fourierdlem63 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem63 43385

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 7221	. . . . . . . . 9 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝐵 − 𝑥) = (𝐵 − (𝑆‘(𝐽 + 1))))
5	4	oveq1d 7228	. . . . . . . 8 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇))
6	5	fveq2d 6721	. . . . . . 7 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)))
7	6	oveq1d 7228	. . . . . 6 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇))
8	3, 7	oveq12d 7231	. . . . 5 ⊢ (𝑥 = (𝑆‘(𝐽 + 1)) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
9	8	adantl 485	. . . 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 43376	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))
22	21	simpld 498	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂‘𝑁)))
23	22	simprd 499	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (𝑂‘𝑁))
24	22	simpld 498	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
25	17	fourierdlem2 43325	. . . . . . . . 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 235	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∈ (ℝ ↑_m (0...𝑁)) ∧ (((𝑆‘0) = 𝐶 ∧ (𝑆‘𝑁) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑁)(𝑆‘𝑖) < (𝑆‘(𝑖 + 1)))))
28	27	simpld 498	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (ℝ ↑_m (0...𝑁)))
29		elmapi 8530	. . . . . 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 13339	. . . . . 6 ⊢ (𝐽 ∈ (0..^𝑁) → (𝐽 + 1) ∈ (0...𝑁))
33	31, 32	syl 17	. . . . 5 ⊢ (𝜑 → (𝐽 + 1) ∈ (0...𝑁))
34	30, 33	ffvelrnd 6905	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℝ)
35	11, 12, 13	fourierdlem11 43334	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
36	35	simp2d 1145	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
37	36, 34	resubcld 11260	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 − (𝑆‘(𝐽 + 1))) ∈ ℝ)
38	35	simp1d 1144	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
39	36, 38	resubcld 11260	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
40	10, 39	eqeltrid 2842	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ ℝ)
41	35	simp3d 1146	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 < 𝐵)
42	38, 36	posdifd 11419	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
43	41, 42	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
44	43, 10	breqtrrdi 5095	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 𝑇)
45	44	gt0ne0d 11396	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ≠ 0)
46	37, 40, 45	redivcld 11660	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇) ∈ ℝ)
47	46	flcld 13373	. . . . . . 7 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℤ)
48	47	zred 12282	. . . . . 6 ⊢ (𝜑 → (⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) ∈ ℝ)
49	48, 40	remulcld 10863	. . . . 5 ⊢ (𝜑 → ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇) ∈ ℝ)
50	34, 49	readdcld 10862	. . . 4 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)) ∈ ℝ)
51	2, 9, 34, 50	fvmptd 6825	. . 3 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) + ((⌊‘((𝐵 − (𝑆‘(𝐽 + 1))) / 𝑇)) · 𝑇)))
52	51, 50	eqeltrd 2838	. 2 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ)
53	11	fourierdlem2 43325	. . . . . . 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 235	. . . . 5 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
56	55	simpld 498	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
57		elmapi 8530	. . . 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	ffvelrnd 6905	. 2 ⊢ (𝜑 → (𝑄‘𝐾) ∈ ℝ)
61	14	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → 𝐶 ∈ ℝ)
62	15	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → 𝐷 ∈ ℝ)
63	38	rexrd 10883	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
64		iocssre 13015	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴(,]𝐵) ⊆ ℝ)
65	63, 36, 64	syl2anc 587	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℝ)
