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 31810
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 12786 . . . . . . . . 9 (0(,)1) ⊆ ℝ
4 ax-resscn 10582 . . . . . . . . 9 ℝ ⊆ ℂ
53, 4sstri 3973 . . . . . . . 8 (0(,)1) ⊆ ℂ
65a1i 11 . . . . . . 7 (𝜑 → (0(,)1) ⊆ ℂ)
76sselda 3964 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → 𝑥 ∈ ℂ)
8 circlemeth.s . . . . . . . 8 (𝜑𝑆 ∈ ℕ)
98nnnn0d 11943 . . . . . . 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 31809 . . . . 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 7160 . . . 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 13329 . . . . 5 ((𝜑𝑥 ∈ (0(,)1)) → (0...(𝑆 · 𝑁)) ∈ Fin)
16 ax-icn 10584 . . . . . . . . 9 i ∈ ℂ
17 2cn 11700 . . . . . . . . . 10 2 ∈ ℂ
18 picn 24972 . . . . . . . . . 10 π ∈ ℂ
1917, 18mulcli 10636 . . . . . . . . 9 (2 · π) ∈ ℂ
2016, 19mulcli 10636 . . . . . . . 8 (i · (2 · π)) ∈ ℂ
2120a1i 11 . . . . . . 7 ((𝜑𝑥 ∈ (0(,)1)) → (i · (2 · π)) ∈ ℂ)
221nn0cnd 11945 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℂ)
2322negcld 10972 . . . . . . . . . 10 (𝜑 → -𝑁 ∈ ℂ)
2423ralrimivw 3180 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ (0(,)1)-𝑁 ∈ ℂ)
2524r19.21bi 3205 . . . . . . . 8 ((𝜑𝑥 ∈ (0(,)1)) → -𝑁 ∈ ℂ)
2625, 7mulcld 10649 . . . . . . 7 ((𝜑𝑥 ∈ (0(,)1)) → (-𝑁 · 𝑥) ∈ ℂ)
2721, 26mulcld 10649 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → ((i · (2 · π)) · (-𝑁 · 𝑥)) ∈ ℂ)
2827efcld 31761 . . . . 5 ((𝜑𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
29 fz1ssnn 12926 . . . . . . . 8 (1...𝑁) ⊆ ℕ
3029a1i 11 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ⊆ ℕ)
31 fzssz 12897 . . . . . . . . 9 (0...(𝑆 · 𝑁)) ⊆ ℤ
32 simpr 485 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
3331, 32sseldi 3962 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
3433adantlr 711 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
3510adantr 481 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ0)
36 fzfid 13329 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
3730, 34, 35, 36reprfi 31786 . . . . . 6 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
38 fzofi 13330 . . . . . . . . 9 (0..^𝑆) ∈ Fin
3938a1i 11 . . . . . . . 8 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
401ad3antrrr 726 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℕ0)
419ad3antrrr 726 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑆 ∈ ℕ0)
4233zcnd 12076 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℂ)
4342ad2antrr 722 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑚 ∈ ℂ)
4411ad3antrrr 726 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
45 simpr 485 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
4629a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (1...𝑁) ⊆ ℕ)
4733adantr 481 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑚 ∈ ℤ)
489ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑆 ∈ ℕ0)
49 simpr 485 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚))
5046, 47, 48, 49reprf 31782 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
5150ffvelrnda 6843 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ (1...𝑁))
5229, 51sseldi 3962 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ ℕ)
5340, 41, 43, 44, 45, 52breprexplemb 31801 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
5453adantl3r 746 . . . . . . . 8 (((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
5539, 54fprodcl 15294 . . . . . . 7 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
5620a1i 11 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (i · (2 · π)) ∈ ℂ)
5734zcnd 12076 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℂ)
587adantr 481 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑥 ∈ ℂ)
5957, 58mulcld 10649 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 · 𝑥) ∈ ℂ)
6056, 59mulcld 10649 . . . . . . . . 9 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · (𝑚 · 𝑥)) ∈ ℂ)
6160efcld 31761 . . . . . . . 8 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘((i · (2 · π)) · (𝑚 · 𝑥))) ∈ ℂ)
6261adantr 481 . . . . . . 7 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · (𝑚 · 𝑥))) ∈ ℂ)
6355, 62mulcld 10649 . . . . . 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)
6437, 63fsumcl 15078 . . . . 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)
6515, 28, 64fsummulc1 15128 . . . 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)))))
6628adantr 481 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
6737, 66, 63fsummulc1 15128 . . . . . 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)))))
6866adantr 481 . . . . . . . . 9 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
6955, 62, 68mulassd 10652 . . . . . . . 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))))))
7027adantr 481 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · (-𝑁 · 𝑥)) ∈ ℂ)
71 efadd 15435 . . . . . . . . . . . 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)))))
7260, 70, 71syl2anc 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)))))
