circlemeth - Mathbox for Thierry Arnoux

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Theorem circlemeth 34617

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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → 𝑁 ∈ ℕ₀)
3		ioossre 13468	. . . . . . . . 9 ⊢ (0(,)1) ⊆ ℝ
4		ax-resscn 11241	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
5	3, 4	sstri 4018	. . . . . . . 8 ⊢ (0(,)1) ⊆ ℂ
6	5	a1i 11	. . . . . . 7 ⊢ (𝜑 → (0(,)1) ⊆ ℂ)
7	6	sselda 4008	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → 𝑥 ∈ ℂ)
8		circlemeth.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℕ)
9	8	nnnn0d 12613	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ ℕ₀)
10	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → 𝑆 ∈ ℕ₀)
11		circlemeth.l	. . . . . . 7 ⊢ (𝜑 → 𝐿:(0..^𝑆)⟶(ℂ ↑_m ℕ))
12	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → 𝐿:(0..^𝑆)⟶(ℂ ↑_m ℕ))
13	2, 7, 10, 12	vtsprod 34616	. . . . 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 7463	. . . 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 14024	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → (0...(𝑆 · 𝑁)) ∈ Fin)
16		ax-icn 11243	. . . . . . . . 9 ⊢ i ∈ ℂ
17		2cn 12368	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
18		picn 26519	. . . . . . . . . 10 ⊢ π ∈ ℂ
19	17, 18	mulcli 11297	. . . . . . . . 9 ⊢ (2 · π) ∈ ℂ
20	16, 19	mulcli 11297	. . . . . . . 8 ⊢ (i · (2 · π)) ∈ ℂ
21	20	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → (i · (2 · π)) ∈ ℂ)
22	1	nn0cnd 12615	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℂ)
23	22	negcld 11634	. . . . . . . . . 10 ⊢ (𝜑 → -𝑁 ∈ ℂ)
24	23	ralrimivw 3156	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (0(,)1)-𝑁 ∈ ℂ)
25	24	r19.21bi 3257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → -𝑁 ∈ ℂ)
26	25, 7	mulcld 11310	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → (-𝑁 · 𝑥) ∈ ℂ)
27	21, 26	mulcld 11310	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → ((i · (2 · π)) · (-𝑁 · 𝑥)) ∈ ℂ)
28	27	efcld 16131	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)1)) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
29		fz1ssnn 13615	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℕ
30	29	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ⊆ ℕ)
31		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
32	31	elfzelzd 13585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
33	32	adantlr 714	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℤ)
34	10	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ₀)
35		fzfid 14024	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
36	30, 33, 34, 35	reprfi 34593	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((1...𝑁)(repr‘𝑆)𝑚) ∈ Fin)
37		fzofi 14025	. . . . . . . . 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 729	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℕ₀)
40	9	ad3antrrr 729	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑆 ∈ ℕ₀)
41	32	zcnd 12748	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℂ)
42	41	ad2antrr 725	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑚 ∈ ℂ)
43	11	ad3antrrr 729	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐿:(0..^𝑆)⟶(ℂ ↑_m ℕ))
44		simpr 484	. . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑚 ∈ ℤ)
47	9	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑆 ∈ ℕ₀)
48		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚))
49	45, 46, 47, 48	reprf 34589	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
50	49	ffvelcdmda 7118	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ (1...𝑁))
51	29, 50	sselid 4006	. . . . . . . . . 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 34608	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
53	52	adantl3r 749	. . . . . . . 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 16000	. . . . . . 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 12748	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑚 ∈ ℂ)
57	7	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑥 ∈ ℂ)
58	56, 57	mulcld 11310	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 · 𝑥) ∈ ℂ)
59	55, 58	mulcld 11310	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · (𝑚 · 𝑥)) ∈ ℂ)
60	59	efcld 16131	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘((i · (2 · π)) · (𝑚 · 𝑥))) ∈ ℂ)
61	60	adantr 480	. . . . . . 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 11310	. . . . . 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 15781	. . . . 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 15833	. . . 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 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (exp‘((i · (2 · π)) · (-𝑁 · 𝑥))) ∈ ℂ)
66	36, 65, 62	fsummulc1 15833	. . . . . 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 480	. . . . . . . . 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 11313	. . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · (-𝑁 · 𝑥)) ∈ ℂ)
70		efadd 16142	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (-𝑁 · 𝑥) ∈ ℂ)
73	55, 58, 72	adddid 11314	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → -𝑁 ∈ ℂ)
75	56, 74, 57	adddird 11315	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 + -𝑁) · 𝑥) = ((𝑚 · 𝑥) + (-𝑁 · 𝑥)))
76	22	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑁 ∈ ℂ)
77	56, 76	negsubd 11653	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 + -𝑁) = (𝑚 − 𝑁))
78	77	oveq1d 7463	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 + -𝑁) · 𝑥) = ((𝑚 − 𝑁) · 𝑥))