66	38, 36, 41, 10, 1	fourierdlem4 43327	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸:ℝ⟶(𝐴(,]𝐵))
67		fourierdlem63.y	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ∈ ((𝑆‘𝐽)[,)(𝑆‘(𝐽 + 1))))
68		elfzofz 13258	. . . . . . . . . . . . . . . . 17 ⊢ (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ (0...𝑁))
69	31, 68	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐽 ∈ (0...𝑁))
70	30, 69	ffvelrnd 6905	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑆‘𝐽) ∈ ℝ)
71	34	rexrd 10883	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℝ^*)
72		elico2 12999	. . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑌 ∈ ((𝑆‘𝐽)[,)(𝑆‘(𝐽 + 1))) ↔ (𝑌 ∈ ℝ ∧ (𝑆‘𝐽) ≤ 𝑌 ∧ 𝑌 < (𝑆‘(𝐽 + 1)))))
74	67, 73	mpbid 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑌 ∈ ℝ ∧ (𝑆‘𝐽) ≤ 𝑌 ∧ 𝑌 < (𝑆‘(𝐽 + 1))))
75	74	simp1d 1144	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ ℝ)
76	66, 75	ffvelrnd 6905	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸‘𝑌) ∈ (𝐴(,]𝐵))
77	65, 76	sseldd 3902	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸‘𝑌) ∈ ℝ)
78	77, 75	resubcld 11260	. . . . . . . . 9 ⊢ (𝜑 → ((𝐸‘𝑌) − 𝑌) ∈ ℝ)
79	60, 78	resubcld 11260	. . . . . . . 8 ⊢ (𝜑 → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ∈ ℝ)
80	79	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ∈ ℝ)
81		icossicc 13024	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝐽)[,)(𝑆‘(𝐽 + 1))) ⊆ ((𝑆‘𝐽)[,](𝑆‘(𝐽 + 1)))
82	14	rexrd 10883	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
83	15	rexrd 10883	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ ℝ^*)
84	17, 24, 23	fourierdlem15 43338	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐶[,]𝐷))
85	82, 83, 84, 31	fourierdlem8 43331	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑆‘𝐽)[,](𝑆‘(𝐽 + 1))) ⊆ (𝐶[,]𝐷))
86	81, 85	sstrid 3912	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆‘𝐽)[,)(𝑆‘(𝐽 + 1))) ⊆ (𝐶[,]𝐷))
87	86, 67	sseldd 3902	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ (𝐶[,]𝐷))
88		elicc2 13000	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝑌 ∈ (𝐶[,]𝐷) ↔ (𝑌 ∈ ℝ ∧ 𝐶 ≤ 𝑌 ∧ 𝑌 ≤ 𝐷)))
89	14, 15, 88	syl2anc 587	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑌 ∈ (𝐶[,]𝐷) ↔ (𝑌 ∈ ℝ ∧ 𝐶 ≤ 𝑌 ∧ 𝑌 ≤ 𝐷)))
90	87, 89	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 ∈ ℝ ∧ 𝐶 ≤ 𝑌 ∧ 𝑌 ≤ 𝐷))
91	90	simp2d 1145	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ≤ 𝑌)
92	60, 77	resubcld 11260	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘𝐾) − (𝐸‘𝑌)) ∈ ℝ)
93		fourierdlem63.eyltqk	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸‘𝑌) < (𝑄‘𝐾))
94	77, 60	posdifd 11419	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐸‘𝑌) < (𝑄‘𝐾) ↔ 0 < ((𝑄‘𝐾) − (𝐸‘𝑌))))
95	93, 94	mpbid 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < ((𝑄‘𝐾) − (𝐸‘𝑌)))
96	92, 95	elrpd 12625	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄‘𝐾) − (𝐸‘𝑌)) ∈ ℝ⁺)
97	75, 96	ltaddrpd 12661	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 < (𝑌 + ((𝑄‘𝐾) − (𝐸‘𝑌))))
98	60	recnd 10861	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄‘𝐾) ∈ ℂ)
99	77	recnd 10861	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸‘𝑌) ∈ ℂ)
100	75	recnd 10861	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ ℂ)
101	98, 99, 100	subsub3d 11219	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) = (((𝑄‘𝐾) + 𝑌) − (𝐸‘𝑌)))
102	98, 100	addcomd 11034	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘𝐾) + 𝑌) = (𝑌 + (𝑄‘𝐾)))
103	102	oveq1d 7228	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑄‘𝐾) + 𝑌) − (𝐸‘𝑌)) = ((𝑌 + (𝑄‘𝐾)) − (𝐸‘𝑌)))
104	100, 98, 99	addsubassd 11209	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑌 + (𝑄‘𝐾)) − (𝐸‘𝑌)) = (𝑌 + ((𝑄‘𝐾) − (𝐸‘𝑌))))
105	101, 103, 104	3eqtrrd 2782	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑌 + ((𝑄‘𝐾) − (𝐸‘𝑌))) = ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)))
106	97, 105	breqtrd 5079	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 < ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)))
