fourierdlem54 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem54 39053

Description: Given a partition 𝑄 and an arbitrary interval [𝐶, 𝐷], a partition 𝑆 on [𝐶, 𝐷] is built such that it preserves any periodic function piecewise continuous on 𝑄 will be piecewise continuous on 𝑆, with the same limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem54.t	⊢ 𝑇 = (𝐵 − 𝐴)
fourierdlem54.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem54.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem54.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem54.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
fourierdlem54.d	⊢ (𝜑 → 𝐷 ∈ ℝ)
fourierdlem54.cd	⊢ (𝜑 → 𝐶 < 𝐷)
fourierdlem54.o	⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem54.h	⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
fourierdlem54.n	⊢ 𝑁 = ((#‘𝐻) − 1)
fourierdlem54.s	⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))

**Assertion**
Ref	Expression
fourierdlem54	⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))

Distinct variable groups: 𝐴,𝑖,𝑚,𝑝 𝐵,𝑖,𝑚,𝑝 𝐶,𝑚,𝑝 𝑥,𝐶 𝐷,𝑚,𝑝 𝑥,𝐷 𝑓,𝐻 𝑥,𝐻 𝑖,𝑀,𝑚,𝑝 𝑓,𝑁 𝑖,𝑁,𝑚,𝑝 𝑥,𝑁,𝑖 𝑄,𝑖,𝑘 𝑄,𝑝 𝑥,𝑄,𝑘 𝑆,𝑓 𝑆,𝑖,𝑝 𝑥,𝑆 𝑇,𝑖,𝑘,𝑥 𝜑,𝑓 𝜑,𝑖,𝑘 Allowed substitution hints: 𝜑(𝑥,𝑚,𝑝) 𝐴(𝑥,𝑓,𝑘) 𝐵(𝑥,𝑓,𝑘) 𝐶(𝑓,𝑖,𝑘) 𝐷(𝑓,𝑖,𝑘) 𝑃(𝑥,𝑓,𝑖,𝑘,𝑚,𝑝) 𝑄(𝑓,𝑚) 𝑆(𝑘,𝑚) 𝑇(𝑓,𝑚,𝑝) 𝐻(𝑖,𝑘,𝑚,𝑝) 𝑀(𝑥,𝑓,𝑘) 𝑁(𝑘) 𝑂(𝑥,𝑓,𝑖,𝑘,𝑚,𝑝)

Proof of Theorem fourierdlem54 Dummy variables 𝑤 ℎ 𝑦 𝑧 𝑗 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem54.n	. . 3 ⊢ 𝑁 = ((#‘𝐻) − 1)
2		2z 11286	. . . . . 6 ⊢ 2 ∈ ℤ
3	2	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℤ)
4		fourierdlem54.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ)
5		prid1g 4239	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ {𝐶, 𝐷})
6		elun1 3742	. . . . . . . . . 10 ⊢ (𝐶 ∈ {𝐶, 𝐷} → 𝐶 ∈ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
7	4, 5, 6	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
8		fourierdlem54.h	. . . . . . . . 9 ⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
9	7, 8	syl6eleqr 2699	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝐻)
10		ne0i 3880	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐻 → 𝐻 ≠ ∅)
11	9, 10	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐻 ≠ ∅)
12		prfi 8120	. . . . . . . . . 10 ⊢ {𝐶, 𝐷} ∈ Fin
13		fourierdlem54.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
14		fourierdlem54.m	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℕ)
15		fourierdlem54.q	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
16	13, 14, 15	fourierdlem11 39011	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
17	16	simp1d 1066	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
18	16	simp2d 1067	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ)
19	16	simp3d 1068	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 < 𝐵)
20		fourierdlem54.t	. . . . . . . . . . 11 ⊢ 𝑇 = (𝐵 − 𝐴)
21	13, 14, 15	fourierdlem15 39015	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
22		frn 5966	. . . . . . . . . . . 12 ⊢ (𝑄:(0...𝑀)⟶(𝐴[,]𝐵) → ran 𝑄 ⊆ (𝐴[,]𝐵))
23	21, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
24	13	fourierdlem2 39002	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
25	14, 24	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
26	15, 25	mpbid 221	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
27	26	simpld 474	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)))
28		elmapi 7765	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ (ℝ ↑_𝑚 (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
29		ffn 5958	. . . . . . . . . . . . . 14 ⊢ (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
