fourierdlem63 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem63 45480

Description: The upper bound of intervals in the moved partition are mapped to points that are not greater than the corresponding upper bounds in the original partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem63.t	⊢ 𝑇 = (𝐵 − 𝐴)
fourierdlem63.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem63.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem63.q	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
fourierdlem63.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
fourierdlem63.d	⊢ (𝜑 → 𝐷 ∈ ℝ)
fourierdlem63.cltd	⊢ (𝜑 → 𝐶 < 𝐷)
fourierdlem63.o	⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem63.h	⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
fourierdlem63.n	⊢ 𝑁 = ((♯‘𝐻) − 1)
fourierdlem63.s	⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
fourierdlem63.e	⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
fourierdlem63.k	⊢ (𝜑 → 𝐾 ∈ (0...𝑀))
fourierdlem63.j	⊢ (𝜑 → 𝐽 ∈ (0..^𝑁))
fourierdlem63.y	⊢ (𝜑 → 𝑌 ∈ ((𝑆‘𝐽)[,)(𝑆‘(𝐽 + 1))))
fourierdlem63.eyltqk	⊢ (𝜑 → (𝐸‘𝑌) < (𝑄‘𝐾))
fourierdlem63.x	⊢ 𝑋 = ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌))

**Assertion**
Ref	Expression
fourierdlem63	⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ≤ (𝑄‘𝐾))

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

Proof of Theorem fourierdlem63 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem63.e	. . . . 5 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
2	1	a1i 11	. . . 4 ⊢ (𝜑 → 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))))
3		id 22	. . . . . 6 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → 𝑥 = (𝑆‘(𝐽 + 1)))
4		oveq2 7422	. . . . . . . . 9 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝐵 − 𝑥) = (𝐵 − (𝑆‘(𝐽 + 1))))
5	4	oveq1d 7429	. . . . . . . 8 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇))
6	5	fveq2d 6895	. . . . . . 7 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
7	6	oveq1d 7429	. . . . . 6 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
8	3, 7	oveq12d 7432	. . . . 5 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
9	8	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = (𝑆‘(𝐽 + 1))) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
10		fourierdlem63.t	. . . . . . . . . . 11 ⊢ 𝑇 = (𝐵 − 𝐴)
11		fourierdlem63.p	. . . . . . . . . . 11 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
12		fourierdlem63.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ)
13		fourierdlem63.q	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
14		fourierdlem63.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ ℝ)
15		fourierdlem63.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ ℝ)
16		fourierdlem63.cltd	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 < 𝐷)
17		fourierdlem63.o	. . . . . . . . . . 11 ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝‘𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
18		fourierdlem63.h	. . . . . . . . . . 11 ⊢ 𝐻 = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
19		fourierdlem63.n	. . . . . . . . . . 11 ⊢ 𝑁 = ((♯‘𝐻) − 1)
20		fourierdlem63.s	. . . . . . . . . . 11 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))
21	10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20	fourierdlem54 45471	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))
22	21	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)))
23	22	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁))
24	22	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
25	17	fourierdlem2 45420	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑆 ∈ (𝑂‘𝑁) ↔ (𝑆 ∈ (ℝ ↑_m (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1))))))
26	24, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑆 ∈ (𝑂‘𝑁) ↔ (𝑆 ∈ (ℝ ↑_m (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1))))))
27	23, 26	mpbid 231	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∈ (ℝ ↑_m (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))))
28	27	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (ℝ ↑_m (0...𝑁)))
29		elmapi 8859	. . . . . 6 ⊢ (𝑆 ∈ (ℝ ↑_m (0...𝑁)) → 𝑆:(0...𝑁)⟶ℝ)
30	28, 29	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶ℝ)
