circlemeth - Mathbox for Thierry Arnoux

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Theorem circlemeth 32620

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	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
circlemeth.s	⊢ (𝜑 → 𝑆 ∈ ℕ)
circlemeth.l	⊢ (𝜑 → 𝐿:(0..^𝑆)⟶(ℂ ↑_m ℕ))

**Assertion**
Ref	Expression
circlemeth	⊢ (𝜑 → Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)(((𝐿‘𝑎)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.

Step	Hyp	Ref	Expression
1		circlemeth.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
2	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → 𝑁 ∈ ℕ₀)
3		ioossre 13140	. . . . . . . . 9 ⊢ (0(,)1) ⊆ ℝ
4		ax-resscn 10928	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
5	3, 4	sstri 3930	. . . . . . . 8 ⊢ (0(,)1) ⊆ ℂ
6	5	a1i 11	. . . . . . 7 ⊢ (𝜑 → (0(,)1) ⊆ ℂ)
7	6	sselda 3921	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → 𝑥 ∈ ℂ)
8		circlemeth.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℕ)
9	8	nnnn0d 12293	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ ℕ₀)
10	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → 𝑆 ∈ ℕ₀)
11		circlemeth.l	. . . . . . 7 ⊢ (𝜑 → 𝐿:(0..^𝑆)⟶(ℂ ↑_m ℕ))
12	11	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → 𝐿:(0..^𝑆)⟶(ℂ ↑_m ℕ))
13	2, 7, 10, 12	vtsprod 32619	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → ∏𝑎 ∈ (0..^𝑆)(((𝐿‘𝑎)vts𝑁)‘𝑥) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))))
14	13	oveq1d 7290	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → (∏𝑎 ∈ (0..^𝑆)(((𝐿‘𝑎)vts𝑁)‘𝑥) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = (Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))))
15		fzfid 13693	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → (0...(𝑆 · 𝑁)) ∈ Fin)
16		ax-icn 10930	. . . . . . . . 9 ⊢ i ∈ ℂ
17		2cn 12048	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
18		picn 25616	. . . . . . . . . 10 ⊢ π ∈ ℂ
19	17, 18	mulcli 10982	. . . . . . . . 9 ⊢ (2 · π) ∈ ℂ
20	16, 19	mulcli 10982	. . . . . . . 8 ⊢ (i · (2 · π)) ∈ ℂ
21	20	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → (i · (2 · π)) ∈ ℂ)
22	1	nn0cnd 12295	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℂ)
23	22	negcld 11319	. . . . . . . . . 10 ⊢ (𝜑 → -𝑁 ∈ ℂ)
24	23	ralrimivw 3104	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (0(,)1)-𝑁 ∈ ℂ)
25	24	r19.21bi 3134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → -𝑁 ∈ ℂ)
26	25, 7	mulcld 10995	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → (-𝑁 · 𝑥) ∈ ℂ)
27	21, 26	mulcld 10995	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → ((i · (2 · π)) · (-𝑁 · 𝑥)) ∈ ℂ)
28	27	efcld 32571	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
29		fz1ssnn 13287	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℕ
30	29	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ⊆ ℕ)
31		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
32	31	elfzelzd 13257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
33	32	adantlr 712	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
34	10	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ₀)
35		fzfid 13693	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
36	30, 33, 34, 35	reprfi 32596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
37		fzofi 13694	. . . . . . . . 9 ⊢ (0..^𝑆) ∈ Fin
38	37	a1i 11	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
39	1	ad3antrrr 727	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℕ₀)
40	9	ad3antrrr 727	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑆 ∈ ℕ₀)
41	32	zcnd 12427	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℂ)
42	41	ad2antrr 723	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑚 ∈ ℂ)
43	11	ad3antrrr 727	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐿:(0..^𝑆)⟶(ℂ ↑_m ℕ))
44		simpr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
45	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (1...𝑁) ⊆ ℕ)
46	32	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑚 ∈ ℤ)
47	9	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑆 ∈ ℕ₀)
48		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚))
49	45, 46, 47, 48	reprf 32592	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
50	49	ffvelrnda 6961	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ (1...𝑁))
51	29, 50	sselid 3919	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ ℕ)
52	39, 40, 42, 43, 44, 51	breprexplemb 32611	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
53	52	adantl3r 747	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
54	38, 53	fprodcl 15662	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
55	20	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (i · (2 · π)) ∈ ℂ)
56	33	zcnd 12427	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℂ)
57	7	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑥 ∈ ℂ)
58	56, 57	mulcld 10995	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 · 𝑥) ∈ ℂ)
