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Theorem circlemeth 30846
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 12273 . . . . . . . . 9 (0(,)1) ⊆ ℝ
4 ax-resscn 10031 . . . . . . . . 9 ℝ ⊆ ℂ
53, 4sstri 3645 . . . . . . . 8 (0(,)1) ⊆ ℂ
65a1i 11 . . . . . . 7 (𝜑 → (0(,)1) ⊆ ℂ)
76sselda 3636 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → 𝑥 ∈ ℂ)
8 circlemeth.s . . . . . . . 8 (𝜑𝑆 ∈ ℕ)
98nnnn0d 11389 . . . . . . 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 30845 . . . . 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 6705 . . . 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 12812 . . . . 5 ((𝜑𝑥 ∈ (0(,)1)) → (0...(𝑆 · 𝑁)) ∈ Fin)
16 ax-icn 10033 . . . . . . . . 9 i ∈ ℂ
17 2cn 11129 . . . . . . . . . 10 2 ∈ ℂ
18 picn 24256 . . . . . . . . . 10 π ∈ ℂ
1917, 18mulcli 10083 . . . . . . . . 9 (2 · π) ∈ ℂ
2016, 19mulcli 10083 . . . . . . . 8 (i · (2 · π)) ∈ ℂ
2120a1i 11 . . . . . . 7 ((𝜑𝑥 ∈ (0(,)1)) → (i · (2 · π)) ∈ ℂ)
221nn0cnd 11391 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℂ)
2322negcld 10417 . . . . . . . . . 10 (𝜑 → -𝑁 ∈ ℂ)
2423ralrimivw 2996 . . . . . . . . 9 (𝜑 → ∀𝑥 ∈ (0(,)1)-𝑁 ∈ ℂ)
2524r19.21bi 2961 . . . . . . . 8 ((𝜑𝑥 ∈ (0(,)1)) → -𝑁 ∈ ℂ)
2625, 7mulcld 10098 . . . . . . 7 ((𝜑𝑥 ∈ (0(,)1)) → (-𝑁 · 𝑥) ∈ ℂ)
2721, 26mulcld 10098 . . . . . 6 ((𝜑𝑥 ∈ (0(,)1)) → ((i · (2 · π)) · (-𝑁 · 𝑥)) ∈ ℂ)
2827efcld 30797 . . . . 5 ((𝜑𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
29 fz1ssnn 12410 . . . . . . . 8 (1...𝑁) ⊆ ℕ
3029a1i 11 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ⊆ ℕ)
31 fzssz 12381 . . . . . . . . 9 (0...(𝑆 · 𝑁)) ⊆ ℤ
32 simpr 476 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
3331, 32sseldi 3634 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
3433adantlr 751 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
3510adantr 480 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ0)
36 fzfid 12812 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
3730, 34, 35, 36reprfi 30822 . . . . . 6 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
38 fzofi 12813 . . . . . . . . 9 (0..^𝑆) ∈ Fin
3938a1i 11 . . . . . . . 8 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
401ad3antrrr 766 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℕ0)
419ad3antrrr 766 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑆 ∈ ℕ0)
4233zcnd 11521 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℂ)
4342ad2antrr 762 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑚 ∈ ℂ)
4411ad3antrrr 766 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))
45 simpr 476 . . . . . . . . . 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 480 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑚 ∈ ℤ)
489ad2antrr 762 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑆 ∈ ℕ0)
49 simpr 476 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚))
5046, 47, 48, 49reprf 30818 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
5150ffvelrnda 6399 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ (1...𝑁))
5229, 51sseldi 3634 . . . . . . . . . 10 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ ℕ)
5340, 41, 43, 44, 45, 52breprexplemb 30837 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
5453adantl3r 801 . . . . . . . 8 (((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
5539, 54fprodcl 14726 . . . . . . 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 11521 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℂ)
587adantr 480 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑥 ∈ ℂ)
5957, 58mulcld 10098 . . . . . . . . . 10 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 · 𝑥) ∈ ℂ)
6056, 59mulcld 10098 . . . . . . . . 9 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · (𝑚 · 𝑥)) ∈ ℂ)
6160efcld 30797 . . . . . . . 8 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘((i · (2 · π)) · (𝑚 · 𝑥))) ∈ ℂ)
6261adantr 480 . . . . . . 7 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · (𝑚 · 𝑥))) ∈ ℂ)
6355, 62mulcld 10098 . . . . . 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 14508 . . . . 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 14561 . . . 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 480 . . . . . . 7 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
6737, 66, 63fsummulc1 14561 . . . . . 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 480 . . . . . . . . 9 ((((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