30	27, 28, 29	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 Fn (0...𝑀))
31		fzfid 12634	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
32		fnfi 8123	. . . . . . . . . . . . 13 ⊢ ((𝑄 Fn (0...𝑀) ∧ (0...𝑀) ∈ Fin) → 𝑄 ∈ Fin)
33	30, 31, 32	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ Fin)
34		rnfi 8132	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ Fin → ran 𝑄 ∈ Fin)
35	33, 34	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝑄 ∈ Fin)
36	26	simprd 478	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
37	36	simpld 474	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
38	37	simpld 474	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
39	14	nnnn0d 11228	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
40		nn0uz 11598	. . . . . . . . . . . . . . 15 ⊢ ℕ₀ = (ℤ_≥‘0)
41	39, 40	syl6eleq 2698	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
42		eluzfz1 12219	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
43	41, 42	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ (0...𝑀))
44		fnfvelrn 6264	. . . . . . . . . . . . 13 ⊢ ((𝑄 Fn (0...𝑀) ∧ 0 ∈ (0...𝑀)) → (𝑄‘0) ∈ ran 𝑄)
45	30, 43, 44	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘0) ∈ ran 𝑄)
46	38, 45	eqeltrrd 2689	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ran 𝑄)
47	37	simprd 478	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
48		eluzfz2 12220	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
49	41, 48	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
50		fnfvelrn 6264	. . . . . . . . . . . . 13 ⊢ ((𝑄 Fn (0...𝑀) ∧ 𝑀 ∈ (0...𝑀)) → (𝑄‘𝑀) ∈ ran 𝑄)
51	30, 49, 50	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ran 𝑄)
52	47, 51	eqeltrrd 2689	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ran 𝑄)
53		eqid 2610	. . . . . . . . . . 11 ⊢ (abs ∘ − ) = (abs ∘ − )
54		eqid 2610	. . . . . . . . . . 11 ⊢ ((ran 𝑄 × ran 𝑄) ∖ I ) = ((ran 𝑄 × ran 𝑄) ∖ I )
55		eqid 2610	. . . . . . . . . . 11 ⊢ ran ((abs ∘ − ) ↾ ((ran 𝑄 × ran 𝑄) ∖ I )) = ran ((abs ∘ − ) ↾ ((ran 𝑄 × ran 𝑄) ∖ I ))
56		eqid 2610	. . . . . . . . . . 11 ⊢ inf(ran ((abs ∘ − ) ↾ ((ran 𝑄 × ran 𝑄) ∖ I )), ℝ, < ) = inf(ran ((abs ∘ − ) ↾ ((ran 𝑄 × ran 𝑄) ∖ I )), ℝ, < )
57		fourierdlem54.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ ℝ)
58		eqid 2610	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
59		eqid 2610	. . . . . . . . . . 11 ⊢ ((topGen‘ran (,)) ↾_t (𝐶[,]𝐷)) = ((topGen‘ran (,)) ↾_t (𝐶[,]𝐷))
60		oveq1 6556	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝑥 + (𝑘 · 𝑇)) = (𝑤 + (𝑘 · 𝑇)))
61	60	eleq1d 2672	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → ((𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄))
62	61	rexbidv 3034	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄))
63	62	cbvrabv 3172	. . . . . . . . . . 11 ⊢ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} = {𝑤 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑤 + (𝑘 · 𝑇)) ∈ ran 𝑄}
64		oveq1 6556	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑗 → (𝑖 · 𝑇) = (𝑗 · 𝑇))
65	64	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → (𝑦 + (𝑖 · 𝑇)) = (𝑦 + (𝑗 · 𝑇)))
66	65	eleq1d 2672	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → ((𝑦 + (𝑖 · 𝑇)) ∈ ran 𝑄 ↔ (𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄))
67	66	anbi1d 737	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (((𝑦 + (𝑖 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄) ↔ ((𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄)))
68		oveq1 6556	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑘 → (𝑙 · 𝑇) = (𝑘 · 𝑇))
69	68	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝑘 → (𝑧 + (𝑙 · 𝑇)) = (𝑧 + (𝑘 · 𝑇)))