107	14, 75, 79, 91, 106	lelttrd 10990	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 < ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)))
108	14, 79, 107	ltled 10980	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ≤ ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)))
109	108	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → 𝐶 ≤ ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)))
110	34	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑆‘(𝐽 + 1)) ∈ ℝ)
111	60	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑄‘𝐾) ∈ ℝ)
112	52, 34	resubcld 11260	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) ∈ ℝ)
113	112	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) ∈ ℝ)
114	111, 113	resubcld 11260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1)))) ∈ ℝ)
115	74	simp3d 1146	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 < (𝑆‘(𝐽 + 1)))
116	75, 34, 115	ltled 10980	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ≤ (𝑆‘(𝐽 + 1)))
117	38, 36, 41, 10, 1, 75, 34, 116	fourierdlem7 43330	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1))) ≤ ((𝐸‘𝑌) − 𝑌))
118	112, 78, 60, 117	lesub2dd 11449	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ≤ ((𝑄‘𝐾) − ((𝐸‘(𝑆‘(𝐽 + 1))) − (𝑆‘(𝐽 + 1)))))
119	118	adantr 484	. . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑄‘𝐾) ∈ ℂ)
121	52	recnd 10861	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℂ)
122	121	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℂ)
123	110	recnd 10861	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑆‘(𝐽 + 1)) ∈ ℂ)
124	120, 122, 123	subsubd 11217	. . . . . . . . . . . 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 11189	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) ∈ ℂ)
126	34	recnd 10861	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ ℂ)
127	125, 126	addcomd 11034	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) + (𝑆‘(𝐽 + 1))) = ((𝑆‘(𝐽 + 1)) + ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1))))))
128	127	adantr 484	. . . . . . . . . . . 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 2777	. . . . . . . . . . 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 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1))))
131	52	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝐸‘(𝑆‘(𝐽 + 1))) ∈ ℝ)
132	111, 131	sublt0d 11458	. . . . . . . . . . . . 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 260	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) < 0)
134	111, 131	resubcld 11260	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1)))) ∈ ℝ)
135		ltaddneg 11047	. . . . . . . . . . . . 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 587	. . . . . . . . . . . 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 235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑆‘(𝐽 + 1)) + ((𝑄‘𝐾) − (𝐸‘(𝑆‘(𝐽 + 1))))) < (𝑆‘(𝐽 + 1)))
138	129, 137	eqbrtrd 5075	. . . . . . . . . 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 10990	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) < (𝑆‘(𝐽 + 1)))
140	84, 33	ffvelrnd 6905	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ∈ (𝐶[,]𝐷))
141		elicc2 13000	. . . . . . . . . . . . 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 587	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) ∈ (𝐶[,]𝐷) ↔ ((𝑆‘(𝐽 + 1)) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘(𝐽 + 1)) ∧ (𝑆‘(𝐽 + 1)) ≤ 𝐷)))
143	140, 142	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑆‘(𝐽 + 1)) ∈ ℝ ∧ 𝐶 ≤ (𝑆‘(𝐽 + 1)) ∧ (𝑆‘(𝐽 + 1)) ≤ 𝐷))
144	143	simp3d 1146	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆‘(𝐽 + 1)) ≤ 𝐷)
145	144	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → (𝑆‘(𝐽 + 1)) ≤ 𝐷)
146	80, 110, 62, 139, 145	ltletrd 10992	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) < 𝐷)
147	80, 62, 146	ltled 10980	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) ≤ 𝐷)
