Theorem fourierdlem71 43608

Description: A periodic piecewise continuous function, possibly undefined on a finite set in each periodic interval, is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem71.dmf	⊢ (𝜑 → dom 𝐹 ⊆ ℝ)
fourierdlem71.f	⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ)
fourierdlem71.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem71.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem71.altb	⊢ (𝜑 → 𝐴 < 𝐵)
fourierdlem71.t	⊢ 𝑇 = (𝐵 − 𝐴)
fourierdlem71.7	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem71.q	⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
fourierdlem71.q0	⊢ (𝜑 → (𝑄‘0) = 𝐴)
fourierdlem71.10	⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
fourierdlem71.fcn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem71.r	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖)))
fourierdlem71.l	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1))))
fourierdlem71.xpt	⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹)
fourierdlem71.fxpt	⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥))
fourierdlem71.i	⊢ 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
fourierdlem71.e	⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))

Assertion

Ref	Expression
fourierdlem71	⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐹(abs‘(𝐹‘𝑥)) ≤ 𝑦)

Distinct variable groups:   𝑥,𝐴,𝑦   𝐵,𝑘,𝑥   𝑦,𝐵   𝑖,𝐹,𝑥,𝑘   𝑦,𝐹   𝑖,𝐼,𝑥   𝑦,𝐼   𝑥,𝐿   𝑖,𝑀,𝑥,𝑘   𝑄,𝑖,𝑥,𝑘   𝑦,𝑄   𝑥,𝑅   𝑇,𝑘,𝑥   𝑦,𝑇   𝜑,𝑖,𝑥,𝑘   𝜑,𝑦

Allowed substitution hints:   𝐴(𝑖,𝑘)   𝐵(𝑖)   𝑅(𝑦,𝑖,𝑘)   𝑇(𝑖)   𝐸(𝑥,𝑦,𝑖,𝑘)   𝐼(𝑘)   𝐿(𝑦,𝑖,𝑘)   𝑀(𝑦)