70	69	eleq1d 2672	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑘 → ((𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄 ↔ (𝑧 + (𝑘 · 𝑇)) ∈ ran 𝑄))
71	70	anbi2d 736	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑘 → (((𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄) ↔ ((𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑘 · 𝑇)) ∈ ran 𝑄)))
72	67, 71	cbvrex2v 3156	. . . . . . . . . . . 12 ⊢ (∃𝑖 ∈ ℤ ∃𝑙 ∈ ℤ ((𝑦 + (𝑖 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄) ↔ ∃𝑗 ∈ ℤ ∃𝑘 ∈ ℤ ((𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑘 · 𝑇)) ∈ ran 𝑄))
73	72	anbi2i 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ 𝑦 < 𝑧)) ∧ ∃𝑖 ∈ ℤ ∃𝑙 ∈ ℤ ((𝑦 + (𝑖 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑙 · 𝑇)) ∈ ran 𝑄)) ↔ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ∧ 𝑦 < 𝑧)) ∧ ∃𝑗 ∈ ℤ ∃𝑘 ∈ ℤ ((𝑦 + (𝑗 · 𝑇)) ∈ ran 𝑄 ∧ (𝑧 + (𝑘 · 𝑇)) ∈ ran 𝑄)))
74	17, 18, 19, 20, 23, 35, 46, 52, 53, 54, 55, 56, 4, 57, 58, 59, 63, 73	fourierdlem42 39042	. . . . . . . . . 10 ⊢ (𝜑 → {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ∈ Fin)
75		unfi 8112	. . . . . . . . . 10 ⊢ (({𝐶, 𝐷} ∈ Fin ∧ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ∈ Fin) → ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) ∈ Fin)
76	12, 74, 75	sylancr 694	. . . . . . . . 9 ⊢ (𝜑 → ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) ∈ Fin)
77	8, 76	syl5eqel 2692	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ Fin)
78		hashnncl 13018	. . . . . . . 8 ⊢ (𝐻 ∈ Fin → ((#‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
79	77, 78	syl 17	. . . . . . 7 ⊢ (𝜑 → ((#‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅))
80	11, 79	mpbird 246	. . . . . 6 ⊢ (𝜑 → (#‘𝐻) ∈ ℕ)
81	80	nnzd 11357	. . . . 5 ⊢ (𝜑 → (#‘𝐻) ∈ ℤ)
82		fourierdlem54.cd	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 < 𝐷)
83	4, 82	ltned 10052	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ≠ 𝐷)
84		hashprg 13043	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶 ≠ 𝐷 ↔ (#‘{𝐶, 𝐷}) = 2))
85	4, 57, 84	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ≠ 𝐷 ↔ (#‘{𝐶, 𝐷}) = 2))
86	83, 85	mpbid 221	. . . . . . 7 ⊢ (𝜑 → (#‘{𝐶, 𝐷}) = 2)
87	86	eqcomd 2616	. . . . . 6 ⊢ (𝜑 → 2 = (#‘{𝐶, 𝐷}))
88		ssun1 3738	. . . . . . . . 9 ⊢ {𝐶, 𝐷} ⊆ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
89	88	a1i 11	. . . . . . . 8 ⊢ (𝜑 → {𝐶, 𝐷} ⊆ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
90	89, 8	syl6sseqr 3615	. . . . . . 7 ⊢ (𝜑 → {𝐶, 𝐷} ⊆ 𝐻)
91		hashssle 38453	. . . . . . 7 ⊢ ((𝐻 ∈ Fin ∧ {𝐶, 𝐷} ⊆ 𝐻) → (#‘{𝐶, 𝐷}) ≤ (#‘𝐻))
92	77, 90, 91	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (#‘{𝐶, 𝐷}) ≤ (#‘𝐻))
93	87, 92	eqbrtrd 4605	. . . . 5 ⊢ (𝜑 → 2 ≤ (#‘𝐻))
94		eluz2 11569	. . . . 5 ⊢ ((#‘𝐻) ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ (#‘𝐻) ∈ ℤ ∧ 2 ≤ (#‘𝐻)))
95	3, 81, 93, 94	syl3anbrc 1239	. . . 4 ⊢ (𝜑 → (#‘𝐻) ∈ (ℤ_≥‘2))
96		uz2m1nn 11639	. . . 4 ⊢ ((#‘𝐻) ∈ (ℤ_≥‘2) → ((#‘𝐻) − 1) ∈ ℕ)
97	95, 96	syl 17	. . 3 ⊢ (𝜑 → ((#‘𝐻) − 1) ∈ ℕ)
98	1, 97	syl5eqel 2692	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ)
99		prssg 4290	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ↔ {𝐶, 𝐷} ⊆ ℝ))
100	4, 57, 99	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) ↔ {𝐶, 𝐷} ⊆ ℝ))
101	4, 57, 100	mpbi2and 958	. . . . . . . . . . 11 ⊢ (𝜑 → {𝐶, 𝐷} ⊆ ℝ)
102		ssrab2 3650	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ⊆ (𝐶[,]𝐷)
103	4, 57	iccssred 38574	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶[,]𝐷) ⊆ ℝ)
104	102, 103	syl5ss 3579	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ⊆ ℝ)