7326adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (-𝑁 · 𝑥) ∈ ℂ)
7456, 59, 73adddid 10653 . . . . . . . . . . . . 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))))
7525adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → -𝑁 ∈ ℂ)
7657, 75, 58adddird 10654 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 + -𝑁) · 𝑥) = ((𝑚 · 𝑥) + (-𝑁 · 𝑥)))
7722ad2antrr 722 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑁 ∈ ℂ)
7857, 77negsubd 10991 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 + -𝑁) = (𝑚𝑁))
7978oveq1d 7160 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 + -𝑁) · 𝑥) = ((𝑚𝑁) · 𝑥))
8076, 79eqtr3d 2855 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 · 𝑥) + (-𝑁 · 𝑥)) = ((𝑚𝑁) · 𝑥))
8180oveq2d 7161 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · ((𝑚 · 𝑥) + (-𝑁 · 𝑥))) = ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))
8274, 81eqtr3d 2855 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (((i · (2 · π)) · (𝑚 · 𝑥)) + ((i · (2 · π)) · (-𝑁 · 𝑥))) = ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))
8382fveq2d 6667 . . . . . . . . . . 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))))
8472, 83eqtr3d 2855 . . . . . . . . . 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))))
8584oveq2d 7161 . . . . . . . . 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)))))
8685adantr 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)))))
8769, 86eqtrd 2853 . . . . . . 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)))))
8887sumeq2dv 15048 . . . . . 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)))))
8967, 88eqtrd 2853 . . . . 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)))))
9089sumeq2dv 15048 . . . 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)))))
9114, 65, 903eqtrd 2857 . . 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)))))
9291itgeq2dv 24309 . 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)
93 ioombl 24093 . . . . 5 (0(,)1) ∈ dom vol
9493a1i 11 . . . 4 (𝜑 → (0(,)1) ∈ dom vol)
95 fzfid 13329 . . . 4 (𝜑 → (0...(𝑆 · 𝑁)) ∈ Fin)
96 sumex 15032 . . . . 5 Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ V
9796a1i 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)
9894adantr 481 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (0(,)1) ∈ dom vol)
9929a1i 11 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ⊆ ℕ)
1009adantr 481 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ0)
101 fzfid 13329 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
10299, 33, 100, 101reprfi 31786 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
10338a1i 11 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
10453adantllr 715 . . . . . . . . 9 (((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
105103, 104fprodcl 15294 . . . . . . . 8 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
10657, 77subcld 10985 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚𝑁) ∈ ℂ)
107106, 58mulcld 10649 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚𝑁) · 𝑥) ∈ ℂ)
10856, 107mulcld 10649 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)) ∈ ℂ)
109108an32s 648 . . . . . . . . . 10 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) → ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)) ∈ ℂ)
110109adantr 481 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)) ∈ ℂ)
111110efcld 31761 . . . . . . . 8 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ ℂ)
112105, 111mulcld 10649 . . . . . . 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)
113112anasss 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)
11438a1i 11 . . . . . . . 8 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
115114, 53fprodcl 15294 . . . . . . 7 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
116 fvex 6676 . . . . . . . 8 (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ V
117116a1i 11 . . . . . . 7 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ V)
118 ioossicc 12810 . . . . . . . . . 10 (0(,)1) ⊆ (0[,]1)
119118a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (0(,)1) ⊆ (0[,]1))
12093a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (0(,)1) ∈ dom vol)
121116a1i 11 . . . . . . . . 9 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0[,]1)) → (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ V)
122 0red 10632 . . . . . . . . . 10 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 0 ∈ ℝ)
123 1red 10630 . . . . . . . . . 10 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 1 ∈ ℝ)
12422adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑁 ∈ ℂ)
12542, 124subcld 10985 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚𝑁) ∈ ℂ)
126 unitsscn 31038 . . . . . . . . . . . . . 14 (0[,]1) ⊆ ℂ
127126a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (0[,]1) ⊆ ℂ)
128 ssidd 3987 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ℂ ⊆ ℂ)
129 cncfmptc 23446 . . . . . . . . . . . . 13 (((𝑚𝑁) ∈ ℂ ∧ (0[,]1) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]1) ↦ (𝑚𝑁)) ∈ ((0[,]1)–cn→ℂ))