6955, 62, 68mulassd 10101 . . . . . . . 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 480 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · (-𝑁 · 𝑥)) ∈ ℂ)
71 efadd 14868 . . . . . . . . . . . 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 694 . . . . . . . . . . 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 480 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (-𝑁 · 𝑥) ∈ ℂ)
7456, 59, 73adddid 10102 . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → -𝑁 ∈ ℂ)
7657, 75, 58adddird 10103 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 + -𝑁) · 𝑥) = ((𝑚 · 𝑥) + (-𝑁 · 𝑥)))
7722ad2antrr 762 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑁 ∈ ℂ)
7857, 77negsubd 10436 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 + -𝑁) = (𝑚𝑁))
7978oveq1d 6705 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 + -𝑁) · 𝑥) = ((𝑚𝑁) · 𝑥))
8076, 79eqtr3d 2687 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 · 𝑥) + (-𝑁 · 𝑥)) = ((𝑚𝑁) · 𝑥))
8180oveq2d 6706 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · ((𝑚 · 𝑥) + (-𝑁 · 𝑥))) = ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))
8274, 81eqtr3d 2687 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (((i · (2 · π)) · (𝑚 · 𝑥)) + ((i · (2 · π)) · (-𝑁 · 𝑥))) = ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))
8382fveq2d 6233 . . . . . . . . . . 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 2687 . . . . . . . . . 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 6706 . . . . . . . . 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 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)))))
8769, 86eqtrd 2685 . . . . . . 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 14477 . . . . . 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 2685 . . . . 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 14477 . . . 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 2689 . . 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 23593 . 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 23379 . . . . 5 (0(,)1) ∈ dom vol
9493a1i 11 . . . 4 (𝜑 → (0(,)1) ∈ dom vol)
95 fzfid 12812 . . . 4 (𝜑 → (0...(𝑆 · 𝑁)) ∈ Fin)
96 sumex 14462 . . . . 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 480 . . . . . 6 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (0(,)1) ∈ dom vol)
9929a1i 11 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ⊆ ℕ)
1009adantr 480 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ0)
101 fzfid 12812 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
10299, 33, 100, 101reprfi 30822 . . . . . 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 755 . . . . . . . . 9 (((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
105103, 104fprodcl 14726 . . . . . . . 8 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
10657, 77subcld 10430 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚𝑁) ∈ ℂ)
107106, 58mulcld 10098 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚𝑁) · 𝑥) ∈ ℂ)
10856, 107mulcld 10098 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)) ∈ ℂ)
109108an32s 863 . . . . . . . . . 10 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) → ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)) ∈ ℂ)
110109adantr 480 . . . . . . . . 9 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ((i · (2 · π)) · ((𝑚𝑁) · 𝑥)) ∈ ℂ)
111110efcld 30797 . . . . . . . 8 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ ℂ)
112105, 111mulcld 10098 . . . . . . 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 680 . . . . . 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 14726 . . . . . . 7 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
116 fvex 6239 . . . . . . . 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 12297 . . . . . . . . . 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 10079 . . . . . . . . . 10 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 0 ∈ ℝ)
123 1red 10093 . . . . . . . . . 10 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 1 ∈ ℝ)
12422adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑁 ∈ ℂ)
12542, 124subcld 10430 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚𝑁) ∈ ℂ)
126 unitsscn 30070 . . . . . . . . . . . . . 14 (0[,]1) ⊆ ℂ
127126a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (0[,]1) ⊆ ℂ)
128 ssid 3657 . . . . . . . . . . . . . 14 ℂ ⊆ ℂ
129128a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ℂ ⊆ ℂ)
130 cncfmptc 22761 . . . . . . . . . . . . 13 (((𝑚𝑁) ∈ ℂ ∧ (0[,]1) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]1) ↦ (𝑚𝑁)) ∈ ((0[,]1)–cn→ℂ))