105	101, 104	unssd 3751	. . . . . . . . . 10 ⊢ (𝜑 → ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) ⊆ ℝ)
106	8, 105	syl5eqss 3612	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ⊆ ℝ)
107		fourierdlem54.s	. . . . . . . . 9 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
108	77, 106, 107, 1	fourierdlem36 39036	. . . . . . . 8 ⊢ (𝜑 → 𝑆 Isom < , < ((0...𝑁), 𝐻))
109		df-isom 5813	. . . . . . . 8 ⊢ (𝑆 Isom < , < ((0...𝑁), 𝐻) ↔ (𝑆:(0...𝑁)–1-1-onto→𝐻 ∧ ∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆‘𝑥) < (𝑆‘𝑦))))
110	108, 109	sylib 207	. . . . . . 7 ⊢ (𝜑 → (𝑆:(0...𝑁)–1-1-onto→𝐻 ∧ ∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆‘𝑥) < (𝑆‘𝑦))))
111	110	simpld 474	. . . . . 6 ⊢ (𝜑 → 𝑆:(0...𝑁)–1-1-onto→𝐻)
112		f1of 6050	. . . . . 6 ⊢ (𝑆:(0...𝑁)–1-1-onto→𝐻 → 𝑆:(0...𝑁)⟶𝐻)
113	111, 112	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶𝐻)
114	113, 106	fssd 5970	. . . 4 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶ℝ)
115		reex 9906	. . . . 5 ⊢ ℝ ∈ V
116		ovex 6577	. . . . . 6 ⊢ (0...𝑁) ∈ V
117	116	a1i 11	. . . . 5 ⊢ (𝜑 → (0...𝑁) ∈ V)
118		elmapg 7757	. . . . 5 ⊢ ((ℝ ∈ V ∧ (0...𝑁) ∈ V) → (𝑆 ∈ (ℝ ↑_𝑚 (0...𝑁)) ↔ 𝑆:(0...𝑁)⟶ℝ))
119	115, 117, 118	sylancr 694	. . . 4 ⊢ (𝜑 → (𝑆 ∈ (ℝ ↑_𝑚 (0...𝑁)) ↔ 𝑆:(0...𝑁)⟶ℝ))
120	114, 119	mpbird 246	. . 3 ⊢ (𝜑 → 𝑆 ∈ (ℝ ↑_𝑚 (0...𝑁)))
121		df-f1o 5811	. . . . . . . . . . 11 ⊢ (𝑆:(0...𝑁)–1-1-onto→𝐻 ↔ (𝑆:(0...𝑁)–1-1→𝐻 ∧ 𝑆:(0...𝑁)–onto→𝐻))
122	111, 121	sylib 207	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆:(0...𝑁)–1-1→𝐻 ∧ 𝑆:(0...𝑁)–onto→𝐻))
123	122	simprd 478	. . . . . . . . 9 ⊢ (𝜑 → 𝑆:(0...𝑁)–onto→𝐻)
124		dffo3 6282	. . . . . . . . 9 ⊢ (𝑆:(0...𝑁)–onto→𝐻 ↔ (𝑆:(0...𝑁)⟶𝐻 ∧ ∀ℎ ∈ 𝐻 ∃𝑦 ∈ (0...𝑁)ℎ = (𝑆‘𝑦)))
125	123, 124	sylib 207	. . . . . . . 8 ⊢ (𝜑 → (𝑆:(0...𝑁)⟶𝐻 ∧ ∀ℎ ∈ 𝐻 ∃𝑦 ∈ (0...𝑁)ℎ = (𝑆‘𝑦)))
126	125	simprd 478	. . . . . . 7 ⊢ (𝜑 → ∀ℎ ∈ 𝐻 ∃𝑦 ∈ (0...𝑁)ℎ = (𝑆‘𝑦))
127		eqeq1 2614	. . . . . . . . . 10 ⊢ (ℎ = 𝐶 → (ℎ = (𝑆‘𝑦) ↔ 𝐶 = (𝑆‘𝑦)))
128		eqcom 2617	. . . . . . . . . 10 ⊢ (𝐶 = (𝑆‘𝑦) ↔ (𝑆‘𝑦) = 𝐶)
129	127, 128	syl6bb 275	. . . . . . . . 9 ⊢ (ℎ = 𝐶 → (ℎ = (𝑆‘𝑦) ↔ (𝑆‘𝑦) = 𝐶))
130	129	rexbidv 3034	. . . . . . . 8 ⊢ (ℎ = 𝐶 → (∃𝑦 ∈ (0...𝑁)ℎ = (𝑆‘𝑦) ↔ ∃𝑦 ∈ (0...𝑁)(𝑆‘𝑦) = 𝐶))
131	130	rspcv 3278	. . . . . . 7 ⊢ (𝐶 ∈ 𝐻 → (∀ℎ ∈ 𝐻 ∃𝑦 ∈ (0...𝑁)ℎ = (𝑆‘𝑦) → ∃𝑦 ∈ (0...𝑁)(𝑆‘𝑦) = 𝐶))
132	9, 126, 131	sylc 63	. . . . . 6 ⊢ (𝜑 → ∃𝑦 ∈ (0...𝑁)(𝑆‘𝑦) = 𝐶)
133		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 0 → (𝑆‘𝑦) = (𝑆‘0))
134	133	eqcomd 2616	. . . . . . . . . . . . 13 ⊢ (𝑦 = 0 → (𝑆‘0) = (𝑆‘𝑦))
135	134	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑆‘𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) = (𝑆‘𝑦))
136		simplr 788	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑆‘𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘𝑦) = 𝐶)
137	135, 136	eqtrd 2644	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑆‘𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) = 𝐶)
138	4	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑆‘𝑦) = 𝐶) ∧ 𝑦 = 0) → 𝐶 ∈ ℝ)
139	137, 138	eqeltrd 2688	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑆‘𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) ∈ ℝ)
140	139, 137	eqled 10019	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑆‘𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) ≤ 𝐶)
141	140	3adantl2 1211	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...𝑁) ∧ (𝑆‘𝑦) = 𝐶) ∧ 𝑦 = 0) → (𝑆‘0) ≤ 𝐶)
142	4	rexrd 9968	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
143	57	rexrd 9968	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐷 ∈ ℝ^*)
144	4, 57, 82	ltled 10064	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐶 ≤ 𝐷)
