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 2774 . . . . . . 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 15734 . . . . . 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 2774 . . . . 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 15734 . . . 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 2778 . . 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 25831 . 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 25613 . . . . 5 (0(,)1) ∈ dom vol
9392a1i 11 . . . 4 (𝜑 → (0(,)1) ∈ dom vol)
94 fzfid 14010 . . . 4 (𝜑 → (0...(𝑆 · 𝑁)) ∈ Fin)
95 sumex 15720 . . . . 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 480 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (0(,)1) ∈ dom vol)
9829a1i 11 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ⊆ ℕ)
999adantr 480 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ0)
100 fzfid 14010 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
10198, 32, 99, 100reprfi 34609 . . . . . 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 719 . . . . . . . . 9 (((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
104102, 103fprodcl 15984 . . . . . . . 8 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
10556, 76subcld 11617 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚𝑁) ∈ ℂ)
106105, 57mulcld 11278 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚𝑁) · 𝑥) ∈ ℂ)
10755, 106mulcld 11278 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)) ∈ ℂ)
108107an32s 652 . . . . . . . . . 10 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) → ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)) ∈ ℂ)
109108adantr 480 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)) ∈ ℂ)
110109efcld 16115 . . . . . . . 8 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ ℂ)
111104, 110mulcld 11278 . . . . . . 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 466 . . . . . 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 15984 . . . . . . 7 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
115 fvex 6919 . . . . . . . 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 13469 . . . . . . . . . 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 11261 . . . . . . . . . 10 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 0 ∈ ℝ)
122 1red 11259 . . . . . . . . . 10 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 1 ∈ ℝ)
12322adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑁 ∈ ℂ)
12441, 123subcld 11617 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚𝑁) ∈ ℂ)
125 unitsscn 13536 . . . . . . . . . . . . . 14 (0[,]1) ⊆ ℂ
126125a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (0[,]1) ⊆ ℂ)
127 ssidd 4018 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ℂ ⊆ ℂ)
128 cncfmptc 24951 . . . . . . . . . . . . 13 (((𝑚𝑁) ∈ ℂ ∧ (0[,]1) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]1) ↦ (𝑚𝑁)) ∈ ((0[,]1)–cn→ℂ))
129124, 126, 127, 128syl3anc 1370 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (𝑚𝑁)) ∈ ((0[,]1)–cn→ℂ))
130 cncfmptid 24952 . . . . . . . . . . . . 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 25493 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ ((𝑚𝑁) · 𝑥)) ∈ ((0[,]1)–cn→ℂ))
133132efmul2picn 34589 . . . . . . . . . 10 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ ((0[,]1)–cn→ℂ))
134 cniccibl 25890 . . . . . . . . . 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 1370 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ 𝐿1)
136118, 119, 120, 135iblss 25854 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0(,)1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ 𝐿1)
137136adantr 480 . . . . . . 7 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (𝑥 ∈ (0(,)1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ 𝐿1)
138114, 116, 137iblmulc2 25880 . . . . . 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 25876 . . . . 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 494 . . . 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 25876 . . 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 495 . 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 7438 . . . . . . 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 7438 . . . . . . 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 15765 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
146145mulridd 11275 . . . . . . 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 11457 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · 0) = 0)
148143, 144, 146, 147ifeq3da 32566 . . . . . 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 4646 . . . . . . . 8 (𝑚 ∈ {𝑁} ↔ 𝑚 = 𝑁)
15041, 123subeq0ad 11627 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚𝑁) = 0 ↔ 𝑚 = 𝑁))
151149, 150bitr4id 290 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 ∈ {𝑁} ↔ (𝑚𝑁) = 0))
152151ifbid 4553 . . . . . 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 12636 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
154153ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑁 ∈ ℤ)
15546, 154zsubcld 12724 . . . . . . . . . 10 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (𝑚𝑁) ∈ ℤ)
156 itgexpif 34599 . . . . . . . . . 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 7446 . . . . . . . 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 15734 . . . . . . 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 11253 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 1 ∈ ℂ)
161 0cnd 11251 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 0 ∈ ℂ)
162160, 161ifcld 4576 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → if((𝑚𝑁) = 0, 1, 0) ∈ ℂ)