31		fourierdlem63.j	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (0..^𝑁))
32		fzofzp1 13753	. . . . . 6 ⊢ (𝐽 ∈ (0..^𝑁) → (𝐽 + 1) ∈ (0...𝑁))
33	31, 32	syl 17	. . . . 5 ⊢ (𝜑 → (𝐽 + 1) ∈ (0...𝑁))
34	30, 33	ffvelcdmd 7089	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℝ)
35	11, 12, 13	fourierdlem11 45429	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
36	35	simp2d 1141	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
37	36, 34	resubcld 11664	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − (𝑆‘(𝐽 + 1))) ∈ ℝ)
38	35	simp1d 1140	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
39	36, 38	resubcld 11664	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
40	10, 39	eqeltrid 2832	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ ℝ)
41	35	simp3d 1142	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 < 𝐵)
42	38, 36	posdifd 11823	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
43	41, 42	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
44	43, 10	breqtrrdi 5184	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝑇)
45	44	gt0ne0d 11800	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ≠ 0)
46	37, 40, 45	redivcld 12064	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℝ)
47	46	flcld 13787	. . . . . . 7 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℤ)
48	47	zred 12688	. . . . . 6 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℝ)
49	48, 40	remulcld 11266	. . . . 5 ⊢ (𝜑 → ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) ∈ ℝ)
50	34, 49	readdcld 11265	. . . 4 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) ∈ ℝ)
51	2, 9, 34, 50	fvmptd 7006	. . 3 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
52	51, 50	eqeltrd 2828	. 2 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ)
53	11	fourierdlem2 45420	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
54	12, 53	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
55	13, 54	mpbid 231	. . . . 5 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
56	55	simpld 494	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
57		elmapi 8859	. . . 4 ⊢ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
58	56, 57	syl 17	. . 3 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
59		fourierdlem63.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ (0...𝑀))
60	58, 59	ffvelcdmd 7089	. 2 ⊢ (𝜑 → (𝑄‘𝐾) ∈ ℝ)
61	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → 𝐶 ∈ ℝ)
62	15	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → 𝐷 ∈ ℝ)
63	38	rexrd 11286	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
64		iocssre 13428	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ)
65	63, 36, 64	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℝ)
66	38, 36, 41, 10, 1	fourierdlem4 45422	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸:ℝ⟶(𝐴(,]𝐵))
67		fourierdlem63.y	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ∈ ((𝑆‘𝐽)[,)(𝑆‘(𝐽 + 1))))
68		elfzofz 13672	. . . . . . . . . . . . . . . . 17 ⊢ (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ (0...𝑁))
69	31, 68	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐽 ∈ (0...𝑁))
70	30, 69	ffvelcdmd 7089	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑆‘𝐽) ∈ ℝ)
71	34	rexrd 11286	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℝ^*)
72		elico2 13412	. . . . . . . . . . . . . . 15 ⊢ (((𝑆‘𝐽) ∈ ℝ ∧ (𝑆‘(𝐽 + 1)) ∈ ℝ^*) → (𝑌 ∈ ((𝑆‘𝐽)[,)(𝑆‘(𝐽 + 1))) ↔ (𝑌 ∈ ℝ ∧ (𝑆‘𝐽) ≤ 𝑌 ∧ 𝑌 < (𝑆‘(𝐽 + 1)))))
73	70, 71, 72	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑌 ∈ ((𝑆‘𝐽)[,)(𝑆‘(𝐽 + 1))) ↔ (𝑌 ∈ ℝ ∧ (𝑆‘𝐽) ≤ 𝑌 ∧ 𝑌 < (𝑆‘(𝐽 + 1)))))
74	67, 73	mpbid 231	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑌 ∈ ℝ ∧ (𝑆‘𝐽) ≤ 𝑌 ∧ 𝑌 < (𝑆‘(𝐽 + 1))))
75	74	simp1d 1140	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ ℝ)
76	66, 75	ffvelcdmd 7089	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸‘𝑌) ∈ (𝐴(,]𝐵))
77	65, 76	sseldd 3979	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸‘𝑌) ∈ ℝ)
78	77, 75	resubcld 11664	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸‘𝑌) − 𝑌) ∈ ℝ)
79	60, 78	resubcld 11664	. . . . . . . 8 ⊢ (𝜑 → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ∈ ℝ)
80	79	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ∈ ℝ)