145		lbicc2 12159	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ≤ 𝐷) → 𝐶 ∈ (𝐶[,]𝐷))
146	142, 143, 144, 145	syl3anc 1318	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐶 ∈ (𝐶[,]𝐷))
147		ubicc2 12160	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^* ∧ 𝐶 ≤ 𝐷) → 𝐷 ∈ (𝐶[,]𝐷))
148	142, 143, 144, 147	syl3anc 1318	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐷 ∈ (𝐶[,]𝐷))
149		prssg 4290	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ (𝐶[,]𝐷) ∧ 𝐷 ∈ (𝐶[,]𝐷)) → ((𝐶 ∈ (𝐶[,]𝐷) ∧ 𝐷 ∈ (𝐶[,]𝐷)) ↔ {𝐶, 𝐷} ⊆ (𝐶[,]𝐷)))
150	146, 148, 149	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐶 ∈ (𝐶[,]𝐷) ∧ 𝐷 ∈ (𝐶[,]𝐷)) ↔ {𝐶, 𝐷} ⊆ (𝐶[,]𝐷)))
151	146, 148, 150	mpbi2and 958	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝐶, 𝐷} ⊆ (𝐶[,]𝐷))
152	102	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ⊆ (𝐶[,]𝐷))
153	151, 152	unssd 3751	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}) ⊆ (𝐶[,]𝐷))
154	8, 153	syl5eqss 3612	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 ⊆ (𝐶[,]𝐷))
155		nnm1nn0 11211	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝐻) ∈ ℕ → ((#‘𝐻) − 1) ∈ ℕ₀)
156	80, 155	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((#‘𝐻) − 1) ∈ ℕ₀)
157	1, 156	syl5eqel 2692	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
158	157, 40	syl6eleq 2698	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
159		eluzfz1 12219	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑁))
160	158, 159	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ (0...𝑁))
161	113, 160	ffvelrnd 6268	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆‘0) ∈ 𝐻)
162	154, 161	sseldd 3569	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘0) ∈ (𝐶[,]𝐷))
163	103, 162	sseldd 3569	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘0) ∈ ℝ)
164	163	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑦 = 0) → (𝑆‘0) ∈ ℝ)
165	164	3ad2antl1 1216	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...𝑁) ∧ (𝑆‘𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆‘0) ∈ ℝ)
166	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑦 = 0) → 𝐶 ∈ ℝ)
167	166	3ad2antl1 1216	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...𝑁) ∧ (𝑆‘𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → 𝐶 ∈ ℝ)
168		elfzelz 12213	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0...𝑁) → 𝑦 ∈ ℤ)
169	168	zred 11358	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0...𝑁) → 𝑦 ∈ ℝ)
170	169	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ (0...𝑁) ∧ ¬ 𝑦 = 0) → 𝑦 ∈ ℝ)
171		elfzle1 12215	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0...𝑁) → 0 ≤ 𝑦)
172	171	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ (0...𝑁) ∧ ¬ 𝑦 = 0) → 0 ≤ 𝑦)
173		neqne 2790	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑦 = 0 → 𝑦 ≠ 0)
174	173	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ (0...𝑁) ∧ ¬ 𝑦 = 0) → 𝑦 ≠ 0)
175	170, 172, 174	ne0gt0d 10053	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ (0...𝑁) ∧ ¬ 𝑦 = 0) → 0 < 𝑦)
176	175	3ad2antl2 1217	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...𝑁) ∧ (𝑆‘𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → 0 < 𝑦)
177		simpl1 1057	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...𝑁) ∧ (𝑆‘𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → 𝜑)
178		simpl2 1058	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...𝑁) ∧ (𝑆‘𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → 𝑦 ∈ (0...𝑁))
179	110	simprd 478	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆‘𝑥) < (𝑆‘𝑦)))
180		breq1 4586	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 0 → (𝑥 < 𝑦 ↔ 0 < 𝑦))