130125, 127, 128, 129syl3anc 1363 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (𝑚𝑁)) ∈ ((0[,]1)–cn→ℂ))
131 cncfmptid 23447 . . . . . . . . . . . . 13 (((0[,]1) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ ((0[,]1)–cn→ℂ))
132127, 128, 131syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ ((0[,]1)–cn→ℂ))
133130, 132mulcncf 23974 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ ((𝑚𝑁) · 𝑥)) ∈ ((0[,]1)–cn→ℂ))
134133efmul2picn 31766 . . . . . . . . . 10 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ ((0[,]1)–cn→ℂ))
135 cniccibl 24368 . . . . . . . . . 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)
136122, 123, 134, 135syl3anc 1363 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ 𝐿1)
137119, 120, 121, 136iblss 24332 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0(,)1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ 𝐿1)
138137adantr 481 . . . . . . 7 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (𝑥 ∈ (0(,)1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ 𝐿1)
139115, 117, 138iblmulc2 24358 . . . . . 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)
14098, 102, 113, 139itgfsum 24354 . . . . 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))
141140simpld 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)
14294, 95, 97, 141itgfsum 24354 . . 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))
143142simprd 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)
144 oveq2 7153 . . . . . . 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))
145 oveq2 7153 . . . . . . 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))
146102, 115fsumcl 15078 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
147146mulid1d 10646 . . . . . . 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)))
148146mul01d 10827 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · 0) = 0)
149144, 145, 147, 148ifeq3da 30228 . . . . . 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)))
15042, 124subeq0ad 10995 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚𝑁) = 0 ↔ 𝑚 = 𝑁))
151 velsn 4573 . . . . . . . 8 (𝑚 ∈ {𝑁} ↔ 𝑚 = 𝑁)
152150, 151syl6rbbr 291 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 ∈ {𝑁} ↔ (𝑚𝑁) = 0))
153152ifbid 4485 . . . . . 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))
1541nn0zd 12073 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
155154ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑁 ∈ ℤ)
15647, 155zsubcld 12080 . . . . . . . . . 10 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (𝑚𝑁) ∈ ℤ)
157 itgexpif 31776 . . . . . . . . . 10 ((𝑚𝑁) ∈ ℤ → ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥 = if((𝑚𝑁) = 0, 1, 0))
158156, 157syl 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))
159158oveq2d 7161 . . . . . . . 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)))
160159sumeq2dv 15048 . . . . . . 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)))
161 1cnd 10624 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 1 ∈ ℂ)
162 0cnd 10622 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 0 ∈ ℂ)
163161, 162ifcld 4508 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → if((𝑚𝑁) = 0, 1, 0) ∈ ℂ)
164102, 163, 115fsummulc1 15128 . . . . . . 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)))
165160, 164eqtr4d 2856 . . . . . 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)))
166149, 153, 1653eqtr4rd 2864 . . . . 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))
167166sumeq2dv 15048 . . . 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))
168 0zd 11981 . . . . . . 7 (𝜑 → 0 ∈ ℤ)
1699nn0zd 12073 . . . . . . . 8 (𝜑𝑆 ∈ ℤ)
170169, 154zmulcld 12081 . . . . . . 7 (𝜑 → (𝑆 · 𝑁) ∈ ℤ)
1711nn0ge0d 11946 . . . . . . 7 (𝜑 → 0 ≤ 𝑁)
172 nnmulge 30400 . . . . . . . 8 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (𝑆 · 𝑁))
1738, 1, 172syl2anc 584 . . . . . . 7 (𝜑𝑁 ≤ (𝑆 · 𝑁))
174 elfz4 12889 . . . . . . 7 (((0 ∈ ℤ ∧ (𝑆 · 𝑁) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝑁𝑁 ≤ (𝑆 · 𝑁))) → 𝑁 ∈ (0...(𝑆 · 𝑁)))
175168, 170, 154, 171, 173, 174syl32anc 1370 . . . . . 6 (𝜑𝑁 ∈ (0...(𝑆 · 𝑁)))
176175snssd 4734 . . . . 5 (𝜑 → {𝑁} ⊆ (0...(𝑆 · 𝑁)))
177176sselda 3964 . . . . . . 7 ((𝜑𝑚 ∈ {𝑁}) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
178177, 146syldan 591 . . . . . 6 ((𝜑𝑚 ∈ {𝑁}) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
179178ralrimiva 3179 . . . . 5 (𝜑 → ∀𝑚 ∈ {𝑁𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
18095olcd 870 . . . . 5 (𝜑 → ((0...(𝑆 · 𝑁)) ⊆ (ℤ‘0) ∨ (0...(𝑆 · 𝑁)) ∈ Fin))
181 sumss2 15071 . . . . 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))
182176, 179, 180, 181syl21anc 833 . . . 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))
18329a1i 11 . . . . . . 7 (𝜑 → (1...𝑁) ⊆ ℕ)
184 fzfid 13329 . . . . . . 7 (𝜑 → (1...𝑁) ∈ Fin)
185183, 154, 9, 184reprfi 31786 . . . . . 6 (𝜑 → ((1...𝑁)(repr‘𝑆)𝑁) ∈ Fin)
18638a1i 11 . . . . . . 7 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → (0..^𝑆) ∈ Fin)
1871ad2antrr 722 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℕ0)