81		icossicc 13437	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝐽)[,)(𝑆‘(𝐽 + 1))) ⊆ ((𝑆‘𝐽)[,](𝑆‘(𝐽 + 1)))
82	14	rexrd 11286	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
83	15	rexrd 11286	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ ℝ^*)
84	17, 24, 23	fourierdlem15 45433	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐶[,]𝐷))
85	82, 83, 84, 31	fourierdlem8 45426	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑆‘𝐽)[,](𝑆‘(𝐽 + 1))) ⊆ (𝐶[,]𝐷))
86	81, 85	sstrid 3989	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆‘𝐽)[,)(𝑆‘(𝐽 + 1))) ⊆ (𝐶[,]𝐷))
87	86, 67	sseldd 3979	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ (𝐶[,]𝐷))
88		elicc2 13413	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝑌 ∈ (𝐶[,]𝐷) ↔ (𝑌 ∈ ℝ ∧ 𝐶 ≤ 𝑌 ∧ 𝑌 ≤ 𝐷)))
89	14, 15, 88	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌 ∈ (𝐶[,]𝐷) ↔ (𝑌 ∈ ℝ ∧ 𝐶 ≤ 𝑌 ∧ 𝑌 ≤ 𝐷)))
90	87, 89	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 ∈ ℝ ∧ 𝐶 ≤ 𝑌 ∧ 𝑌 ≤ 𝐷))
91	90	simp2d 1141	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ≤ 𝑌)
92	60, 77	resubcld 11664	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘𝐾) − (𝐸‘𝑌)) ∈ ℝ)
93		fourierdlem63.eyltqk	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸‘𝑌) < (𝑄‘𝐾))
94	77, 60	posdifd 11823	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐸‘𝑌) < (𝑄‘𝐾) ↔ 0 < ((𝑄‘𝐾) − (𝐸‘𝑌))))
95	93, 94	mpbid 231	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < ((𝑄‘𝐾) − (𝐸‘𝑌)))
96	92, 95	elrpd 13037	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄‘𝐾) − (𝐸‘𝑌)) ∈ ℝ⁺)
97	75, 96	ltaddrpd 13073	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 < (𝑌 + ((𝑄‘𝐾) − (𝐸‘𝑌))))
98	60	recnd 11264	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘𝐾) ∈ ℂ)
99	77	recnd 11264	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸‘𝑌) ∈ ℂ)
100	75	recnd 11264	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ ℂ)
101	98, 99, 100	subsub3d 11623	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) = (((𝑄‘𝐾) + 𝑌) − (𝐸‘𝑌)))
102	98, 100	addcomd 11438	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘𝐾) + 𝑌) = (𝑌 + (𝑄‘𝐾)))
103	102	oveq1d 7429	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑄‘𝐾) + 𝑌) − (𝐸‘𝑌)) = ((𝑌 + (𝑄‘𝐾)) − (𝐸‘𝑌)))
104	100, 98, 99	addsubassd 11613	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑌 + (𝑄‘𝐾)) − (𝐸‘𝑌)) = (𝑌 + ((𝑄‘𝐾) − (𝐸‘𝑌))))
105	101, 103, 104	3eqtrrd 2772	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 + ((𝑄‘𝐾) − (𝐸‘𝑌))) = ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)))
106	97, 105	breqtrd 5168	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 < ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)))
107	14, 75, 79, 91, 106	lelttrd 11394	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 < ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)))
108	14, 79, 107	ltled 11384	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ≤ ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)))
109	108	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → 𝐶 ≤ ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)))
110	34	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑆‘(𝐽 + 1)) ∈ ℝ)
111	60	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑄‘𝐾) ∈ ℝ)
112	52, 34	resubcld 11664	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) ∈ ℝ)
113	112	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) ∈ ℝ)
114	111, 113	resubcld 11664	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1)))) ∈ ℝ)
115	74	simp3d 1142	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 < (𝑆‘(𝐽 + 1)))
116	75, 34, 115	ltled 11384	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ≤ (𝑆‘(𝐽 + 1)))
117	38, 36, 41, 10, 1, 75, 34, 116	fourierdlem7 45425	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) ≤ ((𝐸‘𝑌) − 𝑌))
118	112, 78, 60, 117	lesub2dd 11853	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ≤ ((𝑄‘𝐾) − ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1)))))
119	118	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ≤ ((𝑄‘𝐾) − ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1)))))
120	98	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑄‘𝐾) ∈ ℂ)
121	52	recnd 11264	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℂ)