181		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 0 → (𝑆‘𝑥) = (𝑆‘0))
182	181	breq1d 4593	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 0 → ((𝑆‘𝑥) < (𝑆‘𝑦) ↔ (𝑆‘0) < (𝑆‘𝑦)))
183	180, 182	bibi12d 334	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 0 → ((𝑥 < 𝑦 ↔ (𝑆‘𝑥) < (𝑆‘𝑦)) ↔ (0 < 𝑦 ↔ (𝑆‘0) < (𝑆‘𝑦))))
184	183	ralbidv 2969	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 0 → (∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆‘𝑥) < (𝑆‘𝑦)) ↔ ∀𝑦 ∈ (0...𝑁)(0 < 𝑦 ↔ (𝑆‘0) < (𝑆‘𝑦))))
185	184	rspcv 3278	. . . . . . . . . . . . . 14 ⊢ (0 ∈ (0...𝑁) → (∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆‘𝑥) < (𝑆‘𝑦)) → ∀𝑦 ∈ (0...𝑁)(0 < 𝑦 ↔ (𝑆‘0) < (𝑆‘𝑦))))
186	160, 179, 185	sylc 63	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑦 ∈ (0...𝑁)(0 < 𝑦 ↔ (𝑆‘0) < (𝑆‘𝑦)))
187	186	r19.21bi 2916	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...𝑁)) → (0 < 𝑦 ↔ (𝑆‘0) < (𝑆‘𝑦)))
188	177, 178, 187	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...𝑁) ∧ (𝑆‘𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (0 < 𝑦 ↔ (𝑆‘0) < (𝑆‘𝑦)))
189	176, 188	mpbid 221	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...𝑁) ∧ (𝑆‘𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆‘0) < (𝑆‘𝑦))
190		simpl3 1059	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...𝑁) ∧ (𝑆‘𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆‘𝑦) = 𝐶)
191	189, 190	breqtrd 4609	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...𝑁) ∧ (𝑆‘𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆‘0) < 𝐶)
192	165, 167, 191	ltled 10064	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...𝑁) ∧ (𝑆‘𝑦) = 𝐶) ∧ ¬ 𝑦 = 0) → (𝑆‘0) ≤ 𝐶)
193	141, 192	pm2.61dan 828	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...𝑁) ∧ (𝑆‘𝑦) = 𝐶) → (𝑆‘0) ≤ 𝐶)
194	193	rexlimdv3a 3015	. . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ (0...𝑁)(𝑆‘𝑦) = 𝐶 → (𝑆‘0) ≤ 𝐶))
195	132, 194	mpd 15	. . . . 5 ⊢ (𝜑 → (𝑆‘0) ≤ 𝐶)
196		elicc2 12109	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝑆‘0) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘0) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘0) ∧ (𝑆‘0) ≤ 𝐷)))
197	4, 57, 196	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘0) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘0) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘0) ∧ (𝑆‘0) ≤ 𝐷)))
198	162, 197	mpbid 221	. . . . . 6 ⊢ (𝜑 → ((𝑆‘0) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘0) ∧ (𝑆‘0) ≤ 𝐷))
199	198	simp2d 1067	. . . . 5 ⊢ (𝜑 → 𝐶 ≤ (𝑆‘0))
200	163, 4	letri3d 10058	. . . . 5 ⊢ (𝜑 → ((𝑆‘0) = 𝐶 ↔ ((𝑆‘0) ≤ 𝐶 ∧ 𝐶 ≤ (𝑆‘0))))
201	195, 199, 200	mpbir2and 959	. . . 4 ⊢ (𝜑 → (𝑆‘0) = 𝐶)
202		eluzfz2 12220	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 𝑁 ∈ (0...𝑁))
203	158, 202	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
204	113, 203	ffvelrnd 6268	. . . . . . . 8 ⊢ (𝜑 → (𝑆‘𝑁) ∈ 𝐻)
205	154, 204	sseldd 3569	. . . . . . 7 ⊢ (𝜑 → (𝑆‘𝑁) ∈ (𝐶[,]𝐷))
206		elicc2 12109	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝑆‘𝑁) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘𝑁) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘𝑁) ∧ (𝑆‘𝑁) ≤ 𝐷)))
207	4, 57, 206	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝑁) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘𝑁) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘𝑁) ∧ (𝑆‘𝑁) ≤ 𝐷)))
208	205, 207	mpbid 221	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝑁) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘𝑁) ∧ (𝑆‘𝑁) ≤ 𝐷))
209	208	simp3d 1068	. . . . 5 ⊢ (𝜑 → (𝑆‘𝑁) ≤ 𝐷)
210		prid2g 4240	. . . . . . . . 9 ⊢ (𝐷 ∈ ℝ → 𝐷 ∈ {𝐶, 𝐷})