59	55, 58	mulcld 10995	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · (𝑚 · 𝑥)) ∈ ℂ)
60	59	efcld 32571	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘((i · (2 · π)) · (𝑚 · 𝑥))) ∈ ℂ)
61	60	adantr 481	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · (𝑚 · 𝑥))) ∈ ℂ)
62	54, 61	mulcld 10995	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) ∈ ℂ)
63	36, 62	fsumcl 15445	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) ∈ ℂ)
64	15, 28, 63	fsummulc1 15497	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → (Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (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)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))))
65	28	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
66	36, 65, 62	fsummulc1 15497	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)((∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))))
67	65	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
68	54, 61, 67	mulassd 10998	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ((∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · ((exp‘((i · (2 · π)) · (𝑚 · 𝑥))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))))))
69	27	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · (-𝑁 · 𝑥)) ∈ ℂ)
70		efadd 15803	. . . . . . . . . . . 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)))))
71	59, 69, 70	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (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)))))
72	26	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (-𝑁 · 𝑥) ∈ ℂ)
73	55, 58, 72	adddid 10999	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · ((𝑚 · 𝑥) + (-𝑁 · 𝑥))) = (((i · (2 · π)) · (𝑚 · 𝑥)) + ((i · (2 · π)) · (-𝑁 · 𝑥))))
74	25	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → -𝑁 ∈ ℂ)
75	56, 74, 57	adddird 11000	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 + -𝑁) · 𝑥) = ((𝑚 · 𝑥) + (-𝑁 · 𝑥)))
76	22	ad2antrr 723	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑁 ∈ ℂ)
77	56, 76	negsubd 11338	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 + -𝑁) = (𝑚 − 𝑁))
78	77	oveq1d 7290	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 + -𝑁) · 𝑥) = ((𝑚 − 𝑁) · 𝑥))
79	75, 78	eqtr3d 2780	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 · 𝑥) + (-𝑁 · 𝑥)) = ((𝑚 − 𝑁) · 𝑥))
80	79	oveq2d 7291	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · ((𝑚 · 𝑥) + (-𝑁 · 𝑥))) = ((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))
81	73, 80	eqtr3d 2780	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (((i · (2 · π)) · (𝑚 · 𝑥)) + ((i · (2 · π)) · (-𝑁 · 𝑥))) = ((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))
82	81	fveq2d 6778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘(((i · (2 · π)) · (𝑚 · 𝑥)) + ((i · (2 · π)) · (-𝑁 · 𝑥)))) = (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))))
83	71, 82	eqtr3d 2780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((exp‘((i · (2 · π)) · (𝑚 · 𝑥))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))))
84	83	oveq2d 7291	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · ((exp‘((i · (2 · π)) · (𝑚 · 𝑥))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))))) = (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))))
85	84	adantr 481	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · ((exp‘((i · (2 · π)) · (𝑚 · 𝑥))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))))) = (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))))
86	68, 85	eqtrd 2778	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ((∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))))
87	86	sumeq2dv 15415	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)((∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))))
88	66, 87	eqtrd 2778	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑥)))) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))))
89	88	sumeq2dv 15415	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → Σ𝑚 ∈ (0...(𝑆 · 𝑁))(Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (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)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))))
90	14, 64, 89	3eqtrd 2782	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → (∏𝑎 ∈ (0..^𝑆)(((𝐿‘𝑎)vts𝑁)‘𝑥) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))))
91	90	itgeq2dv 24946	. 2 ⊢ (𝜑 → ∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)(((𝐿‘𝑎)vts𝑁)‘𝑥) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥 = ∫(0(,)1)Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) d𝑥)
92		ioombl 24729	. . . . 5 ⊢ (0(,)1) ∈ dom vol
93	92	a1i 11	. . . 4 ⊢ (𝜑 → (0(,)1) ∈ dom vol)
94		fzfid 13693	. . . 4 ⊢ (𝜑 → (0...(𝑆 · 𝑁)) ∈ Fin)
95		sumex 15399	. . . . 5 ⊢ Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) ∈ V