122	121	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℂ)
123	110	recnd 11264	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑆‘(𝐽 + 1)) ∈ ℂ)
124	120, 122, 123	subsubd 11621	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1)))) = (((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) + (𝑆‘(𝐽 + 1))))
125	98, 121	subcld 11593	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) ∈ ℂ)
126	34	recnd 11264	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℂ)
127	125, 126	addcomd 11438	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) + (𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) + ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1))))))
128	127	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) + (𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) + ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1))))))
129	124, 128	eqtrd 2767	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1)))) = ((𝑆‘(𝐽 + 1)) + ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1))))))
130		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1))))
131	52	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ)
132	111, 131	sublt0d 11862	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) < 0 ↔ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))))
133	130, 132	mpbird 257	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) < 0)
134	111, 131	resubcld 11664	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) ∈ ℝ)
135		ltaddneg 11451	. . . . . . . . . . . . 13 ⊢ ((((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) ∈ ℝ ∧ (𝑆‘(𝐽 + 1)) ∈ ℝ) → (((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) < 0 ↔ ((𝑆‘(𝐽 + 1)) + ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1))))) < (𝑆‘(𝐽 + 1))))
136	134, 110, 135	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) < 0 ↔ ((𝑆‘(𝐽 + 1)) + ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1))))) < (𝑆‘(𝐽 + 1))))
137	133, 136	mpbid 231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑆‘(𝐽 + 1)) + ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1))))) < (𝑆‘(𝐽 + 1)))
138	129, 137	eqbrtrd 5164	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1)))) < (𝑆‘(𝐽 + 1)))
139	80, 114, 110, 119, 138	lelttrd 11394	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) < (𝑆‘(𝐽 + 1)))
140	84, 33	ffvelcdmd 7089	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ (𝐶[,]𝐷))
141		elicc2 13413	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → ((𝑆‘(𝐽 + 1)) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘(𝐽 + 1)) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘(𝐽 + 1)) ∧ (𝑆‘(𝐽 + 1)) ≤ 𝐷)))
142	14, 15, 141	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘(𝐽 + 1)) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘(𝐽 + 1)) ∧ (𝑆‘(𝐽 + 1)) ≤ 𝐷)))
143	140, 142	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘(𝐽 + 1)) ∧ (𝑆‘(𝐽 + 1)) ≤ 𝐷))
144	143	simp3d 1142	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ≤ 𝐷)
145	144	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑆‘(𝐽 + 1)) ≤ 𝐷)
146	80, 110, 62, 139, 145	ltletrd 11396	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) < 𝐷)
147	80, 62, 146	ltled 11384	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ≤ 𝐷)
148	61, 62, 80, 109, 147	eliccd 44812	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ∈ (𝐶[,]𝐷))
149		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑌 → 𝑥 = 𝑌)
150		oveq2 7422	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑌 → (𝐵 − 𝑥) = (𝐵 − 𝑌))
151	150	oveq1d 7429	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑌 → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − 𝑌) / 𝑇))
152	151	fveq2d 6895	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑌 → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − 𝑌) / 𝑇)))
153	152	oveq1d 7429	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑌 → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇))
154	149, 153	oveq12d 7432	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑌 → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = (𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)))
155	154	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = (𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)))