131125, 127, 129, 130syl3anc 1366 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (𝑚𝑁)) ∈ ((0[,]1)–cn→ℂ))
132 cncfmptid 22762 . . . . . . . . . . . . 13 (((0[,]1) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ ((0[,]1)–cn→ℂ))
133127, 129, 132syl2anc 694 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ ((0[,]1)–cn→ℂ))
134131, 133mulcncf 23261 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ ((𝑚𝑁) · 𝑥)) ∈ ((0[,]1)–cn→ℂ))
135134efmul2picn 30802 . . . . . . . . . 10 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ ((0[,]1)–cn→ℂ))
136 cniccibl 23652 . . . . . . . . . 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)
137122, 123, 135, 136syl3anc 1366 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ 𝐿1)
138119, 120, 121, 137iblss 23616 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0(,)1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ 𝐿1)
139138adantr 480 . . . . . . 7 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (𝑥 ∈ (0(,)1) ↦ (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥)))) ∈ 𝐿1)
140115, 117, 139iblmulc2 23642 . . . . . 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)
14198, 102, 113, 140itgfsum 23638 . . . . 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))
142141simpld 474 . . . 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)
14394, 95, 97, 142itgfsum 23638 . . 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))
144143simprd 478 . 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)
145 oveq2 6698 . . . . . . 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))
146 oveq2 6698 . . . . . . 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))
147102, 115fsumcl 14508 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
148147mulid1d 10095 . . . . . . 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)))
149147mul01d 10273 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · 0) = 0)
150145, 146, 148, 149ifeq3da 29491 . . . . . 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)))
15142, 124subeq0ad 10440 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚𝑁) = 0 ↔ 𝑚 = 𝑁))
152 velsn 4226 . . . . . . . 8 (𝑚 ∈ {𝑁} ↔ 𝑚 = 𝑁)
153151, 152syl6rbbr 279 . . . . . . 7 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 ∈ {𝑁} ↔ (𝑚𝑁) = 0))
154153ifbid 4141 . . . . . 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))
1551nn0zd 11518 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
156155ad2antrr 762 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑁 ∈ ℤ)
15747, 156zsubcld 11525 . . . . . . . . . 10 (((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (𝑚𝑁) ∈ ℤ)
158 itgexpif 30812 . . . . . . . . . 10 ((𝑚𝑁) ∈ ℤ → ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) d𝑥 = if((𝑚𝑁) = 0, 1, 0))
159157, 158syl 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))
160159oveq2d 6706 . . . . . . . 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)))
161160sumeq2dv 14477 . . . . . . 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)))
162 1cnd 10094 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 1 ∈ ℂ)
163 0cnd 10071 . . . . . . . . 9 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → 0 ∈ ℂ)
164162, 163ifcld 4164 . . . . . . . 8 ((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) → if((𝑚𝑁) = 0, 1, 0) ∈ ℂ)
165102, 164, 115fsummulc1 14561 . . . . . . 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)))
166161, 165eqtr4d 2688 . . . . . 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)))
167150, 154, 1663eqtr4rd 2696 . . . . 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))
168167sumeq2dv 14477 . . . 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))
169 0zd 11427 . . . . . . 7 (𝜑 → 0 ∈ ℤ)
1709nn0zd 11518 . . . . . . . 8 (𝜑𝑆 ∈ ℤ)
171170, 155zmulcld 11526 . . . . . . 7 (𝜑 → (𝑆 · 𝑁) ∈ ℤ)
1721nn0ge0d 11392 . . . . . . 7 (𝜑 → 0 ≤ 𝑁)
173 nnmulge 29643 . . . . . . . 8 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (𝑆 · 𝑁))
1748, 1, 173syl2anc 694 . . . . . . 7 (𝜑𝑁 ≤ (𝑆 · 𝑁))
175 elfz4 12373 . . . . . . 7 (((0 ∈ ℤ ∧ (𝑆 · 𝑁) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (0 ≤ 𝑁𝑁 ≤ (𝑆 · 𝑁))) → 𝑁 ∈ (0...(𝑆 · 𝑁)))
176169, 171, 155, 172, 174, 175syl32anc 1374 . . . . . 6 (𝜑𝑁 ∈ (0...(𝑆 · 𝑁)))
177176snssd 4372 . . . . 5 (𝜑 → {𝑁} ⊆ (0...(𝑆 · 𝑁)))
178177sselda 3636 . . . . . . 7 ((𝜑𝑚 ∈ {𝑁}) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
179178, 147syldan 486 . . . . . 6 ((𝜑𝑚 ∈ {𝑁}) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
