circlemeth - Mathbox for Thierry Arnoux

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Theorem circlemeth 34833

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 13352	. . . . . . . . 9 ⊢ (0(,)1) ⊆ ℝ
4		ax-resscn 11087	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
5	3, 4	sstri 3924	. . . . . . . 8 ⊢ (0(,)1) ⊆ ℂ
6	5	a1i 11	. . . . . . 7 ⊢ (𝜑 → (0(,)1) ⊆ ℂ)
7	6	sselda 3915	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → 𝑥 ∈ ℂ)
8		circlemeth.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℕ)
9	8	nnnn0d 12490	. . . . . . 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 34832	. . . . 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 7372	. . . 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 13927	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → (0...(𝑆 · 𝑁)) ∈ Fin)
16		ax-icn 11089	. . . . . . . . 9 ⊢ i ∈ ℂ
17		2cn 12248	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
18		picn 26441	. . . . . . . . . 10 ⊢ π ∈ ℂ
19	17, 18	mulcli 11144	. . . . . . . . 9 ⊢ (2 · π) ∈ ℂ
20	16, 19	mulcli 11144	. . . . . . . 8 ⊢ (i · (2 · π)) ∈ ℂ
21	20	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → (i · (2 · π)) ∈ ℂ)
22	1	nn0cnd 12492	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℂ)
23	22	negcld 11484	. . . . . . . . . 10 ⊢ (𝜑 → -𝑁 ∈ ℂ)
24	23	ralrimivw 3135	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (0(,)1)-𝑁 ∈ ℂ)
25	24	r19.21bi 3231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → -𝑁 ∈ ℂ)
26	25, 7	mulcld 11157	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → (-𝑁 · 𝑥) ∈ ℂ)
27	21, 26	mulcld 11157	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → ((i · (2 · π)) · (-𝑁 · 𝑥)) ∈ ℂ)
28	27	efcld 16040	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
29		fz1ssnn 13501	. . . . . . . 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 13471	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
33	32	adantlr 721	. . . . . . 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 13927	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
36	30, 33, 34, 35	reprfi 34809	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
37		fzofi 13928	. . . . . . . . 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 736	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℕ₀)
40	9	ad3antrrr 736	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑆 ∈ ℕ₀)
41	32	zcnd 12626	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℂ)
42	41	ad2antrr 732	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑚 ∈ ℂ)
43	11	ad3antrrr 736	. . . . . . . . . 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 732	. . . . . . . . . . . . 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 34805	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
50	49	ffvelcdmda 7026	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ (1...𝑁))
51	29, 50	sselid 3913	. . . . . . . . . 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 34824	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
53	52	adantl3r 756	. . . . . . . 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 15909	. . . . . . 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 12626	. . . . . . . . . . 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 11157	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 · 𝑥) ∈ ℂ)
59	55, 58	mulcld 11157	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · (𝑚 · 𝑥)) ∈ ℂ)
60	59	efcld 16040	. . . . . . . 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 11157	. . . . . 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 15687	. . . . 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 15739	. . . 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 15739	. . . . . 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 11160	. . . . . . . 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 16051	. . . . . . . . . . . 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 590	. . . . . . . . . . 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 11161	. . . . . . . . . . . . 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 11162	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 + -𝑁) · 𝑥) = ((𝑚 · 𝑥) + (-𝑁 · 𝑥)))
76	22	ad2antrr 732	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑁 ∈ ℂ)
77	56, 76	negsubd 11503	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 + -𝑁) = (𝑚 − 𝑁))
78	77	oveq1d 7372	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 + -𝑁) · 𝑥) = ((𝑚 − 𝑁) · 𝑥))
79	75, 78	eqtr3d 2776	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 · 𝑥) + (-𝑁 · 𝑥)) = ((𝑚 − 𝑁) · 𝑥))
80	79	oveq2d 7373	. . . . . . . . . . . . 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 2776	. . . . . . . . . . . 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 6832	. . . . . . . . . . 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 2776	. . . . . . . . . 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 7373	. . . . . . . . 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 2774	. . . . . . 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 15656	. . . . . 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 2774	. . . . 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 15656	. . . 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 2778	. . 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 25768	. 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 25551	. . . . 5 ⊢ (0(,)1) ∈ dom vol
93	92	a1i 11	. . . 4 ⊢ (𝜑 → (0(,)1) ∈ dom vol)
94		fzfid 13927	. . . 4 ⊢ (𝜑 → (0...(𝑆 · 𝑁)) ∈ Fin)
95		sumex 15642	. . . . 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 13927	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
101	98, 32, 99, 100	reprfi 34809	. . . . . 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 725	. . . . . . . . 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 15909	. . . . . . . 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 11497	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 − 𝑁) ∈ ℂ)
106	105, 57	mulcld 11157	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 − 𝑁) · 𝑥) ∈ ℂ)
107	55, 106	mulcld 11157	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)) ∈ ℂ)
108	107	an32s 658	. . . . . . . . . 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 16040	. . . . . . . 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 11157	. . . . . . 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 15909	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
115		fvex 6841	. . . . . . . 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 13378	. . . . . . . . . 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 11139	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 0 ∈ ℝ)
122		1red 11137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 1 ∈ ℝ)
123	22	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑁 ∈ ℂ)
124	41, 123	subcld 11497	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 − 𝑁) ∈ ℂ)