156	36, 75	resubcld 11664	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐵 − 𝑌) ∈ ℝ)
157	156, 40, 45	redivcld 12064	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐵 − 𝑌) / 𝑇) ∈ ℝ)
158	157	flcld 13787	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘((𝐵 − 𝑌) / 𝑇)) ∈ ℤ)
159	158	zred 12688	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (⌊‘((𝐵 − 𝑌) / 𝑇)) ∈ ℝ)
160	159, 40	remulcld 11266	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇) ∈ ℝ)
161	75, 160	readdcld 11265	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)) ∈ ℝ)
162	2, 155, 75, 161	fvmptd 7006	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸‘𝑌) = (𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)))
163	162	oveq1d 7429	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸‘𝑌) − 𝑌) = ((𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)) − 𝑌))
164	163	oveq1d 7429	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐸‘𝑌) − 𝑌) / 𝑇) = (((𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)) − 𝑌) / 𝑇))
165	160	recnd 11264	. . . . . . . . . . . 12 ⊢ (𝜑 → ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇) ∈ ℂ)
166	100, 165	pncan2d 11595	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)) − 𝑌) = ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇))
167	166	oveq1d 7429	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)) − 𝑌) / 𝑇) = (((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇) / 𝑇))
168	159	recnd 11264	. . . . . . . . . . 11 ⊢ (𝜑 → (⌊‘((𝐵 − 𝑌) / 𝑇)) ∈ ℂ)
169	40	recnd 11264	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ ℂ)
170	168, 169, 45	divcan4d 12018	. . . . . . . . . 10 ⊢ (𝜑 → (((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇) / 𝑇) = (⌊‘((𝐵 − 𝑌) / 𝑇)))
171	164, 167, 170	3eqtrd 2771	. . . . . . . . 9 ⊢ (𝜑 → (((𝐸‘𝑌) − 𝑌) / 𝑇) = (⌊‘((𝐵 − 𝑌) / 𝑇)))
172	171, 158	eqeltrd 2828	. . . . . . . 8 ⊢ (𝜑 → (((𝐸‘𝑌) − 𝑌) / 𝑇) ∈ ℤ)
173	78	recnd 11264	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘𝑌) − 𝑌) ∈ ℂ)
174	173, 169, 45	divcan1d 12013	. . . . . . . . . . 11 ⊢ (𝜑 → ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇) = ((𝐸‘𝑌) − 𝑌))
175	174	oveq2d 7430	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇)) = (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((𝐸‘𝑌) − 𝑌)))
176	98, 173	npcand 11597	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((𝐸‘𝑌) − 𝑌)) = (𝑄‘𝐾))
177	175, 176	eqtrd 2767	. . . . . . . . 9 ⊢ (𝜑 → (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇)) = (𝑄‘𝐾))
178		ffun 6719	. . . . . . . . . . 11 ⊢ (𝑄:(0...𝑀)⟶ℝ → Fun 𝑄)
179	58, 178	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝑄)
180	58	fdmd 6727	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝑄 = (0...𝑀))
181	59, 180	eleqtrrd 2831	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ dom 𝑄)
182		fvelrn 7080	. . . . . . . . . 10 ⊢ ((Fun 𝑄 ∧ 𝐾 ∈ dom 𝑄) → (𝑄‘𝐾) ∈ ran 𝑄)
183	179, 181, 182	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘𝐾) ∈ ran 𝑄)
184	177, 183	eqeltrd 2828	. . . . . . . 8 ⊢ (𝜑 → (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇)) ∈ ran 𝑄)
185		oveq1 7421	. . . . . . . . . . 11 ⊢ (𝑘 = (((𝐸‘𝑌) − 𝑌) / 𝑇) → (𝑘 · 𝑇) = ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇))
186	185	oveq2d 7430	. . . . . . . . . 10 ⊢ (𝑘 = (((𝐸‘𝑌) − 𝑌) / 𝑇) → (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + (𝑘 · 𝑇)) = (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇)))
187	186	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑘 = (((𝐸‘𝑌) − 𝑌) / 𝑇) → ((((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇)) ∈ ran 𝑄))
188	187	rspcev 3607	. . . . . . . 8 ⊢ (((((𝐸‘𝑌) − 𝑌) / 𝑇) ∈ ℤ ∧ (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇)) ∈ ran 𝑄) → ∃𝑘 ∈ ℤ (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + (𝑘 · 𝑇)) ∈ ran 𝑄)
189	172, 184, 188	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ∃𝑘 ∈ ℤ (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + (𝑘 · 𝑇)) ∈ ran 𝑄)
190	189	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ∃𝑘 ∈ ℤ (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + (𝑘 · 𝑇)) ∈ ran 𝑄)
191		oveq1 7421	. . . . . . . . 9 ⊢ (𝑥 = ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) → (𝑥 + (𝑘 · 𝑇)) = (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + (𝑘 · 𝑇)))