180179ralrimiva 2995 . . . . 5 (𝜑 → ∀𝑚 ∈ {𝑁𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
18195olcd 407 . . . . 5 (𝜑 → ((0...(𝑆 · 𝑁)) ⊆ (ℤ‘0) ∨ (0...(𝑆 · 𝑁)) ∈ Fin))
182 sumss2 14501 . . . . 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))
183177, 180, 181, 182syl21anc 1365 . . . 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))
18429a1i 11 . . . . . . 7 (𝜑 → (1...𝑁) ⊆ ℕ)
185 fzfid 12812 . . . . . . 7 (𝜑 → (1...𝑁) ∈ Fin)
186184, 155, 9, 185reprfi 30822 . . . . . 6 (𝜑 → ((1...𝑁)(repr‘𝑆)𝑁) ∈ Fin)
18738a1i 11 . . . . . . 7 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → (0..^𝑆) ∈ Fin)
1881ad2antrr 762 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℕ0)
1899ad2antrr 762 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑆 ∈ ℕ0)
19022ad2antrr 762 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℂ)
19111ad2antrr 762 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))
192 simpr 476 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
19329a1i 11 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → (1...𝑁) ⊆ ℕ)
194155adantr 480 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑁 ∈ ℤ)
1959adantr 480 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑆 ∈ ℕ0)
196 simpr 476 . . . . . . . . . . 11 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁))
197193, 194, 195, 196reprf 30818 . . . . . . . . . 10 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
198197ffvelrnda 6399 . . . . . . . . 9 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ (1...𝑁))
19929, 198sseldi 3634 . . . . . . . 8 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐𝑎) ∈ ℕ)
200188, 189, 190, 191, 192, 199breprexplemb 30837 . . . . . . 7 (((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
201187, 200fprodcl 14726 . . . . . 6 ((𝜑𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
202186, 201fsumcl 14508 . . . . 5 (𝜑 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) ∈ ℂ)
203 oveq2 6698 . . . . . . 7 (𝑚 = 𝑁 → ((1...𝑁)(repr‘𝑆)𝑚) = ((1...𝑁)(repr‘𝑆)𝑁))
204203sumeq1d 14475 . . . . . 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)))
205204sumsn 14519 . . . . 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)))
2061, 202, 205syl2anc 694 . . . 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)))
207168, 183, 2063eqtr2d 2691 . . 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)))
208141simprd 478 . . . . 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)
209111an32s 863 . . . . . . 7 ((((𝜑𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · ((𝑚𝑁) · 𝑥))) ∈ ℂ)
210115, 209, 139itgmulc2 23645 . . . . . 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)
211210sumeq2dv 14477 . . . . 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)
212208, 211eqtr4d 2688 . . . 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))
213212sumeq2dv 14477 . . 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))
2141, 9reprfz1 30830 . . . 4 (𝜑 → (ℕ(repr‘𝑆)𝑁) = ((1...𝑁)(repr‘𝑆)𝑁))
215214sumeq1d 14475 . . 3 (𝜑 → Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)))
216207, 213, 2153eqtr4d 2695 . 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)))
21792, 144, 2163eqtrrd 2690 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  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  wss 3607  ifcif 4119  {csn 4210   class class class wbr 4685  cmpt 4762  dom cdm 5143  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  Fincfn 7997  cc 9972  cr 9973  0cc0 9974  1c1 9975  ici 9976   + caddc 9977   · cmul 9979  cle 10113  cmin 10304  -cneg 10305  cn 11058  2c2 11108  0cn0 11330  cz 11415  cuz 11725  (,)cioo 12213  [,]cicc 12216  ...cfz 12364  ..^cfzo 12504  Σcsu 14460  cprod 14679  expce 14836  πcpi 14841  cnccncf 22726  volcvol 23278  𝐿1cibl 23431  citg 23432  reprcrepr 30814  vtscvts 30841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cc 9295  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-sum 14461  df-prod 14680  df-ef 14842  df-sin 14844  df-cos 14845  df-pi 14847  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-cmp 21238  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-ovol 23279  df-vol 23280  df-mbf 23433  df-itg1 23434  df-itg2 23435  df-ibl 23436  df-itg 23437  df-0p 23482  df-limc 23675  df-dv 23676  df-repr 30815  df-vts 30842
This theorem is referenced by:  circlemethnat  30847  circlevma  30848  circlemethhgt  30849
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