96	95	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (0(,)1) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁)))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) ∈ V)
97	93	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (0(,)1) ∈ dom vol)
98	29	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ⊆ ℕ)
99	9	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ₀)
100		fzfid 13693	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
101	98, 32, 99, 100	reprfi 32596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
102	37	a1i 11	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
103	52	adantllr 716	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
104	102, 103	fprodcl 15662	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
105	56, 76	subcld 11332	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 − 𝑁) ∈ ℂ)
106	105, 57	mulcld 10995	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 − 𝑁) · 𝑥) ∈ ℂ)
107	55, 106	mulcld 10995	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)) ∈ ℂ)
108	107	an32s 649	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) → ((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)) ∈ ℂ)
109	108	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)) ∈ ℂ)
110	109	efcld 32571	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))) ∈ ℂ)
111	104, 110	mulcld 10995	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) ∈ ℂ)
112	111	anasss 467	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ (𝑥 ∈ (0(,)1) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚))) → (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) ∈ ℂ)
113	37	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (0..^𝑆) ∈ Fin)
114	113, 52	fprodcl 15662	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
115		fvex 6787	. . . . . . . 8 ⊢ (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))) ∈ V
116	115	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))) ∈ V)
117		ioossicc 13165	. . . . . . . . . 10 ⊢ (0(,)1) ⊆ (0[,]1)
118	117	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (0(,)1) ⊆ (0[,]1))
119	92	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (0(,)1) ∈ dom vol)
120	115	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0[,]1)) → (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))) ∈ V)
121		0red 10978	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 0 ∈ ℝ)
122		1red 10976	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 1 ∈ ℝ)
123	22	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑁 ∈ ℂ)
124	41, 123	subcld 11332	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 − 𝑁) ∈ ℂ)
125		unitsscn 13232	. . . . . . . . . . . . . 14 ⊢ (0[,]1) ⊆ ℂ
126	125	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (0[,]1) ⊆ ℂ)
127		ssidd 3944	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ℂ ⊆ ℂ)
128		cncfmptc 24075	. . . . . . . . . . . . 13 ⊢ (((𝑚 − 𝑁) ∈ ℂ ∧ (0[,]1) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]1) ↦ (𝑚 − 𝑁)) ∈ ((0[,]1)–cn→ℂ))
129	124, 126, 127, 128	syl3anc 1370	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (𝑚 − 𝑁)) ∈ ((0[,]1)–cn→ℂ))
130		cncfmptid 24076	. . . . . . . . . . . . 13 ⊢ (((0[,]1) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ ((0[,]1)–cn→ℂ))
131	126, 127, 130	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ ((0[,]1)–cn→ℂ))
132	129, 131	mulcncf 24610	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ ((𝑚 − 𝑁) · 𝑥)) ∈ ((0[,]1)–cn→ℂ))
133	132	efmul2picn 32576	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) ∈ ((0[,]1)–cn→ℂ))
134		cniccibl 25005	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) ∈ ((0[,]1)–cn→ℂ)) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) ∈ 𝐿¹)
135	121, 122, 133, 134	syl3anc 1370	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) ∈ 𝐿¹)
136	118, 119, 120, 135	iblss 24969	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0(,)1) ↦ (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) ∈ 𝐿¹)
137	136	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (𝑥 ∈ (0(,)1) ↦ (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) ∈ 𝐿¹)
138	114, 116, 137	iblmulc2 24995	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (𝑥 ∈ (0(,)1) ↦ (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))))) ∈ 𝐿¹)
139	97, 101, 112, 138	itgfsum 24991	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑥 ∈ (0(,)1) ↦ Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))))) ∈ 𝐿¹ ∧ ∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) d𝑥 = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) d𝑥))
140	139	simpld 495	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0(,)1) ↦ Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))))) ∈ 𝐿¹)
141	93, 94, 96, 140	itgfsum 24991	. . 3 ⊢ (𝜑 → ((𝑥 ∈ (0(,)1) ↦ Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))))) ∈ 𝐿¹ ∧ ∫(0(,)1)Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) d𝑥 = Σ𝑚 ∈ (0...(𝑆 · 𝑁))∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) d𝑥))