192	191	eleq1d 2813	. . . . . . . 8 ⊢ (𝑥 = ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) → ((𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + (𝑘 · 𝑇)) ∈ ran 𝑄))
193	192	rexbidv 3173	. . . . . . 7 ⊢ (𝑥 = ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) → (∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ ∃𝑘 ∈ ℤ (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + (𝑘 · 𝑇)) ∈ ran 𝑄))
194	193	elrab 3680	. . . . . 6 ⊢ (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ∈ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} ↔ (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ∈ (𝐶[,]𝐷) ∧ ∃𝑘 ∈ ℤ (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + (𝑘 · 𝑇)) ∈ ran 𝑄))
195	148, 190, 194	sylanbrc 582	. . . . 5 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ∈ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})
196		elun2 4173	. . . . 5 ⊢ (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ∈ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄} → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ∈ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
197	195, 196	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ∈ ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄}))
198		fourierdlem63.x	. . . 4 ⊢ 𝑋 = ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌))
199	197, 198, 18	3eltr4g 2845	. . 3 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → 𝑋 ∈ 𝐻)
200		elfzelz 13525	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
201	200	ad2antlr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))) → 𝑗 ∈ ℤ)
202		elfzoelz 13656	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ ℤ)
203	31, 202	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℤ)
204	203	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))) → 𝐽 ∈ ℤ)
205		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → (𝑆‘𝐽) < (𝑆‘𝑗))
206	21	simprd 495	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 Isom < , < ((0...𝑁), 𝐻))
207	206	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → 𝑆 Isom < , < ((0...𝑁), 𝐻))
208	69	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → 𝐽 ∈ (0...𝑁))
209		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → 𝑗 ∈ (0...𝑁))
210		isorel 7328	. . . . . . . . . . . 12 ⊢ ((𝑆 Isom < , < ((0...𝑁), 𝐻) ∧ (𝐽 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑁))) → (𝐽 < 𝑗 ↔ (𝑆‘𝐽) < (𝑆‘𝑗)))
211	207, 208, 209, 210	syl12anc 836	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → (𝐽 < 𝑗 ↔ (𝑆‘𝐽) < (𝑆‘𝑗)))
212	205, 211	mpbird 257	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → 𝐽 < 𝑗)
213	212	adantrr 716	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))) → 𝐽 < 𝑗)
214		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))
215	206	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → 𝑆 Isom < , < ((0...𝑁), 𝐻))
216		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → 𝑗 ∈ (0...𝑁))
217	33	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → (𝐽 + 1) ∈ (0...𝑁))
218		isorel 7328	. . . . . . . . . . . 12 ⊢ ((𝑆 Isom < , < ((0...𝑁), 𝐻) ∧ (𝑗 ∈ (0...𝑁) ∧ (𝐽 + 1) ∈ (0...𝑁))) → (𝑗 < (𝐽 + 1) ↔ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))))
219	215, 216, 217, 218	syl12anc 836	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → (𝑗 < (𝐽 + 1) ↔ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))))
220	214, 219	mpbird 257	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → 𝑗 < (𝐽 + 1))
221	220	adantrl 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))) → 𝑗 < (𝐽 + 1))
222		btwnnz 12660	. . . . . . . . 9 ⊢ ((𝐽 ∈ ℤ ∧ 𝐽 < 𝑗 ∧ 𝑗 < (𝐽 + 1)) → ¬ 𝑗 ∈ ℤ)
223	204, 213, 221, 222	syl3anc 1369	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))) → ¬ 𝑗 ∈ ℤ)
224	201, 223	pm2.65da 816	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ¬ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))))
225	224	adantlr 714	. . . . . 6 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ 𝑗 ∈ (0...𝑁)) → ¬ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))))
226	70	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝐽) ∈ ℝ)
227	75	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → 𝑌 ∈ ℝ)
