fourierdlem63 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem63 45616

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 7421	. . . . . . . . 9 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝐵 − 𝑥) = (𝐵 − (𝑆‘(𝐽 + 1))))
5	4	oveq1d 7428	. . . . . . . 8 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇))
6	5	fveq2d 6894	. . . . . . 7 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
7	6	oveq1d 7428	. . . . . 6 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
8	3, 7	oveq12d 7431	. . . . 5 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
9	8	adantl 480	. . . 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 45607	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))
22	21	simpld 493	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)))
23	22	simprd 494	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁))
24	22	simpld 493	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
25	17	fourierdlem2 45556	. . . . . . . . 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 493	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (ℝ ↑_m (0...𝑁)))
29		elmapi 8861	. . . . . 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 13756	. . . . . 6 ⊢ (𝐽 ∈ (0..^𝑁) → (𝐽 + 1) ∈ (0...𝑁))
33	31, 32	syl 17	. . . . 5 ⊢ (𝜑 → (𝐽 + 1) ∈ (0...𝑁))
34	30, 33	ffvelcdmd 7088	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℝ)
35	11, 12, 13	fourierdlem11 45565	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
36	35	simp2d 1140	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
37	36, 34	resubcld 11667	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − (𝑆‘(𝐽 + 1))) ∈ ℝ)
38	35	simp1d 1139	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
39	36, 38	resubcld 11667	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
40	10, 39	eqeltrid 2829	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ ℝ)
41	35	simp3d 1141	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 < 𝐵)
42	38, 36	posdifd 11826	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
43	41, 42	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
44	43, 10	breqtrrdi 5186	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝑇)
45	44	gt0ne0d 11803	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ≠ 0)
46	37, 40, 45	redivcld 12067	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℝ)
47	46	flcld 13790	. . . . . . 7 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℤ)
48	47	zred 12691	. . . . . 6 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℝ)
49	48, 40	remulcld 11269	. . . . 5 ⊢ (𝜑 → ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) ∈ ℝ)
50	34, 49	readdcld 11268	. . . 4 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) ∈ ℝ)
51	2, 9, 34, 50	fvmptd 7005	. . 3 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
52	51, 50	eqeltrd 2825	. 2 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ)
53	11	fourierdlem2 45556	. . . . . . 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 493	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
57		elmapi 8861	. . . 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 7088	. 2 ⊢ (𝜑 → (𝑄‘𝐾) ∈ ℝ)
61	14	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → 𝐶 ∈ ℝ)
62	15	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → 𝐷 ∈ ℝ)
63	38	rexrd 11289	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
64		iocssre 13431	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ)
65	63, 36, 64	syl2anc 582	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℝ)
66	38, 36, 41, 10, 1	fourierdlem4 45558	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸:ℝ⟶(𝐴(,]𝐵))
67		fourierdlem63.y	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ∈ ((𝑆‘𝐽)[,)(𝑆‘(𝐽 + 1))))
68		elfzofz 13675	. . . . . . . . . . . . . . . . 17 ⊢ (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ (0...𝑁))
69	31, 68	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐽 ∈ (0...𝑁))
70	30, 69	ffvelcdmd 7088	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑆‘𝐽) ∈ ℝ)
71	34	rexrd 11289	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℝ^*)
72		elico2 13415	. . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . 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 1139	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ ℝ)
76	66, 75	ffvelcdmd 7088	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸‘𝑌) ∈ (𝐴(,]𝐵))
77	65, 76	sseldd 3974	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸‘𝑌) ∈ ℝ)
78	77, 75	resubcld 11667	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸‘𝑌) − 𝑌) ∈ ℝ)