211		elun1 3742	. . . . . . . . 9 ⊢ (𝐷 ∈ {𝐶, 𝐷} → 𝐷 ∈ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
212	57, 210, 211	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
213	212, 8	syl6eleqr 2699	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝐻)
214		eqeq1 2614	. . . . . . . . . 10 ⊢ (ℎ = 𝐷 → (ℎ = (𝑆‘𝑦) ↔ 𝐷 = (𝑆‘𝑦)))
215		eqcom 2617	. . . . . . . . . 10 ⊢ (𝐷 = (𝑆‘𝑦) ↔ (𝑆‘𝑦) = 𝐷)
216	214, 215	syl6bb 275	. . . . . . . . 9 ⊢ (ℎ = 𝐷 → (ℎ = (𝑆‘𝑦) ↔ (𝑆‘𝑦) = 𝐷))
217	216	rexbidv 3034	. . . . . . . 8 ⊢ (ℎ = 𝐷 → (∃𝑦 ∈ (0...𝑁)ℎ = (𝑆‘𝑦) ↔ ∃𝑦 ∈ (0...𝑁)(𝑆‘𝑦) = 𝐷))
218	217	rspcv 3278	. . . . . . 7 ⊢ (𝐷 ∈ 𝐻 → (∀ℎ ∈ 𝐻 ∃𝑦 ∈ (0...𝑁)ℎ = (𝑆‘𝑦) → ∃𝑦 ∈ (0...𝑁)(𝑆‘𝑦) = 𝐷))
219	213, 126, 218	sylc 63	. . . . . 6 ⊢ (𝜑 → ∃𝑦 ∈ (0...𝑁)(𝑆‘𝑦) = 𝐷)
220	215	biimpri 217	. . . . . . . . 9 ⊢ ((𝑆‘𝑦) = 𝐷 → 𝐷 = (𝑆‘𝑦))
221	220	3ad2ant3 1077	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...𝑁) ∧ (𝑆‘𝑦) = 𝐷) → 𝐷 = (𝑆‘𝑦))
222	114	ffvelrnda 6267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...𝑁)) → (𝑆‘𝑦) ∈ ℝ)
223	103, 205	sseldd 3569	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆‘𝑁) ∈ ℝ)
224	223	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...𝑁)) → (𝑆‘𝑁) ∈ ℝ)
225	169	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...𝑁)) → 𝑦 ∈ ℝ)
226		elfzel2 12211	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0...𝑁) → 𝑁 ∈ ℤ)
227	226	zred 11358	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...𝑁) → 𝑁 ∈ ℝ)
228	227	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...𝑁)) → 𝑁 ∈ ℝ)
229		elfzle2 12216	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...𝑁) → 𝑦 ≤ 𝑁)
230	229	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...𝑁)) → 𝑦 ≤ 𝑁)
231	225, 228, 230	lensymd 10067	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...𝑁)) → ¬ 𝑁 < 𝑦)
232		breq1 4586	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑁 → (𝑥 < 𝑦 ↔ 𝑁 < 𝑦))
233		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑁 → (𝑆‘𝑥) = (𝑆‘𝑁))
234	233	breq1d 4593	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑁 → ((𝑆‘𝑥) < (𝑆‘𝑦) ↔ (𝑆‘𝑁) < (𝑆‘𝑦)))
235	232, 234	bibi12d 334	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑁 → ((𝑥 < 𝑦 ↔ (𝑆‘𝑥) < (𝑆‘𝑦)) ↔ (𝑁 < 𝑦 ↔ (𝑆‘𝑁) < (𝑆‘𝑦))))
236	235	ralbidv 2969	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑁 → (∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆‘𝑥) < (𝑆‘𝑦)) ↔ ∀𝑦 ∈ (0...𝑁)(𝑁 < 𝑦 ↔ (𝑆‘𝑁) < (𝑆‘𝑦))))
237	236	rspcv 3278	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (0...𝑁) → (∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆‘𝑥) < (𝑆‘𝑦)) → ∀𝑦 ∈ (0...𝑁)(𝑁 < 𝑦 ↔ (𝑆‘𝑁) < (𝑆‘𝑦))))
238	203, 179, 237	sylc 63	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑦 ∈ (0...𝑁)(𝑁 < 𝑦 ↔ (𝑆‘𝑁) < (𝑆‘𝑦)))
239	238	r19.21bi 2916	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...𝑁)) → (𝑁 < 𝑦 ↔ (𝑆‘𝑁) < (𝑆‘𝑦)))
240	231, 239	mtbid 313	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...𝑁)) → ¬ (𝑆‘𝑁) < (𝑆‘𝑦))
241	222, 224, 240	nltled 10066	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...𝑁)) → (𝑆‘𝑦) ≤ (𝑆‘𝑁))
242	241	3adant3 1074	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...𝑁) ∧ (𝑆‘𝑦) = 𝐷) → (𝑆‘𝑦) ≤ (𝑆‘𝑁))
243	221, 242	eqbrtrd 4605	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...𝑁) ∧ (𝑆‘𝑦) = 𝐷) → 𝐷 ≤ (𝑆‘𝑁))
244	243	rexlimdv3a 3015	. . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ (0...𝑁)(𝑆‘𝑦) = 𝐷 → 𝐷 ≤ (𝑆‘𝑁)))
245	219, 244	mpd 15	. . . . 5 ⊢ (𝜑 → 𝐷 ≤ (𝑆‘𝑁))
246	223, 57	letri3d 10058	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑁) = 𝐷 ↔ ((𝑆‘𝑁) ≤ 𝐷 ∧ 𝐷 ≤ (𝑆‘𝑁))))
247	209, 245, 246	mpbir2and 959	. . . 4 ⊢ (𝜑 → (𝑆‘𝑁) = 𝐷)
248		elfzoelz 12339	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℤ)