Proof of Theorem fourierdlem71

Dummy variables 𝑤 𝑏 𝑡 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prfi 9019	. . . 4 ⊢ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} ∈ Fin
2	1	a1i 11	. . 3 ⊢ (𝜑 → {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} ∈ Fin)
3		fourierdlem71.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ)
4	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → 𝐹:dom 𝐹⟶ℝ)
5		simpl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → 𝜑)
6		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼})
7		fourierdlem71.q	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
8		ovex 7288	. . . . . . . . . . . . 13 ⊢ (0...𝑀) ∈ V
9	8	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝑀) ∈ V)
10	7, 9	fexd 7085	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ V)
11		rnexg 7725	. . . . . . . . . . 11 ⊢ (𝑄 ∈ V → ran 𝑄 ∈ V)
12		inex1g 5238	. . . . . . . . . . 11 ⊢ (ran 𝑄 ∈ V → (ran 𝑄 ∩ dom 𝐹) ∈ V)
13	10, 11, 12	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (ran 𝑄 ∩ dom 𝐹) ∈ V)
14	13	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → (ran 𝑄 ∩ dom 𝐹) ∈ V)
15		fourierdlem71.i	. . . . . . . . . . . . . 14 ⊢ 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
16		ovex 7288	. . . . . . . . . . . . . . 15 ⊢ (0..^𝑀) ∈ V
17	16	mptex 7081	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^𝑀) ↦ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ V
18	15, 17	eqeltri 2835	. . . . . . . . . . . . 13 ⊢ 𝐼 ∈ V
19	18	rnex 7733	. . . . . . . . . . . 12 ⊢ ran 𝐼 ∈ V
20	19	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐼 ∈ V)
21	20	uniexd 7573	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran 𝐼 ∈ V)
22	21	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → ∪ ran 𝐼 ∈ V)
23		uniprg 4853	. . . . . . . . 9 ⊢ (((ran 𝑄 ∩ dom 𝐹) ∈ V ∧ ∪ ran 𝐼 ∈ V) → ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
24	14, 22, 23	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
25	6, 24	eleqtrd 2841	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
26		elinel2 4126	. . . . . . . . 9 ⊢ (𝑥 ∈ (ran 𝑄 ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
27	26	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼)) ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
28		simpll 763	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝜑)
29		elunnel1 4080	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ ∪ ran 𝐼)
30	29	adantll 710	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ ∪ ran 𝐼)
31	15	funmpt2 6457	. . . . . . . . . . . . 13 ⊢ Fun 𝐼
32		elunirn 7106	. . . . . . . . . . . . 13 ⊢ (Fun 𝐼 → (𝑥 ∈ ∪ ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖)))
33	31, 32	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖))
34	33	biimpi 215	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ ran 𝐼 → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖))
35	34	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖))
36		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ dom 𝐼 → 𝑖 ∈ dom 𝐼)
37		ovex 7288	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V
38	37, 15	dmmpti 6561	. . . . . . . . . . . . . . . . . . 19 ⊢ dom 𝐼 = (0..^𝑀)
39	36, 38	eleqtrdi 2849	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ dom 𝐼 → 𝑖 ∈ (0..^𝑀))
40	39	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → 𝑖 ∈ (0..^𝑀))
41	37	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V)
42	15	fvmpt2 6868	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
43	40, 41, 42	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
44		fourierdlem71.fcn	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
45		cncff 23962	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ)
46		fdm 6593	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))):((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))⟶ℂ → dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
47	44, 45, 46	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
48	39, 47	sylan2 592	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
49		ssdmres 5903	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
50	48, 49	sylibr 233	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
51	43, 50	eqsstrd 3955	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → (𝐼‘𝑖) ⊆ dom 𝐹)
52	51	3adant3 1130	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)) → (𝐼‘𝑖) ⊆ dom 𝐹)
53		simp3 1136	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)) → 𝑥 ∈ (𝐼‘𝑖))
54	52, 53	sseldd 3918	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)) → 𝑥 ∈ dom 𝐹)
55	54	3exp 1117	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑖 ∈ dom 𝐼 → (𝑥 ∈ (𝐼‘𝑖) → 𝑥 ∈ dom 𝐹)))
56	55	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → (𝑖 ∈ dom 𝐼 → (𝑥 ∈ (𝐼‘𝑖) → 𝑥 ∈ dom 𝐹)))
57	56	rexlimdv 3211	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖) → 𝑥 ∈ dom 𝐹))
58	35, 57	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → 𝑥 ∈ dom 𝐹)
59	28, 30, 58	syl2anc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
60	27, 59	pm2.61dan 809	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼)) → 𝑥 ∈ dom 𝐹)
61	5, 25, 60	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → 𝑥 ∈ dom 𝐹)
62	4, 61	ffvelrnd 6944	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → (𝐹‘𝑥) ∈ ℝ)
63	62	recnd 10934	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → (𝐹‘𝑥) ∈ ℂ)
64	63	abscld 15076	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
65		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = (ran 𝑄 ∩ dom 𝐹))
66		fzfid 13621	. . . . . . . . . 10 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
67		rnffi 42600	. . . . . . . . . 10 ⊢ ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
68	7, 66, 67	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑄 ∈ Fin)
69		infi 8972	. . . . . . . . 9 ⊢ (ran 𝑄 ∈ Fin → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
70	68, 69	syl 17	. . . . . . . 8 ⊢ (𝜑 → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
71	70	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
72	65, 71	eqeltrd 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 ∈ Fin)
73		simpll 763	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥 ∈ 𝑤) → 𝜑)
74		simpr 484	. . . . . . . . . 10 ⊢ ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥 ∈ 𝑤) → 𝑥 ∈ 𝑤)
75		simpl 482	. . . . . . . . . 10 ⊢ ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥 ∈ 𝑤) → 𝑤 = (ran 𝑄 ∩ dom 𝐹))
76	74, 75	eleqtrd 2841	. . . . . . . . 9 ⊢ ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥 ∈ 𝑤) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
77	76	adantll 710	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥 ∈ 𝑤) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
78	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝐹:dom 𝐹⟶ℝ)
79	26	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
80	78, 79	ffvelrnd 6944	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (𝐹‘𝑥) ∈ ℝ)
81	80	recnd 10934	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (𝐹‘𝑥) ∈ ℂ)
82	81	abscld 15076	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
83	73, 77, 82	syl2anc 583	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥 ∈ 𝑤) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
84	83	ralrimiva 3107	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ∈ ℝ)
85		fimaxre3 11851	. . . . . 6 ⊢ ((𝑤 ∈ Fin ∧ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦)
86	72, 84, 85	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦)
87	86	adantlr 711	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦)
88		simpll 763	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝜑)
89		neqne 2950	. . . . . . 7 ⊢ (¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹) → 𝑤 ≠ (ran 𝑄 ∩ dom 𝐹))