79	60, 78	resubcld 11667	. . . . . . . 8 ⊢ (𝜑 → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ∈ ℝ)
80	79	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ∈ ℝ)
81		icossicc 13440	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝐽)[,)(𝑆‘(𝐽 + 1))) ⊆ ((𝑆‘𝐽)[,](𝑆‘(𝐽 + 1)))
82	14	rexrd 11289	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
83	15	rexrd 11289	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ ℝ^*)
84	17, 24, 23	fourierdlem15 45569	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐶[,]𝐷))
85	82, 83, 84, 31	fourierdlem8 45562	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑆‘𝐽)[,](𝑆‘(𝐽 + 1))) ⊆ (𝐶[,]𝐷))
86	81, 85	sstrid 3985	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆‘𝐽)[,)(𝑆‘(𝐽 + 1))) ⊆ (𝐶[,]𝐷))
87	86, 67	sseldd 3974	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ (𝐶[,]𝐷))
88		elicc2 13416	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝑌 ∈ (𝐶[,]𝐷) ↔ (𝑌 ∈ ℝ ∧ 𝐶 ≤ 𝑌 ∧ 𝑌 ≤ 𝐷)))
89	14, 15, 88	syl2anc 582	. . . . . . . . . . . 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 1140	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ≤ 𝑌)
92	60, 77	resubcld 11667	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘𝐾) − (𝐸‘𝑌)) ∈ ℝ)
93		fourierdlem63.eyltqk	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸‘𝑌) < (𝑄‘𝐾))
94	77, 60	posdifd 11826	. . . . . . . . . . . . . 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 13040	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄‘𝐾) − (𝐸‘𝑌)) ∈ ℝ⁺)
97	75, 96	ltaddrpd 13076	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 < (𝑌 + ((𝑄‘𝐾) − (𝐸‘𝑌))))
98	60	recnd 11267	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘𝐾) ∈ ℂ)
99	77	recnd 11267	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸‘𝑌) ∈ ℂ)
100	75	recnd 11267	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ ℂ)
101	98, 99, 100	subsub3d 11626	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) = (((𝑄‘𝐾) + 𝑌) − (𝐸‘𝑌)))
102	98, 100	addcomd 11441	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘𝐾) + 𝑌) = (𝑌 + (𝑄‘𝐾)))
103	102	oveq1d 7428	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑄‘𝐾) + 𝑌) − (𝐸‘𝑌)) = ((𝑌 + (𝑄‘𝐾)) − (𝐸‘𝑌)))
104	100, 98, 99	addsubassd 11616	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑌 + (𝑄‘𝐾)) − (𝐸‘𝑌)) = (𝑌 + ((𝑄‘𝐾) − (𝐸‘𝑌))))
105	101, 103, 104	3eqtrrd 2770	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 + ((𝑄‘𝐾) − (𝐸‘𝑌))) = ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)))
106	97, 105	breqtrd 5170	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 < ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)))
107	14, 75, 79, 91, 106	lelttrd 11397	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 < ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)))
108	14, 79, 107	ltled 11387	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ≤ ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)))
109	108	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → 𝐶 ≤ ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)))
110	34	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑆‘(𝐽 + 1)) ∈ ℝ)
111	60	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑄‘𝐾) ∈ ℝ)
112	52, 34	resubcld 11667	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) ∈ ℝ)
113	112	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) ∈ ℝ)
114	111, 113	resubcld 11667	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1)))) ∈ ℝ)
115	74	simp3d 1141	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 < (𝑆‘(𝐽 + 1)))
116	75, 34, 115	ltled 11387	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ≤ (𝑆‘(𝐽 + 1)))
117	38, 36, 41, 10, 1, 75, 34, 116	fourierdlem7 45561	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) ≤ ((𝐸‘𝑌) − 𝑌))
118	112, 78, 60, 117	lesub2dd 11856	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ≤ ((𝑄‘𝐾) − ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1)))))
119	118	adantr 479	. . . . . . . . . 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 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑄‘𝐾) ∈ ℂ)
121	52	recnd 11267	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℂ)
122	121	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℂ)
123	110	recnd 11267	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑆‘(𝐽 + 1)) ∈ ℂ)
124	120, 122, 123	subsubd 11624	. . . . . . . . . . . 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 11596	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) ∈ ℂ)