1889ad2antrr 722 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑆 ∈ ℕ0)
18922ad2antrr 722 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℂ)
19011ad2antrr 722 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐿:(0..^𝑆)⟶(ℂ ↑m ℕ))
191 simpr 485 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
19229a1i 11 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → (1...𝑁) ⊆ ℕ)
193154adantr 481 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑁 ∈ ℤ)
1949adantr 481 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑆 ∈ ℕ0)
195 simpr 485 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁))
196192, 193, 194, 195reprf 31782 . . . . . . . . . 10 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
197196ffvelrnda 6843 . . . . . . . . 9 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ (1...𝑁))
19829, 197sseldi 3962 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ ℕ)
199187, 188, 189, 190, 191, 198breprexplemb 31801 . . . . . . 7 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
200186, 199fprodcl 15294 . . . . . 6 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
201185, 200fsumcl 15078 . . . . 5 (𝜑 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
202 oveq2 7153 . . . . . . 7 (𝑚 = 𝑁 → ((1...𝑁)(repr‘𝑆)𝑚) = ((1...𝑁)(repr‘𝑆)𝑁))
203202sumeq1d 15046 . . . . . 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)))
204203sumsn 15089 . . . . 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)))
2051, 201, 204syl2anc 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)))
206167, 182, 2053eqtr2d 2859 . . 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)))
207140simprd 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)
208111an32s 648 . . . . . . 7 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ ℂ)
209115, 208, 138itgmulc2 24361 . . . . . 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)
210209sumeq2dv 15048 . . . . 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)
211207, 210eqtr4d 2856 . . . 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))
212211sumeq2dv 15048 . . 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))
2131, 9reprfz1 31794 . . . 4 (𝜑 → (ℕ(repr‘𝑆)𝑁) = ((1...𝑁)(repr‘𝑆)𝑁))
214213sumeq1d 15046 . . 3 (𝜑 → Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
215206, 212, 2143eqtr4d 2863 . 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)))
21692, 143, 2153eqtrrd 2858 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 841   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  wss 3933  ifcif 4463  {csn 4557   class class class wbr 5057  cmpt 5137  dom cdm 5548  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395  Fincfn 8497  cc 10523  cr 10524  0cc0 10525  1c1 10526  ici 10527   + caddc 10528   · cmul 10530  cle 10664  cmin 10858  -cneg 10859  cn 11626  2c2 11680  0cn0 11885  cz 11969  cuz 12231  (,)cioo 12726  [,]cicc 12729  ...cfz 12880  ..^cfzo 13021  Σcsu 15030  cprod 15247  expce 15403  πcpi 15408  cnccncf 23411  volcvol 23991  𝐿1cibl 24145  citg 24146  reprcrepr 31778  vtscvts 31805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cc 9845  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-addf 10604  ax-mulf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-symdif 4216  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-disj 5023  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-ofr 7399  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-omul 8096  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-acn 9359  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ioc 12731  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418  df-fac 13622  df-bc 13651  df-hash 13679  df-shft 14414  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-limsup 14816  df-clim 14833  df-rlim 14834  df-sum 15031  df-prod 15248  df-ef 15409  df-sin 15411  df-cos 15412  df-pi 15414  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-starv 16568  df-sca 16569  df-vsca 16570  df-ip 16571  df-tset 16572  df-ple 16573  df-ds 16575  df-unif 16576  df-hom 16577  df-cco 16578  df-rest 16684  df-topn 16685  df-0g 16703  df-gsum 16704  df-topgen 16705  df-pt 16706  df-prds 16709  df-xrs 16763  df-qtop 16768  df-imas 16769  df-xps 16771  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-submnd 17945  df-mulg 18163  df-cntz 18385  df-cmn 18837  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-fbas 20470  df-fg 20471  df-cnfld 20474  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-lp 21672  df-perf 21673  df-cn 21763  df-cnp 21764  df-haus 21851  df-cmp 21923  df-tx 22098  df-hmeo 22291  df-fil 22382  df-fm 22474  df-flim 22475  df-flf 22476  df-xms 22857  df-ms 22858  df-tms 22859  df-cncf 23413  df-ovol 23992  df-vol 23993  df-mbf 24147  df-itg1 24148  df-itg2 24149  df-ibl 24150  df-itg 24151  df-0p 24198  df-limc 24391  df-dv 24392  df-repr 31779  df-vts 31806
This theorem is referenced by:  circlemethnat  31811  circlevma  31812  circlemethhgt  31813
  Copyright terms: Public domain W3C validator