142	141	simprd 496	. 2 ⊢ (𝜑 → ∫(0(,)1)Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) d𝑥 = Σ𝑚 ∈ (0...(𝑆 · 𝑁))∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) d𝑥)
143		oveq2 7283	. . . . . . 7 ⊢ (if((𝑚 − 𝑁) = 0, 1, 0) = 1 → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · if((𝑚 − 𝑁) = 0, 1, 0)) = (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · 1))
144		oveq2 7283	. . . . . . 7 ⊢ (if((𝑚 − 𝑁) = 0, 1, 0) = 0 → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · if((𝑚 − 𝑁) = 0, 1, 0)) = (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · 0))
145	101, 114	fsumcl 15445	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
146	145	mulid1d 10992	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · 1) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)))
147	145	mul01d 11174	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · 0) = 0)
148	143, 144, 146, 147	ifeq3da 30889	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → if((𝑚 − 𝑁) = 0, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)), 0) = (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · if((𝑚 − 𝑁) = 0, 1, 0)))
149		velsn 4577	. . . . . . . 8 ⊢ (𝑚 ∈ {𝑁} ↔ 𝑚 = 𝑁)
150	41, 123	subeq0ad 11342	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 − 𝑁) = 0 ↔ 𝑚 = 𝑁))
151	149, 150	bitr4id 290	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 ∈ {𝑁} ↔ (𝑚 − 𝑁) = 0))
152	151	ifbid 4482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → if(𝑚 ∈ {𝑁}, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)), 0) = if((𝑚 − 𝑁) = 0, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)), 0))
153	1	nn0zd 12424	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
154	153	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑁 ∈ ℤ)
155	46, 154	zsubcld 12431	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (𝑚 − 𝑁) ∈ ℤ)
156		itgexpif 32586	. . . . . . . . . 10 ⊢ ((𝑚 − 𝑁) ∈ ℤ → ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))) d𝑥 = if((𝑚 − 𝑁) = 0, 1, 0))
157	155, 156	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))) d𝑥 = if((𝑚 − 𝑁) = 0, 1, 0))
158	157	oveq2d 7291	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))) d𝑥) = (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · if((𝑚 − 𝑁) = 0, 1, 0)))
159	158	sumeq2dv 15415	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))) d𝑥) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · if((𝑚 − 𝑁) = 0, 1, 0)))
160		1cnd 10970	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 1 ∈ ℂ)
161		0cnd 10968	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 0 ∈ ℂ)
162	160, 161	ifcld 4505	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → if((𝑚 − 𝑁) = 0, 1, 0) ∈ ℂ)
163	101, 162, 114	fsummulc1 15497	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · if((𝑚 − 𝑁) = 0, 1, 0)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · if((𝑚 − 𝑁) = 0, 1, 0)))
164	159, 163	eqtr4d 2781	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))) d𝑥) = (Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · if((𝑚 − 𝑁) = 0, 1, 0)))
165	148, 152, 164	3eqtr4rd 2789	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))) d𝑥) = if(𝑚 ∈ {𝑁}, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)), 0))
166	165	sumeq2dv 15415	. . . 4 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))) d𝑥) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))if(𝑚 ∈ {𝑁}, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)), 0))
167		0zd 12331	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ)
168	9	nn0zd 12424	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℤ)
169	168, 153	zmulcld 12432	. . . . . . 7 ⊢ (𝜑 → (𝑆 · 𝑁) ∈ ℤ)
170	1	nn0ge0d 12296	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑁)
171		nnmulge 31073	. . . . . . . 8 ⊢ ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ≤ (𝑆 · 𝑁))
172	8, 1, 171	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑁 ≤ (𝑆 · 𝑁))
173	167, 169, 153, 170, 172	elfzd 13247	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (0...(𝑆 · 𝑁)))
174	173	snssd 4742	. . . . 5 ⊢ (𝜑 → {𝑁} ⊆ (0...(𝑆 · 𝑁)))
175	174	sselda 3921	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑁}) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
176	175, 145	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑁}) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
177	176	ralrimiva 3103	. . . . 5 ⊢ (𝜑 → ∀𝑚 ∈ {𝑁}Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
178	94	olcd 871	. . . . 5 ⊢ (𝜑 → ((0...(𝑆 · 𝑁)) ⊆ (ℤ_≥‘0) ∨ (0...(𝑆 · 𝑁)) ∈ Fin))
179		sumss2 15438	. . . . 5 ⊢ ((({𝑁} ⊆ (0...(𝑆 · 𝑁)) ∧ ∀𝑚 ∈ {𝑁}Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ) ∧ ((0...(𝑆 · 𝑁)) ⊆ (ℤ_≥‘0) ∨ (0...(𝑆 · 𝑁)) ∈ Fin)) → Σ𝑚 ∈ {𝑁}Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))if(𝑚 ∈ {𝑁}, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)), 0))