79	75, 78	eqtr3d 2782	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 · 𝑥) + (-𝑁 · 𝑥)) = ((𝑚 − 𝑁) · 𝑥))
80	79	oveq2d 7464	. . . . . . . . . . . . 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 2782	. . . . . . . . . . . 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 6924	. . . . . . . . . . 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 2782	. . . . . . . . . 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 7464	. . . . . . . . 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 480	. . . . . . . 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 2780	. . . . . . 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 15750	. . . . . 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 2780	. . . . 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 15750	. . . 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 2784	. . 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 25837	. 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 25619	. . . . 5 ⊢ (0(,)1) ∈ dom vol
93	92	a1i 11	. . . 4 ⊢ (𝜑 → (0(,)1) ∈ dom vol)
94		fzfid 14024	. . . 4 ⊢ (𝜑 → (0...(𝑆 · 𝑁)) ∈ Fin)
95		sumex 15736	. . . . 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 480	. . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑆 ∈ ℕ₀)
100		fzfid 14024	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (1...𝑁) ∈ Fin)
101	98, 32, 99, 100	reprfi 34593	. . . . . 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 718	. . . . . . . . 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 16000	. . . . . . . 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 11647	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 − 𝑁) ∈ ℂ)
106	105, 57	mulcld 11310	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((𝑚 − 𝑁) · 𝑥) ∈ ℂ)
107	55, 106	mulcld 11310	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (0(,)1)) ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)) ∈ ℂ)
108	107	an32s 651	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑥 ∈ (0(,)1)) → ((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)) ∈ ℂ)
109	108	adantr 480	. . . . . . . . 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 16131	. . . . . . . 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 11310	. . . . . . 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 466	. . . . . 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 16000	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
115		fvex 6933	. . . . . . . 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 13493	. . . . . . . . . 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 11293	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 0 ∈ ℝ)
122		1red 11291	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 1 ∈ ℝ)
123	22	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 𝑁 ∈ ℂ)
124	41, 123	subcld 11647	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑚 − 𝑁) ∈ ℂ)
125		unitsscn 13560	. . . . . . . . . . . . . 14 ⊢ (0[,]1) ⊆ ℂ
126	125	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (0[,]1) ⊆ ℂ)
127		ssidd 4032	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → ℂ ⊆ ℂ)
128		cncfmptc 24957	. . . . . . . . . . . . 13 ⊢ (((𝑚 − 𝑁) ∈ ℂ ∧ (0[,]1) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]1) ↦ (𝑚 − 𝑁)) ∈ ((0[,]1)–cn→ℂ))
129	124, 126, 127, 128	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (𝑚 − 𝑁)) ∈ ((0[,]1)–cn→ℂ))
130		cncfmptid 24958	. . . . . . . . . . . . 13 ⊢ (((0[,]1) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ ((0[,]1)–cn→ℂ))
131	126, 127, 130	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ ((0[,]1)–cn→ℂ))
132	129, 131	mulcncf 25499	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ ((𝑚 − 𝑁) · 𝑥)) ∈ ((0[,]1)–cn→ℂ))
133	132	efmul2picn 34573	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) ∈ ((0[,]1)–cn→ℂ))
134		cniccibl 25896	. . . . . . . . . 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 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0[,]1) ↦ (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) ∈ 𝐿¹)
136	118, 119, 120, 135	iblss 25860	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → (𝑥 ∈ (0(,)1) ↦ (exp‘((i · (2 · π)) · ((𝑚 − 𝑁) · 𝑥)))) ∈ 𝐿¹)
137	136	adantr 480	. . . . . . 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 25886	. . . . . 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 25882	. . . . 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 494	. . . 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 25882	. . 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 495	. 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 7456	. . . . . . 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 7456	. . . . . . 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 15781	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
146	145	mulridd 11307	. . . . . . 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 11489	. . . . . . 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 32569	. . . . . 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 4664	. . . . . . . 8 ⊢ (𝑚 ∈ {𝑁} ↔ 𝑚 = 𝑁)
150	41, 123	subeq0ad 11657	. . . . . . . 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 4571	. . . . . 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 12665	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
154	153	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → 𝑁 ∈ ℤ)
155	46, 154	zsubcld 12752	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)) → (𝑚 − 𝑁) ∈ ℤ)