126	34	recnd 11267	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℂ)
127	125, 126	addcomd 11441	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) + (𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) + ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1))))))
128	127	adantr 479	. . . . . . . . . . . 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 2765	. . . . . . . . . . 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 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1))))
131	52	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ)
132	111, 131	sublt0d 11865	. . . . . . . . . . . . 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 256	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) < 0)
134	111, 131	resubcld 11667	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) ∈ ℝ)
135		ltaddneg 11454	. . . . . . . . . . . . 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 582	. . . . . . . . . . . 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 5166	. . . . . . . . . 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 11397	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) < (𝑆‘(𝐽 + 1)))
140	84, 33	ffvelcdmd 7088	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ (𝐶[,]𝐷))
141		elicc2 13416	. . . . . . . . . . . . 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 582	. . . . . . . . . . . 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 1141	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ≤ 𝐷)
145	144	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑆‘(𝐽 + 1)) ≤ 𝐷)
146	80, 110, 62, 139, 145	ltletrd 11399	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) < 𝐷)
147	80, 62, 146	ltled 11387	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ≤ 𝐷)
148	61, 62, 80, 109, 147	eliccd 44948	. . . . . 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 7421	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑌 → (𝐵 − 𝑥) = (𝐵 − 𝑌))
151	150	oveq1d 7428	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑌 → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − 𝑌) / 𝑇))
152	151	fveq2d 6894	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑌 → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − 𝑌) / 𝑇)))
153	152	oveq1d 7428	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑌 → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇))
154	149, 153	oveq12d 7431	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑌 → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = (𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)))
155	154	adantl 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = (𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)))
156	36, 75	resubcld 11667	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐵 − 𝑌) ∈ ℝ)
157	156, 40, 45	redivcld 12067	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐵 − 𝑌) / 𝑇) ∈ ℝ)
158	157	flcld 13790	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘((𝐵 − 𝑌) / 𝑇)) ∈ ℤ)
159	158	zred 12691	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (⌊‘((𝐵 − 𝑌) / 𝑇)) ∈ ℝ)
160	159, 40	remulcld 11269	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇) ∈ ℝ)
161	75, 160	readdcld 11268	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)) ∈ ℝ)
162	2, 155, 75, 161	fvmptd 7005	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸‘𝑌) = (𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)))
163	162	oveq1d 7428	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸‘𝑌) − 𝑌) = ((𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)) − 𝑌))
164	163	oveq1d 7428	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐸‘𝑌) − 𝑌) / 𝑇) = (((𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)) − 𝑌) / 𝑇))
165	160	recnd 11267	. . . . . . . . . . . 12 ⊢ (𝜑 → ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇) ∈ ℂ)
166	100, 165	pncan2d 11598	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)) − 𝑌) = ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇))
167	166	oveq1d 7428	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)) − 𝑌) / 𝑇) = (((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇) / 𝑇))
168	159	recnd 11267	. . . . . . . . . . 11 ⊢ (𝜑 → (⌊‘((𝐵 − 𝑌) / 𝑇)) ∈ ℂ)
169	40	recnd 11267	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ ℂ)
170	168, 169, 45	divcan4d 12021	. . . . . . . . . 10 ⊢ (𝜑 → (((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇) / 𝑇) = (⌊‘((𝐵 − 𝑌) / 𝑇)))
171	164, 167, 170	3eqtrd 2769	. . . . . . . . 9 ⊢ (𝜑 → (((𝐸‘𝑌) − 𝑌) / 𝑇) = (⌊‘((𝐵 − 𝑌) / 𝑇)))