148	61, 62, 80, 109, 147	eliccd 42717	. . . . . 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 7221	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑌 → (𝐵 − 𝑥) = (𝐵 − 𝑌))
151	150	oveq1d 7228	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑌 → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − 𝑌) / 𝑇))
152	151	fveq2d 6721	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑌 → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − 𝑌) / 𝑇)))
153	152	oveq1d 7228	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑌 → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇))
154	149, 153	oveq12d 7231	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑌 → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = (𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)))
155	154	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 = 𝑌) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = (𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)))
156	36, 75	resubcld 11260	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐵 − 𝑌) ∈ ℝ)
157	156, 40, 45	redivcld 11660	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐵 − 𝑌) / 𝑇) ∈ ℝ)
158	157	flcld 13373	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘((𝐵 − 𝑌) / 𝑇)) ∈ ℤ)
159	158	zred 12282	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (⌊‘((𝐵 − 𝑌) / 𝑇)) ∈ ℝ)
160	159, 40	remulcld 10863	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇) ∈ ℝ)
161	75, 160	readdcld 10862	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)) ∈ ℝ)
162	2, 155, 75, 161	fvmptd 6825	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸‘𝑌) = (𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)))
163	162	oveq1d 7228	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸‘𝑌) − 𝑌) = ((𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)) − 𝑌))
164	163	oveq1d 7228	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐸‘𝑌) − 𝑌) / 𝑇) = (((𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)) − 𝑌) / 𝑇))
165	160	recnd 10861	. . . . . . . . . . . 12 ⊢ (𝜑 → ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇) ∈ ℂ)
166	100, 165	pncan2d 11191	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)) − 𝑌) = ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇))
167	166	oveq1d 7228	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑌 + ((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇)) − 𝑌) / 𝑇) = (((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇) / 𝑇))
168	159	recnd 10861	. . . . . . . . . . 11 ⊢ (𝜑 → (⌊‘((𝐵 − 𝑌) / 𝑇)) ∈ ℂ)
169	40	recnd 10861	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ ℂ)
170	168, 169, 45	divcan4d 11614	. . . . . . . . . 10 ⊢ (𝜑 → (((⌊‘((𝐵 − 𝑌) / 𝑇)) · 𝑇) / 𝑇) = (⌊‘((𝐵 − 𝑌) / 𝑇)))
171	164, 167, 170	3eqtrd 2781	. . . . . . . . 9 ⊢ (𝜑 → (((𝐸‘𝑌) − 𝑌) / 𝑇) = (⌊‘((𝐵 − 𝑌) / 𝑇)))
172	171, 158	eqeltrd 2838	. . . . . . . 8 ⊢ (𝜑 → (((𝐸‘𝑌) − 𝑌) / 𝑇) ∈ ℤ)
173	78	recnd 10861	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐸‘𝑌) − 𝑌) ∈ ℂ)
174	173, 169, 45	divcan1d 11609	. . . . . . . . . . 11 ⊢ (𝜑 → ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇) = ((𝐸‘𝑌) − 𝑌))
175	174	oveq2d 7229	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇)) = (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((𝐸‘𝑌) − 𝑌)))
176	98, 173	npcand 11193	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((𝐸‘𝑌) − 𝑌)) = (𝑄‘𝐾))
177	175, 176	eqtrd 2777	. . . . . . . . 9 ⊢ (𝜑 → (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇)) = (𝑄‘𝐾))
178		ffun 6548	. . . . . . . . . . 11 ⊢ (𝑄:(0...𝑀)⟶ℝ → Fun 𝑄)
179	58, 178	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝑄)
180	58	fdmd 6556	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝑄 = (0...𝑀))
181	59, 180	eleqtrrd 2841	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ dom 𝑄)
182		fvelrn 6897	. . . . . . . . . 10 ⊢ ((Fun 𝑄 ∧ 𝐾 ∈ dom 𝑄) → (𝑄‘𝐾) ∈ ran 𝑄)
183	179, 181, 182	syl2anc 587	. . . . . . . . 9 ⊢ (𝜑 → (𝑄‘𝐾) ∈ ran 𝑄)