249	248	zred 11358	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℝ)
250	249	ltp1d 10833	. . . . . . 7 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 < (𝑖 + 1))
251	250	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 < (𝑖 + 1))
252	179	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆‘𝑥) < (𝑆‘𝑦)))
253		elfzofz 12354	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ (0...𝑁))
254	253	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0...𝑁))
255		fzofzp1 12431	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^𝑁) → (𝑖 + 1) ∈ (0...𝑁))
256	255	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 + 1) ∈ (0...𝑁))
257		breq1 4586	. . . . . . . . . 10 ⊢ (𝑥 = 𝑖 → (𝑥 < 𝑦 ↔ 𝑖 < 𝑦))
258		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑖 → (𝑆‘𝑥) = (𝑆‘𝑖))
259	258	breq1d 4593	. . . . . . . . . 10 ⊢ (𝑥 = 𝑖 → ((𝑆‘𝑥) < (𝑆‘𝑦) ↔ (𝑆‘𝑖) < (𝑆‘𝑦)))
260	257, 259	bibi12d 334	. . . . . . . . 9 ⊢ (𝑥 = 𝑖 → ((𝑥 < 𝑦 ↔ (𝑆‘𝑥) < (𝑆‘𝑦)) ↔ (𝑖 < 𝑦 ↔ (𝑆‘𝑖) < (𝑆‘𝑦))))
261		breq2 4587	. . . . . . . . . 10 ⊢ (𝑦 = (𝑖 + 1) → (𝑖 < 𝑦 ↔ 𝑖 < (𝑖 + 1)))
262		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑖 + 1) → (𝑆‘𝑦) = (𝑆‘(𝑖 + 1)))
263	262	breq2d 4595	. . . . . . . . . 10 ⊢ (𝑦 = (𝑖 + 1) → ((𝑆‘𝑖) < (𝑆‘𝑦) ↔ (𝑆‘𝑖) < (𝑆‘(𝑖 + 1))))
264	261, 263	bibi12d 334	. . . . . . . . 9 ⊢ (𝑦 = (𝑖 + 1) → ((𝑖 < 𝑦 ↔ (𝑆‘𝑖) < (𝑆‘𝑦)) ↔ (𝑖 < (𝑖 + 1) ↔ (𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))))
265	260, 264	rspc2v 3293	. . . . . . . 8 ⊢ ((𝑖 ∈ (0...𝑁) ∧ (𝑖 + 1) ∈ (0...𝑁)) → (∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆‘𝑥) < (𝑆‘𝑦)) → (𝑖 < (𝑖 + 1) ↔ (𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))))
266	254, 256, 265	syl2anc 691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (∀𝑥 ∈ (0...𝑁)∀𝑦 ∈ (0...𝑁)(𝑥 < 𝑦 ↔ (𝑆‘𝑥) < (𝑆‘𝑦)) → (𝑖 < (𝑖 + 1) ↔ (𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))))
267	252, 266	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑖 < (𝑖 + 1) ↔ (𝑆‘𝑖) < (𝑆‘(𝑖 + 1))))
268	251, 267	mpbid 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))
269	268	ralrimiva 2949	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))
270	201, 247, 269	jca31 555	. . 3 ⊢ (𝜑 → (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1))))
271		fourierdlem54.o	. . . . 5 ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
272	271	fourierdlem2 39002	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑆 ∈ (𝑂‘𝑁) ↔ (𝑆 ∈ (ℝ ↑_𝑚 (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1))))))
273	98, 272	syl 17	. . 3 ⊢ (𝜑 → (𝑆 ∈ (𝑂‘𝑁) ↔ (𝑆 ∈ (ℝ ↑_𝑚 (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1))))))
274	120, 270, 273	mpbir2and 959	. 2 ⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁))
275	98, 274, 108	jca31 555	1 ⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 ∅c0 3874 {cpr 4127 class class class wbr 4583 ↦ cmpt 4643 I cid 4948 × cxp 5036 ran crn 5039 ↾ cres 5040 ∘ ccom 5042 ℩cio 5766 Fn wfn 5799 ⟶wf 5800 –1-1→wf1 5801 –onto→wfo 5802 –1-1-onto→wf1o 5803 ‘cfv 5804 Isom wiso 5805 (class class class)co 6549 ↑_𝑚 cmap 7744 Fincfn 7841 infcinf 8230 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 (,)cioo 12046 [,]cicc 12049 ...cfz 12197 ..^cfzo 12334 #chash 12979 abscabs 13822 ↾_t crest 15904 topGenctg 15921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-rest 15906 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-cmp 21000

This theorem is referenced by: fourierdlem63 39062 fourierdlem64 39063 fourierdlem65 39064 fourierdlem79 39078 fourierdlem89 39088 fourierdlem90 39089 fourierdlem91 39090 fourierdlem100 39099 fourierdlem107 39106 fourierdlem109 39108 fourierdlem112 39111

W3C validator