172	171, 158	eqeltrd 2825	. . . . . . . 8 ⊢ (𝜑 → (((𝐸‘𝑌) − 𝑌) / 𝑇) ∈ ℤ)
173	78	recnd 11267	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘𝑌) − 𝑌) ∈ ℂ)
174	173, 169, 45	divcan1d 12016	. . . . . . . . . . 11 ⊢ (𝜑 → ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇) = ((𝐸‘𝑌) − 𝑌))
175	174	oveq2d 7429	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇)) = (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((𝐸‘𝑌) − 𝑌)))
176	98, 173	npcand 11600	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((𝐸‘𝑌) − 𝑌)) = (𝑄‘𝐾))
177	175, 176	eqtrd 2765	. . . . . . . . 9 ⊢ (𝜑 → (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇)) = (𝑄‘𝐾))
178		ffun 6720	. . . . . . . . . . 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 2828	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ dom 𝑄)
182		fvelrn 7079	. . . . . . . . . 10 ⊢ ((Fun 𝑄 ∧ 𝐾 ∈ dom 𝑄) → (𝑄‘𝐾) ∈ ran 𝑄)
183	179, 181, 182	syl2anc 582	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘𝐾) ∈ ran 𝑄)
184	177, 183	eqeltrd 2825	. . . . . . . 8 ⊢ (𝜑 → (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇)) ∈ ran 𝑄)
185		oveq1 7420	. . . . . . . . . . 11 ⊢ (𝑘 = (((𝐸‘𝑌) − 𝑌) / 𝑇) → (𝑘 · 𝑇) = ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇))
186	185	oveq2d 7429	. . . . . . . . . 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 2810	. . . . . . . . 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 3603	. . . . . . . 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 582	. . . . . . 7 ⊢ (𝜑 → ∃𝑘 ∈ ℤ (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + (𝑘 · 𝑇)) ∈ ran 𝑄)
190	189	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ∃𝑘 ∈ ℤ (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + (𝑘 · 𝑇)) ∈ ran 𝑄)
191		oveq1 7420	. . . . . . . . 9 ⊢ (𝑥 = ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) → (𝑥 + (𝑘 · 𝑇)) = (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + (𝑘 · 𝑇)))
192	191	eleq1d 2810	. . . . . . . 8 ⊢ (𝑥 = ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) → ((𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + (𝑘 · 𝑇)) ∈ ran 𝑄))
193	192	rexbidv 3169	. . . . . . 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 3676	. . . . . 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 581	. . . . 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 4172	. . . . 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 2842	. . 3 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → 𝑋 ∈ 𝐻)
200		elfzelz 13528	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
201	200	ad2antlr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))) → 𝑗 ∈ ℤ)
202		elfzoelz 13659	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ ℤ)
203	31, 202	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℤ)
204	203	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))) → 𝐽 ∈ ℤ)
205		simpr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → (𝑆‘𝐽) < (𝑆‘𝑗))
206	21	simprd 494	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 Isom < , < ((0...𝑁), 𝐻))
207	206	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → 𝑆 Isom < , < ((0...𝑁), 𝐻))
208	69	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → 𝐽 ∈ (0...𝑁))
209		simplr 767	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → 𝑗 ∈ (0...𝑁))
210		isorel 7327	. . . . . . . . . . . 12 ⊢ ((𝑆 Isom < , < ((0...𝑁), 𝐻) ∧ (𝐽 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑁))) → (𝐽 < 𝑗 ↔ (𝑆‘𝐽) < (𝑆‘𝑗)))
211	207, 208, 209, 210	syl12anc 835	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → (𝐽 < 𝑗 ↔ (𝑆‘𝐽) < (𝑆‘𝑗)))
212	205, 211	mpbird 256	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → 𝐽 < 𝑗)
213	212	adantrr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))) → 𝐽 < 𝑗)
214		simpr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))
215	206	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → 𝑆 Isom < , < ((0...𝑁), 𝐻))
216		simplr 767	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → 𝑗 ∈ (0...𝑁))
217	33	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → (𝐽 + 1) ∈ (0...𝑁))
218		isorel 7327	. . . . . . . . . . . 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 835	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → (𝑗 < (𝐽 + 1) ↔ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))))
220	214, 219	mpbird 256	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → 𝑗 < (𝐽 + 1))
221	220	adantrl 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))) → 𝑗 < (𝐽 + 1))
222		btwnnz 12663	. . . . . . . . 9 ⊢ ((𝐽 ∈ ℤ ∧ 𝐽 < 𝑗 ∧ 𝑗 < (𝐽 + 1)) → ¬ 𝑗 ∈ ℤ)