125		unitsscn 13445	. . . . . . . . . . . . . 14 ⊢ (0[,]1) ⊆ ℂ
126	125	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (0[,]1) ⊆ ℂ)
127		ssidd 3938	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ℂ ⊆ ℂ)
128		cncfmptc 24898	. . . . . . . . . . . . 13 ⊢ (((𝑚 − 𝑁) ∈ ℂ ∧ (0[,]1) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]1) ↦ (𝑚 − 𝑁)) ∈ ((0[,]1)–cn→ℂ))
129	124, 126, 127, 128	syl3anc 1379	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (𝑚 − 𝑁)) ∈ ((0[,]1)–cn→ℂ))
130		cncfmptid 24899	. . . . . . . . . . . . 13 ⊢ (((0[,]1) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ ((0[,]1)–cn→ℂ))
131	126, 127, 130	syl2anc 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ ((0[,]1)–cn→ℂ))
132	129, 131	mulcncf 25432	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ ((𝑚 − 𝑁) · 𝑥)) ∈ ((0[,]1)–cn→ℂ))
133	132	efmul2picn 34789	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) ∈ ((0[,]1)–cn→ℂ))
134		cniccibl 25827	. . . . . . . . . 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 1379	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) ∈ 𝐿¹)
136	118, 119, 120, 135	iblss 25791	. . . . . . . 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 25817	. . . . . 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 25813	. . . . 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 25813	. . 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 7365	. . . . . . 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 7365	. . . . . . 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 15687	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
146	145	mulridd 11154	. . . . . . 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 11337	. . . . . . 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 32635	. . . . . 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 4572	. . . . . . . 8 ⊢ (𝑚 ∈ {𝑁} ↔ 𝑚 = 𝑁)
150	41, 123	subeq0ad 11507	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 − 𝑁) = 0 ↔ 𝑚 = 𝑁))
151	149, 150	bitr4id 291	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 ∈ {𝑁} ↔ (𝑚 − 𝑁) = 0))
152	151	ifbid 4479	. . . . . 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 12541	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
154	153	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑁 ∈ ℤ)
155	46, 154	zsubcld 12630	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (𝑚 − 𝑁) ∈ ℤ)
156		itgexpif 34799	. . . . . . . . . 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 7373	. . . . . . . 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 15656	. . . . . . 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 11131	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 1 ∈ ℂ)
161		0cnd 11129	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 0 ∈ ℂ)
162	160, 161	ifcld 4502	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → if((𝑚 − 𝑁) = 0, 1, 0) ∈ ℂ)
163	101, 162, 114	fsummulc1 15739	. . . . . . 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 2777	. . . . . 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 2785	. . . . 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 15656	. . . 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 12528	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ)
168	9	nn0zd 12541	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℤ)
169	168, 153	zmulcld 12631	. . . . . . 7 ⊢ (𝜑 → (𝑆 · 𝑁) ∈ ℤ)
170	1	nn0ge0d 12493	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑁)
171		nnmulge 32832	. . . . . . . 8 ⊢ ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ≤ (𝑆 · 𝑁))
172	8, 1, 171	syl2anc 590	. . . . . . 7 ⊢ (𝜑 → 𝑁 ≤ (𝑆 · 𝑁))
173	167, 169, 153, 170, 172	elfzd 13461	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (0...(𝑆 · 𝑁)))
174	173	snssd 4719	. . . . 5 ⊢ (𝜑 → {𝑁} ⊆ (0...(𝑆 · 𝑁)))
175	174	sselda 3915	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑁}) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
176	175, 145	syldan 597	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑁}) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
177	176	ralrimiva 3131	. . . . 5 ⊢ (𝜑 → ∀𝑚 ∈ {𝑁}Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
178	94	olcd 880	. . . . 5 ⊢ (𝜑 → ((0...(𝑆 · 𝑁)) ⊆ (ℤ_≥‘0) ∨ (0...(𝑆 · 𝑁)) ∈ Fin))
179		sumss2 15680	. . . . 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 843	. . . 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 13927	. . . . . . 7 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
183	181, 153, 9, 182	reprfi 34809	. . . . . 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 732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℕ₀)
186	9	ad2antrr 732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑆 ∈ ℕ₀)
187	22	ad2antrr 732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℂ)
188	11	ad2antrr 732	. . . . . . . 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 34805	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
195	194	ffvelcdmda 7026	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ (1...𝑁))
196	29, 195	sselid 3913	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ ℕ)
197	185, 186, 187, 188, 189, 196	breprexplemb 34824	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
198	184, 197	fprodcl 15909	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
199	183, 198	fsumcl 15687	. . . . 5 ⊢ (𝜑 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
200		oveq2 7365	. . . . . . 7 ⊢ (𝑚 = 𝑁 → ((1...𝑁)(repr‘𝑆)𝑚) = ((1...𝑁)(repr‘𝑆)𝑁))
201	200	sumeq1d 15654	. . . . . 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 15700	. . . . 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 590	. . . 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 2780	. . 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 658	. . . . . . 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 25820	. . . . . 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 15656	. . . . 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 2777	. . . 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 15656	. . 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 34817	. . . 4 ⊢ (𝜑 → (ℕ(repr‘𝑆)𝑁) = ((1...𝑁)(repr‘𝑆)𝑁))
212	211	sumeq1d 15654	. . 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 2784	. 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 2779	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 853 = wceq 1547 ∈ wcel 2119 ∀wral 3053 Vcvv 3431 ⊆ wss 3883 ifcif 4455 {csn 4556 class class class wbr 5073 ↦ cmpt 5154 dom cdm 5619 ⟶wf 6482 ‘cfv 6486 (class class class)co 7357 ↑_m cmap 8764 Fincfn 8884 ℂcc 11028 ℝcr 11029 0cc0 11030 1c1 11031 ici 11032 + caddc 11033 · cmul 11035 ≤ cle 11172 − cmin 11369 -cneg 11370 ℕcn 12166 2c2 12228 ℕ₀cn0 12429 ℤcz 12516 ℤ_≥cuz 12780 (,)cioo 13290 [,]cicc 13293 ...cfz 13453 ..^cfzo 13600 Σcsu 15640 ∏cprod 15860 expce 16018 πcpi 16023 –cn→ccncf 24862 volcvol 25449 𝐿¹cibl 25603 ∫citg 25604 reprcrepr 34801 vtscvts 34828