156		itgexpif 34583	. . . . . . . . . 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 7464	. . . . . . . 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 15750	. . . . . . 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 11285	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 1 ∈ ℂ)
161		0cnd 11283	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → 0 ∈ ℂ)
162	160, 161	ifcld 4594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(𝑆 · 𝑁))) → if((𝑚 − 𝑁) = 0, 1, 0) ∈ ℂ)
163	101, 162, 114	fsummulc1 15833	. . . . . . 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 2783	. . . . . 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 2791	. . . . 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 15750	. . . 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 12651	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ)
168	9	nn0zd 12665	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℤ)
169	168, 153	zmulcld 12753	. . . . . . 7 ⊢ (𝜑 → (𝑆 · 𝑁) ∈ ℤ)
170	1	nn0ge0d 12616	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑁)
171		nnmulge 32752	. . . . . . . 8 ⊢ ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ≤ (𝑆 · 𝑁))
172	8, 1, 171	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → 𝑁 ≤ (𝑆 · 𝑁))
173	167, 169, 153, 170, 172	elfzd 13575	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (0...(𝑆 · 𝑁)))
174	173	snssd 4834	. . . . 5 ⊢ (𝜑 → {𝑁} ⊆ (0...(𝑆 · 𝑁)))
175	174	sselda 4008	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑁}) → 𝑚 ∈ (0...(𝑆 · 𝑁)))
176	175, 145	syldan 590	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑁}) → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
177	176	ralrimiva 3152	. . . . 5 ⊢ (𝜑 → ∀𝑚 ∈ {𝑁}Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
178	94	olcd 873	. . . . 5 ⊢ (𝜑 → ((0...(𝑆 · 𝑁)) ⊆ (ℤ_≥‘0) ∨ (0...(𝑆 · 𝑁)) ∈ Fin))
179		sumss2 15774	. . . . 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 837	. . . 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 14024	. . . . . . 7 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
183	181, 153, 9, 182	reprfi 34593	. . . . . 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 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℕ₀)
186	9	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑆 ∈ ℕ₀)
187	22	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑁 ∈ ℂ)
188	11	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐿:(0..^𝑆)⟶(ℂ ↑_m ℕ))
189		simpr 484	. . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑁 ∈ ℤ)
192	9	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑆 ∈ ℕ₀)
193		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁))
194	190, 191, 192, 193	reprf 34589	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → 𝑐:(0..^𝑆)⟶(1...𝑁))
195	194	ffvelcdmda 7118	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ (1...𝑁))
196	29, 195	sselid 4006	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑐‘𝑎) ∈ ℕ)
197	185, 186, 187, 188, 189, 196	breprexplemb 34608	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
198	184, 197	fprodcl 16000	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)) → ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
199	183, 198	fsumcl 15781	. . . . 5 ⊢ (𝜑 → Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) ∈ ℂ)
200		oveq2 7456	. . . . . . 7 ⊢ (𝑚 = 𝑁 → ((1...𝑁)(repr‘𝑆)𝑚) = ((1...𝑁)(repr‘𝑆)𝑁))
201	200	sumeq1d 15748	. . . . . 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 15794	. . . . 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 583	. . . 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 2786	. . 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 495	. . . . 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 651	. . . . . . 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 25889	. . . . . 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 15750	. . . . 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 2783	. . . 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 15750	. . 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 34601	. . . 4 ⊢ (𝜑 → (ℕ(repr‘𝑆)𝑁) = ((1...𝑁)(repr‘𝑆)𝑁))
212	211	sumeq1d 15748	. . 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 2790	. 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 2785	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 395 ∨ wo 846 = wceq 1537 ∈ wcel 2108 ∀wral 3067 Vcvv 3488 ⊆ wss 3976 ifcif 4548 {csn 4648 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ↑_m cmap 8884 Fincfn 9003 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 ici 11186 + caddc 11187 · cmul 11189 ≤ cle 11325 − cmin 11520 -cneg 11521 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 (,)cioo 13407 [,]cicc 13410 ...cfz 13567 ..^cfzo 13711 Σcsu 15734 ∏cprod 15951 expce 16109 πcpi 16114 –cn→ccncf 24921 volcvol 25517 𝐿¹cibl 25671 ∫citg 25672 reprcrepr 34585 vtscvts 34612

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cc 10504 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-symdif 4272 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-omul 8527 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-acn 10011 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-prod 15952 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-ovol 25518 df-vol 25519 df-mbf 25673 df-itg1 25674 df-itg2 25675 df-ibl 25676 df-itg 25677 df-0p 25724 df-limc 25921 df-dv 25922 df-repr 34586 df-vts 34613

This theorem is referenced by: circlemethnat 34618 circlevma 34619 circlemethhgt 34620

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