223	204, 213, 221, 222	syl3anc 1368	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))) → ¬ 𝑗 ∈ ℤ)
224	201, 223	pm2.65da 815	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ¬ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))))
225	224	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ 𝑗 ∈ (0...𝑁)) → ¬ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))))
226	70	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝐽) ∈ ℝ)
227	75	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → 𝑌 ∈ ℝ)
228	30	ffvelcdmda 7087	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑆‘𝑗) ∈ ℝ)
229	228	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝑗) ∈ ℝ)
230	74	simp2d 1140	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆‘𝐽) ≤ 𝑌)
231	230	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝐽) ≤ 𝑌)
232	106, 198	breqtrrdi 5186	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 < 𝑋)
233	232	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆‘𝑗) = 𝑋) → 𝑌 < 𝑋)
234		eqcom 2732	. . . . . . . . . . . . 13 ⊢ (𝑋 = (𝑆‘𝑗) ↔ (𝑆‘𝑗) = 𝑋)
235	234	biimpri 227	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑗) = 𝑋 → 𝑋 = (𝑆‘𝑗))
236	235	adantl 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆‘𝑗) = 𝑋) → 𝑋 = (𝑆‘𝑗))
237	233, 236	breqtrd 5170	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑆‘𝑗) = 𝑋) → 𝑌 < (𝑆‘𝑗))
238	237	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → 𝑌 < (𝑆‘𝑗))
239	226, 227, 229, 231, 238	lelttrd 11397	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝐽) < (𝑆‘𝑗))
240	239	adantllr 717	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝐽) < (𝑆‘𝑗))
241		simpr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝑗) = 𝑋)
242	198, 139	eqbrtrid 5179	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → 𝑋 < (𝑆‘(𝐽 + 1)))
243	242	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ (𝑆‘𝑗) = 𝑋) → 𝑋 < (𝑆‘(𝐽 + 1)))
244	241, 243	eqbrtrd 5166	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))
245	244	adantlr 713	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))
246	240, 245	jca 510	. . . . . 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 814	. . . . 5 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ 𝑗 ∈ (0...𝑁)) → ¬ (𝑆‘𝑗) = 𝑋)
248	247	nrexdv 3139	. . . 4 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ¬ ∃𝑗 ∈ (0...𝑁)(𝑆‘𝑗) = 𝑋)
249		isof1o 7324	. . . . . . . . 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 6839	. . . . . . . 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 7110	. . . . . . 7 ⊢ ((𝑆:(0...𝑁)–onto→𝐻 ∧ 𝑋 ∈ 𝐻) → ∃𝑗 ∈ (0...𝑁)𝑋 = (𝑆‘𝑗))
254	252, 253	sylan 578	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐻) → ∃𝑗 ∈ (0...𝑁)𝑋 = (𝑆‘𝑗))
255	234	rexbii 3084	. . . . . 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 713	. . . 4 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ 𝑋 ∈ 𝐻) → ∃𝑗 ∈ (0...𝑁)(𝑆‘𝑗) = 𝑋)
258	248, 257	mtand 814	. . 3 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ¬ 𝑋 ∈ 𝐻)
259	199, 258	pm2.65da 815	. 2 ⊢ (𝜑 → ¬ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1))))
260	52, 60, 259	nltled 11389	1 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ≤ (𝑄‘𝐾))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ∃wrex 3060 {crab 3419 ∪ cun 3939 ⊆ wss 3941 {cpr 4627 class class class wbr 5144 ↦ cmpt 5227 dom cdm 5673 ran crn 5674 ℩cio 6493 Fun wfun 6537 ⟶wf 6539 –onto→wfo 6541 –1-1-onto→wf1o 6542 ‘cfv 6543 Isom wiso 6544 (class class class)co 7413 ↑_m cmap 8838 ℂcc 11131 ℝcr 11132 0cc0 11133 1c1 11134 + caddc 11136 · cmul 11138 ℝ^*cxr 11272 < clt 11273 ≤ cle 11274 − cmin 11469 / cdiv 11896 ℕcn 12237 ℤcz 12583 (,]cioc 13352 [,)cico 13353 [,]cicc 13354 ...cfz 13511 ..^cfzo 13654 ⌊cfl 13782 ♯chash 14316

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 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 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fi 9429 df-sup 9460 df-inf 9461 df-oi 9528 df-dju 9919 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-xnn0 12570 df-z 12584 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-ioo 13355 df-ioc 13356 df-ico 13357 df-icc 13358 df-fz 13512 df-fzo 13655 df-fl 13784 df-seq 13994 df-exp 14054 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-rest 17398 df-topgen 17419 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-top 22809 df-topon 22826 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lp 23053 df-cmp 23304

This theorem is referenced by: fourierdlem79 45632

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