184	177, 183	eqeltrd 2838	. . . . . . . 8 ⊢ (𝜑 → (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇)) ∈ ran 𝑄)
185		oveq1 7220	. . . . . . . . . . 11 ⊢ (𝑘 = (((𝐸‘𝑌) − 𝑌) / 𝑇) → (𝑘 · 𝑇) = ((((𝐸‘𝑌) − 𝑌) / 𝑇) · 𝑇))
186	185	oveq2d 7229	. . . . . . . . . 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 2822	. . . . . . . . 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 3537	. . . . . . . 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 587	. . . . . . 7 ⊢ (𝜑 → ∃𝑘 ∈ ℤ (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + (𝑘 · 𝑇)) ∈ ran 𝑄)
190	189	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ∃𝑘 ∈ ℤ (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + (𝑘 · 𝑇)) ∈ ran 𝑄)
191		oveq1 7220	. . . . . . . . 9 ⊢ (𝑥 = ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) → (𝑥 + (𝑘 · 𝑇)) = (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + (𝑘 · 𝑇)))
192	191	eleq1d 2822	. . . . . . . 8 ⊢ (𝑥 = ((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) → ((𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄 ↔ (((𝑄‘𝐾) − ((𝐸‘𝑌) − 𝑌)) + (𝑘 · 𝑇)) ∈ ran 𝑄))
193	192	rexbidv 3216	. . . . . . 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 3602	. . . . . 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 586	. . . . 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 4091	. . . . 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 2855	. . 3 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → 𝑋 ∈ 𝐻)
200		elfzelz 13112	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
201	200	ad2antlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))) → 𝑗 ∈ ℤ)
202		elfzoelz 13243	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ ℤ)
203	31, 202	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℤ)
204	203	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))) → 𝐽 ∈ ℤ)
205		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → (𝑆‘𝐽) < (𝑆‘𝑗))
206	21	simprd 499	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 Isom < , < ((0...𝑁), 𝐻))
207	206	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → 𝑆 Isom < , < ((0...𝑁), 𝐻))
208	69	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → 𝐽 ∈ (0...𝑁))
209		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → 𝑗 ∈ (0...𝑁))
210		isorel 7135	. . . . . . . . . . . 12 ⊢ ((𝑆 Isom < , < ((0...𝑁), 𝐻) ∧ (𝐽 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑁))) → (𝐽 < 𝑗 ↔ (𝑆‘𝐽) < (𝑆‘𝑗)))
211	207, 208, 209, 210	syl12anc 837	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → (𝐽 < 𝑗 ↔ (𝑆‘𝐽) < (𝑆‘𝑗)))
212	205, 211	mpbird 260	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝐽) < (𝑆‘𝑗)) → 𝐽 < 𝑗)
213	212	adantrr 717	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))) → 𝐽 < 𝑗)
214		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))
215	206	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → 𝑆 Isom < , < ((0...𝑁), 𝐻))
216		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → 𝑗 ∈ (0...𝑁))
217	33	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → (𝐽 + 1) ∈ (0...𝑁))
218		isorel 7135	. . . . . . . . . . . 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 837	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → (𝑗 < (𝐽 + 1) ↔ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))))
220	214, 219	mpbird 260	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))) → 𝑗 < (𝐽 + 1))
221	220	adantrl 716	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))) → 𝑗 < (𝐽 + 1))
222		btwnnz 12253	. . . . . . . . 9 ⊢ ((𝐽 ∈ ℤ ∧ 𝐽 < 𝑗 ∧ 𝑗 < (𝐽 + 1)) → ¬ 𝑗 ∈ ℤ)
223	204, 213, 221, 222	syl3anc 1373	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))) → ¬ 𝑗 ∈ ℤ)
224	201, 223	pm2.65da 817	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ¬ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))))
225	224	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ 𝑗 ∈ (0...𝑁)) → ¬ ((𝑆‘𝐽) < (𝑆‘𝑗) ∧ (𝑆‘𝑗) < (𝑆‘(𝐽 + 1))))