180	174, 177, 178, 179	syl21anc 835	. . . 4 ⊢ (𝜑 → Σ𝑚 ∈ {𝑁}Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))if(𝑚 ∈ {𝑁}, Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)), 0))
181	29	a1i 11	. . . . . . 7 ⊢ (𝜑 → (1...𝑁) ⊆ ℕ)
182		fzfid 13693	. . . . . . 7 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
183	181, 153, 9, 182	reprfi 32596	. . . . . 6 ⊢ (𝜑 → ((1...𝑁)(repr‘𝑆)𝑁) ∈ Fin)
184	37	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → (0..^𝑆) ∈ Fin)
185	1	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℕ₀)
186	9	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑆 ∈ ℕ₀)
187	22	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℂ)
188	11	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐿:(0..^𝑆)⟶(ℂ ↑_m ℕ))
189		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
190	29	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → (1...𝑁) ⊆ ℕ)
191	153	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑁 ∈ ℤ)
192	9	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑆 ∈ ℕ₀)
193		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁))
194	190, 191, 192, 193	reprf 32592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
195	194	ffvelrnda 6961	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ (1...𝑁))
196	29, 195	sselid 3919	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ ℕ)
197	185, 186, 187, 188, 189, 196	breprexplemb 32611	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
198	184, 197	fprodcl 15662	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
199	183, 198	fsumcl 15445	. . . . 5 ⊢ (𝜑 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
200		oveq2 7283	. . . . . . 7 ⊢ (𝑚 = 𝑁 → ((1...𝑁)(repr‘𝑆)𝑚) = ((1...𝑁)(repr‘𝑆)𝑁))
201	200	sumeq1d 15413	. . . . . 6 ⊢ (𝑚 = 𝑁 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)))
202	201	sumsn 15458	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ) → Σ𝑚 ∈ {𝑁}Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)))
203	1, 199, 202	syl2anc 584	. . . 4 ⊢ (𝜑 → Σ𝑚 ∈ {𝑁}Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)))
204	166, 180, 203	3eqtr2d 2784	. . 3 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))) d𝑥) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)))
205	139	simprd 496	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) d𝑥 = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) d𝑥)
206	110	an32s 649	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))) ∈ ℂ)
207	114, 206, 137	itgmulc2 24998	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))) d𝑥) = ∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) d𝑥)
208	207	sumeq2dv 15415	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))) d𝑥) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) d𝑥)
209	205, 208	eqtr4d 2781	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) d𝑥 = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))) d𝑥))
210	209	sumeq2dv 15415	. . 3 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑆 · 𝑁))∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) d𝑥 = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · ∫(0(,)1)(exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥))) d𝑥))
211	1, 9	reprfz1 32604	. . . 4 ⊢ (𝜑 → (ℕ(repr‘𝑆)𝑁) = ((1...𝑁)(repr‘𝑆)𝑁))
212	211	sumeq1d 15413	. . 3 ⊢ (𝜑 → Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) = Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)))
213	204, 210, 212	3eqtr4d 2788	. 2 ⊢ (𝜑 → Σ𝑚 ∈ (0...(𝑆 · 𝑁))∫(0(,)1)Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) d𝑥 = Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)))
214	91, 142, 213	3eqtrrd 2783	1 ⊢ (𝜑 → Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)(((𝐿‘𝑎)vts𝑁)‘𝑥) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 ⊆ wss 3887 ifcif 4459 {csn 4561 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 Fincfn 8733 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 ici 10873 + caddc 10874 · cmul 10876 ≤ cle 11010 − cmin 11205 -cneg 11206 ℕcn 11973 2c2 12028 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 (,)cioo 13079 [,]cicc 13082 ...cfz 13239 ..^cfzo 13382 Σcsu 15397 ∏cprod 15615 expce 15771 πcpi 15776 –cn→ccncf 24039 volcvol 24627 𝐿¹cibl 24781 ∫citg 24782 reprcrepr 32588 vtscvts 32615

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cc 10191 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-symdif 4176 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-omul 8302 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-prod 15616 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-ovol 24628 df-vol 24629 df-mbf 24783 df-itg1 24784 df-itg2 24785 df-ibl 24786 df-itg 24787 df-0p 24834 df-limc 25030 df-dv 25031 df-repr 32589 df-vts 32616

This theorem is referenced by: circlemethnat 32621 circlevma 32622 circlemethhgt 32623

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