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-inf2 9554 ax-cc 10349 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-symdif 4182 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-iin 4925 df-disj 5041 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7621 df-ofr 7622 df-om 7808 df-1st 7932 df-2nd 7933 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-omul 8401 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-dju 9817 df-card 9855 df-acn 9858 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-7 12241 df-8 12242 df-9 12243 df-n0 12430 df-z 12517 df-dec 12637 df-uz 12781 df-q 12891 df-rp 12935 df-xneg 13055 df-xadd 13056 df-xmul 13057 df-ioo 13294 df-ioc 13295 df-ico 13296 df-icc 13297 df-fz 13454 df-fzo 13601 df-fl 13743 df-mod 13821 df-seq 13956 df-exp 14016 df-fac 14228 df-bc 14257 df-hash 14285 df-shft 15021 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-limsup 15425 df-clim 15442 df-rlim 15443 df-sum 15641 df-prod 15861 df-ef 16024 df-sin 16026 df-cos 16027 df-pi 16029 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17172 df-ress 17193 df-plusg 17225 df-mulr 17226 df-starv 17227 df-sca 17228 df-vsca 17229 df-ip 17230 df-tset 17231 df-ple 17232 df-ds 17234 df-unif 17235 df-hom 17236 df-cco 17237 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17458 df-qtop 17463 df-imas 17464 df-xps 17466 df-mre 17540 df-mrc 17541 df-acs 17543 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18744 df-mulg 19036 df-cntz 19284 df-cmn 19749 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-fbas 21345 df-fg 21346 df-cnfld 21349 df-top 22878 df-topon 22895 df-topsp 22917 df-bases 22930 df-cld 23003 df-ntr 23004 df-cls 23005 df-nei 23082 df-lp 23120 df-perf 23121 df-cn 23211 df-cnp 23212 df-haus 23299 df-cmp 23371 df-tx 23546 df-hmeo 23739 df-fil 23830 df-fm 23922 df-flim 23923 df-flf 23924 df-xms 24304 df-ms 24305 df-tms 24306 df-cncf 24864 df-ovol 25450 df-vol 25451 df-mbf 25605 df-itg1 25606 df-itg2 25607 df-ibl 25608 df-itg 25609 df-0p 25656 df-limc 25852 df-dv 25853 df-repr 34802 df-vts 34829

This theorem is referenced by: circlemethnat 34834 circlevma 34835 circlemethhgt 34836

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