226	70	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝐽) ∈ ℝ)
227	75	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → 𝑌 ∈ ℝ)
228	30	ffvelrnda 6904	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑆‘𝑗) ∈ ℝ)
229	228	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝑗) ∈ ℝ)
230	74	simp2d 1145	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆‘𝐽) ≤ 𝑌)
231	230	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝐽) ≤ 𝑌)
232	106, 198	breqtrrdi 5095	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 < 𝑋)
233	232	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆‘𝑗) = 𝑋) → 𝑌 < 𝑋)
234		eqcom 2744	. . . . . . . . . . . . 13 ⊢ (𝑋 = (𝑆‘𝑗) ↔ (𝑆‘𝑗) = 𝑋)
235	234	biimpri 231	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑗) = 𝑋 → 𝑋 = (𝑆‘𝑗))
236	235	adantl 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑆‘𝑗) = 𝑋) → 𝑋 = (𝑆‘𝑗))
237	233, 236	breqtrd 5079	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑆‘𝑗) = 𝑋) → 𝑌 < (𝑆‘𝑗))
238	237	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → 𝑌 < (𝑆‘𝑗))
239	226, 227, 229, 231, 238	lelttrd 10990	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝐽) < (𝑆‘𝑗))
240	239	adantllr 719	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝐽) < (𝑆‘𝑗))
241		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝑗) = 𝑋)
242	198, 139	eqbrtrid 5088	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → 𝑋 < (𝑆‘(𝐽 + 1)))
243	242	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ (𝑆‘𝑗) = 𝑋) → 𝑋 < (𝑆‘(𝐽 + 1)))
244	241, 243	eqbrtrd 5075	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))
245	244	adantlr 715	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ 𝑗 ∈ (0...𝑁)) ∧ (𝑆‘𝑗) = 𝑋) → (𝑆‘𝑗) < (𝑆‘(𝐽 + 1)))
246	240, 245	jca 515	. . . . . 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 816	. . . . 5 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ 𝑗 ∈ (0...𝑁)) → ¬ (𝑆‘𝑗) = 𝑋)
248	247	nrexdv 3189	. . . 4 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ¬ ∃𝑗 ∈ (0...𝑁)(𝑆‘𝑗) = 𝑋)
249		isof1o 7132	. . . . . . . . 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 6668	. . . . . . . 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 6925	. . . . . . 7 ⊢ ((𝑆:(0...𝑁)–onto→𝐻 ∧ 𝑋 ∈ 𝐻) → ∃𝑗 ∈ (0...𝑁)𝑋 = (𝑆‘𝑗))
254	252, 253	sylan 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐻) → ∃𝑗 ∈ (0...𝑁)𝑋 = (𝑆‘𝑗))
255	234	rexbii 3170	. . . . . 6 ⊢ (∃𝑗 ∈ (0...𝑁)𝑋 = (𝑆‘𝑗) ↔ ∃𝑗 ∈ (0...𝑁)(𝑆‘𝑗) = 𝑋)
256	254, 255	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐻) → ∃𝑗 ∈ (0...𝑁)(𝑆‘𝑗) = 𝑋)
257	256	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) ∧ 𝑋 ∈ 𝐻) → ∃𝑗 ∈ (0...𝑁)(𝑆‘𝑗) = 𝑋)
258	248, 257	mtand 816	. . 3 ⊢ ((𝜑 ∧ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1)))) → ¬ 𝑋 ∈ 𝐻)
259	199, 258	pm2.65da 817	. 2 ⊢ (𝜑 → ¬ (𝑄‘𝐾) < (𝐸‘(𝑆‘(𝐽 + 1))))
260	52, 60, 259	nltled 10982	1 ⊢ (𝜑 → (𝐸‘(𝑆‘(𝐽 + 1))) ≤ (𝑄‘𝐾))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ∃wrex 3062 {crab 3065 ∪ cun 3864 ⊆ wss 3866 {cpr 4543 class class class wbr 5053 ↦ cmpt 5135 dom cdm 5551 ran crn 5552 ℩cio 6336 Fun wfun 6374 ⟶wf 6376 –onto→wfo 6378 –1-1-onto→wf1o 6379 ‘cfv 6380 Isom wiso 6381 (class class class)co 7213 ↑_m cmap 8508 ℂcc 10727 ℝcr 10728 0cc0 10729 1c1 10730 + caddc 10732 · cmul 10734 ℝ^*cxr 10866 < clt 10867 ≤ cle 10868 − cmin 11062 / cdiv 11489 ℕcn 11830 ℤcz 12176 (,]cioc 12936 [,)cico 12937 [,]cicc 12938 ...cfz 13095 ..^cfzo 13238 ⌊cfl 13365 ♯chash 13896

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-oadd 8206 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-dju 9517 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-xnn0 12163 df-z 12177 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-rest 16927 df-topgen 16948 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-top 21791 df-topon 21808 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-lp 22033 df-cmp 22284

This theorem is referenced by: fourierdlem79 43401

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