163101, 162, 114fsummulc1 15817 . . . . . . 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 2777 . . . . . 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 2785 . . . . 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 15734 . . . 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 12622 . . . . . . 7 (𝜑 → 0 ∈ ℤ)
1689nn0zd 12636 . . . . . . . 8 (𝜑𝑆 ∈ ℤ)
169168, 153zmulcld 12725 . . . . . . 7 (𝜑 → (𝑆 · 𝑁) ∈ ℤ)
1701nn0ge0d 12587 . . . . . . 7 (𝜑 → 0 ≤ 𝑁)
171 nnmulge 32755 . . . . . . . 8 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (𝑆 · 𝑁))
1728, 1, 171syl2anc 584 . . . . . . 7 (𝜑𝑁 ≤ (𝑆 · 𝑁))
173167, 169, 153, 170, 172elfzd 13551 . . . . . 6 (𝜑𝑁 ∈ (0...(𝑆 · 𝑁)))
174173snssd 4813 . . . . 5 (𝜑 → {𝑁} ⊆ (0...(𝑆 · 𝑁)))
175174sselda 3994 . . . . . . 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 874 . . . . 5 (𝜑 → ((0...(𝑆 · 𝑁)) ⊆ (ℤ‘0) ∨ (0...(𝑆 · 𝑁)) ∈ Fin))
179 sumss2 15758 . . . . 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 838 . . . 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 14010 . . . . . . 7 (𝜑 → (1...𝑁) ∈ Fin)
183181, 153, 9, 182reprfi 34609 . . . . . 6 (𝜑 → ((1...𝑁)(repr‘𝑆)𝑁) ∈ Fin)
18437a1i 11 . . . . . . 7 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → (0..^𝑆) ∈ Fin)
1851ad2antrr 726 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℕ0)
1869ad2antrr 726 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑆 ∈ ℕ0)
18722ad2antrr 726 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℂ)
18811ad2antrr 726 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
189 simpr 484 . . . . . . . 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 480 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑁 ∈ ℤ)
1929adantr 480 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑆 ∈ ℕ0)
193 simpr 484 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁))
194190, 191, 192, 193reprf 34605 . . . . . . . . . 10 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
195194ffvelcdmda 7103 . . . . . . . . 9 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ (1...𝑁))
19629, 195sselid 3992 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ ℕ)
197185, 186, 187, 188, 189, 196breprexplemb 34624 . . . . . . 7 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
198184, 197fprodcl 15984 . . . . . 6 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
199183, 198fsumcl 15765 . . . . 5 (𝜑 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
200 oveq2 7438 . . . . . . 7 (𝑚 = 𝑁 → ((1...𝑁)(repr‘𝑆)𝑚) = ((1...𝑁)(repr‘𝑆)𝑁))
201200sumeq1d 15732 . . . . . 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 15778 . . . . 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 2780 . . 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 495 . . . . 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 652 . . . . . . 7 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ ℂ)
207114, 206, 137itgmulc2 25883 . . . . . 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 15734 . . . . 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 2777 . . . 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 15734 . . 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 34617 . . . 4 (𝜑 → (ℕ(repr‘𝑆)𝑁) = ((1...𝑁)(repr‘𝑆)𝑁))
212211sumeq1d 15732 . . 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 2784 . 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 2779 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 395  wo 847   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  wss 3962  ifcif 4530  {csn 4630   class class class wbr 5147  cmpt 5230  dom cdm 5688  wf 6558  cfv 6562  (class class class)co 7430  m cmap 8864  Fincfn 8983  cc 11150  cr 11151  0cc0 11152  1c1 11153  ici 11154   + caddc 11155   · cmul 11157  cle 11293  cmin 11489  -cneg 11490  cn 12263  2c2 12318  0cn0 12523  cz 12610  cuz 12875  (,)cioo 13383  [,]cicc 13386  ...cfz 13543  ..^cfzo 13690  Σcsu 15718  cprod 15935  expce 16093  πcpi 16098  cnccncf 24915  volcvol 25511  𝐿1cibl 25665  citg 25666  reprcrepr 34601  vtscvts 34628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cc 10472  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-symdif 4258  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-disj 5115  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-omul 8509  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-dju 9938  df-card 9976  df-acn 9979  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ioo 13387  df-ioc 13388  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-fac 14309  df-bc 14338  df-hash 14366  df-shft 15102  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-limsup 15503  df-clim 15520  df-rlim 15521  df-sum 15719  df-prod 15936  df-ef 16099  df-sin 16101  df-cos 16102  df-pi 16104  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-topgen 17489  df-pt 17490  df-prds 17493  df-xrs 17548  df-qtop 17553  df-imas 17554  df-xps 17556  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-mulg 19098  df-cntz 19347  df-cmn 19814  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-fbas 21378  df-fg 21379  df-cnfld 21382  df-top 22915  df-topon 22932  df-topsp 22954  df-bases 22968  df-cld 23042  df-ntr 23043  df-cls 23044  df-nei 23121  df-lp 23159  df-perf 23160  df-cn 23250  df-cnp 23251  df-haus 23338  df-cmp 23410  df-tx 23585  df-hmeo 23778  df-fil 23869  df-fm 23961  df-flim 23962  df-flf 23963  df-xms 24345  df-ms 24346  df-tms 24347  df-cncf 24917  df-ovol 25512  df-vol 25513  df-mbf 25667  df-itg1 25668  df-itg2 25669  df-ibl 25670  df-itg 25671  df-0p 25718  df-limc 25915  df-dv 25916  df-repr 34602  df-vts 34629
This theorem is referenced by:  circlemethnat  34634  circlevma  34635  circlemethhgt  34636
  Copyright terms: Public domain W3C validator