228	30	ffvelcdmda 7088	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑆‘𝑗) ∈ ℝ)
229	228	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝑗) ∈ ℝ)
230	74	simp2d 1141	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆‘𝐽) ≤ 𝑌)
231	230	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝐽) ≤ 𝑌)
232	106, 198	breqtrrdi 5184	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 < 𝑋)
233	232	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆‘𝑗) = 𝑋) → 𝑌 < 𝑋)
234		eqcom 2734	. . . . . . . . . . . . 13 ⊢ (𝑋 = (𝑆‘𝑗) ↔ (𝑆‘𝑗) = 𝑋)
235	234	biimpri 227	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑗) = 𝑋 → 𝑋 = (𝑆‘𝑗))
236	235	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆‘𝑗) = 𝑋) → 𝑋 = (𝑆‘𝑗))
237	233, 236	breqtrd 5168	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑆‘𝑗) = 𝑋) → 𝑌 < (𝑆‘𝑗))
238	237	adantlr 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → 𝑌 < (𝑆‘𝑗))
239	226, 227, 229, 231, 238	lelttrd 11394	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝐽) < (𝑆‘𝑗))
240	239	adantllr 718	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝐽) < (𝑆‘𝑗))
241		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝑗) = 𝑋)
242	198, 139	eqbrtrid 5177	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → 𝑋 < (𝑆‘(𝐽 + 1)))
243	242	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ (𝑆‘𝑗) = 𝑋) → 𝑋 < (𝑆‘(𝐽 + 1)))
244	241, 243	eqbrtrd 5164	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))
245	244	adantlr 714	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))
246	240, 245	jca 511	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))))
247	225, 246	mtand 815	. . . . 5 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ 𝑗 ∈ (0...𝑁)) → ¬ (𝑆‘𝑗) = 𝑋)
248	247	nrexdv 3144	. . . 4 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ¬ ∃𝑗 ∈ (0...𝑁)(𝑆‘𝑗) = 𝑋)
249		isof1o 7325	. . . . . . . . 9 ⊢ (𝑆 Isom < , < ((0...𝑁), 𝐻) → 𝑆:(0...𝑁)–1-1-onto→𝐻)
250	206, 249	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆:(0...𝑁)–1-1-onto→𝐻)
251		f1ofo 6840	. . . . . . . 8 ⊢ (𝑆:(0...𝑁)–1-1-onto→𝐻 → 𝑆:(0...𝑁)–onto→𝐻)
252	250, 251	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆:(0...𝑁)–onto→𝐻)
253		foelrn 7111	. . . . . . 7 ⊢ ((𝑆:(0...𝑁)–onto→𝐻 ∧ 𝑋 ∈ 𝐻) → ∃𝑗 ∈ (0...𝑁)𝑋 = (𝑆‘𝑗))
254	252, 253	sylan 579	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐻) → ∃𝑗 ∈ (0...𝑁)𝑋 = (𝑆‘𝑗))
255	234	rexbii 3089	. . . . . 6 ⊢ (∃𝑗 ∈ (0...𝑁)𝑋 = (𝑆‘𝑗) ↔ ∃𝑗 ∈ (0...𝑁)(𝑆‘𝑗) = 𝑋)
256	254, 255	sylib 217	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐻) → ∃𝑗 ∈ (0...𝑁)(𝑆‘𝑗) = 𝑋)
257	256	adantlr 714	. . . 4 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ 𝑋 ∈ 𝐻) → ∃𝑗 ∈ (0...𝑁)(𝑆‘𝑗) = 𝑋)
258	248, 257	mtand 815	. . 3 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ¬ 𝑋 ∈ 𝐻)
259	199, 258	pm2.65da 816	. 2 ⊢ (𝜑 → ¬ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1))))
260	52, 60, 259	nltled 11386	1 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ≤ (𝑄‘𝐾))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 {crab 3427 ∪ cun 3942 ⊆ wss 3944 {cpr 4626 class class class wbr 5142 ↦ cmpt 5225 dom cdm 5672 ran crn 5673 ℩cio 6492 Fun wfun 6536 ⟶wf 6538 –onto→wfo 6540 –1-1-onto→wf1o 6541 ‘cfv 6542 Isom wiso 6543 (class class class)co 7414 ↑_m cmap 8836 ℂcc 11128 ℝcr 11129 0cc0 11130 1c1 11131 + caddc 11133 · cmul 11135 ℝ^*cxr 11269 < clt 11270 ≤ cle 11271 − cmin 11466 / cdiv 11893 ℕcn 12234 ℤcz 12580 (,]cioc 13349 [,)cico 13350 [,]cicc 13351 ...cfz 13508 ..^cfzo 13651 ⌊cfl 13779 ♯chash 14313

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fi 9426 df-sup 9457 df-inf 9458 df-oi 9525 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-xnn0 12567 df-z 12581 df-uz 12845 df-q 12955 df-rp 12999 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13352 df-ioc 13353 df-ico 13354 df-icc 13355 df-fz 13509 df-fzo 13652 df-fl 13781 df-seq 13991 df-exp 14051 df-hash 14314 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-rest 17395 df-topgen 17416 df-psmet 21258 df-xmet 21259 df-met 21260 df-bl 21261 df-mopn 21262 df-top 22783 df-topon 22800 df-bases 22836 df-cld 22910 df-ntr 22911 df-cls 22912 df-nei 22989 df-lp 23027 df-cmp 23278

This theorem is referenced by: fourierdlem79 45496

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