90		elprn1 43064	. . . . . . 7 ⊢ ((𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} ∧ 𝑤 ≠ (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ∪ ran 𝐼)
91	89, 90	sylan2 592	. . . . . 6 ⊢ ((𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ∪ ran 𝐼)
92	91	adantll 710	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ∪ ran 𝐼)
93		fzofi 13622	. . . . . . . 8 ⊢ (0..^𝑀) ∈ Fin
94	15	rnmptfi 42596	. . . . . . . 8 ⊢ ((0..^𝑀) ∈ Fin → ran 𝐼 ∈ Fin)
95	93, 94	ax-mp 5	. . . . . . 7 ⊢ ran 𝐼 ∈ Fin
96	95	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → ran 𝐼 ∈ Fin)
97	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → 𝐹:dom 𝐹⟶ℝ)
98	97, 58	ffvelrnd 6944	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → (𝐹‘𝑥) ∈ ℝ)
99	98	recnd 10934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → (𝐹‘𝑥) ∈ ℂ)
100	99	adantlr 711	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑥 ∈ ∪ ran 𝐼) → (𝐹‘𝑥) ∈ ℂ)
101	100	abscld 15076	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑥 ∈ ∪ ran 𝐼) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
102	37, 15	fnmpti 6560	. . . . . . . . . . 11 ⊢ 𝐼 Fn (0..^𝑀)
103		fvelrnb 6812	. . . . . . . . . . 11 ⊢ (𝐼 Fn (0..^𝑀) → (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡))
104	102, 103	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡)
105	104	biimpi 215	. . . . . . . . 9 ⊢ (𝑡 ∈ ran 𝐼 → ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡)
106	105	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡)
107	7	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
108		elfzofz 13331	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
109	108	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
110	107, 109	ffvelrnd 6944	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
111		fzofzp1 13412	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
112	111	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
113	107, 112	ffvelrnd 6944	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
114		fourierdlem71.l	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1))))
115		fourierdlem71.r	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖)))
116	110, 113, 44, 114, 115	cncfioobd 43328	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏)
117	116	3adant3 1130	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏)
118		fvres 6775	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥) = (𝐹‘𝑥))
119	118	fveq2d 6760	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) = (abs‘(𝐹‘𝑥)))
120	119	breq1d 5080	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑥)) ≤ 𝑏))
121	120	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑥)) ≤ 𝑏))
122	121	ralbidva 3119	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑥)) ≤ 𝑏))
123	122	rexbidv 3225	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑥)) ≤ 𝑏))
124	123	3adant3 1130	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑥)) ≤ 𝑏))
125	37, 42	mpan2 687	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0..^𝑀) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
126		id 22	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼‘𝑖) = 𝑡 → (𝐼‘𝑖) = 𝑡)
127	125, 126	sylan9req 2800	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
128	127	3adant1 1128	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
129	128	raleqdv 3339	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → (∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑥)) ≤ 𝑏 ↔ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏))
130	129	rexbidv 3225	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑥)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏))
131	124, 130	bitrd 278	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏))
132	117, 131	mpbid 231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏)
133	132	3exp 1117	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) → ((𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏)))
134	133	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → (𝑖 ∈ (0..^𝑀) → ((𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏)))
135	134	rexlimdv 3211	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → (∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏))
136	106, 135	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏)
137	136	adantlr 711	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏)
138		eqimss 3973	. . . . . . 7 ⊢ (𝑤 = ∪ ran 𝐼 → 𝑤 ⊆ ∪ ran 𝐼)
139	138	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → 𝑤 ⊆ ∪ ran 𝐼)
140	96, 101, 137, 139	ssfiunibd 42738	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦)
141	88, 92, 140	syl2anc 583	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦)
142	87, 141	pm2.61dan 809	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦)
143		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ran 𝑄)
144		elinel2 4126	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
145	144	ad2antlr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ dom 𝐹)
146	143, 145	elind 4124	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
147		elun1 4106	. . . . . . . 8 ⊢ (𝑥 ∈ (ran 𝑄 ∩ dom 𝐹) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
148	146, 147	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
149		fourierdlem71.7	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℕ)
150	149	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑀 ∈ ℕ)
151	7	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
152		elinel1 4125	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹) → 𝑥 ∈ (𝐴[,]𝐵))
153	152	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ (𝐴[,]𝐵))
154		fourierdlem71.q0	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
155	154	eqcomd 2744	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 = (𝑄‘0))
156	155	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝐴 = (𝑄‘0))
157		fourierdlem71.10	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
158	157	eqcomd 2744	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 = (𝑄‘𝑀))
159	158	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝐵 = (𝑄‘𝑀))
160	156, 159	oveq12d 7273	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄‘𝑀)))
161	153, 160	eleqtrd 2841	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
162	161	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
163		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ¬ 𝑥 ∈ ran 𝑄)
164		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝑄‘𝑘) = (𝑄‘𝑗))
165	164	breq1d 5080	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → ((𝑄‘𝑘) < 𝑥 ↔ (𝑄‘𝑗) < 𝑥))
166	165	cbvrabv 3416	. . . . . . . . . . . 12 ⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝑥} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < 𝑥}
167	166	supeq1i 9136	. . . . . . . . . . 11 ⊢ sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝑥}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < 𝑥}, ℝ, < )
168	150, 151, 162, 163, 167	fourierdlem25 43563	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
169	39	ad2antrl 724	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) → 𝑖 ∈ (0..^𝑀))
170		simprr 769	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) → 𝑥 ∈ (𝐼‘𝑖))
171	169, 125	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
172	170, 171	eleqtrd 2841	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) → 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
173	169, 172	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) → (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
174		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0..^𝑀))
175	174, 38	eleqtrrdi 2850	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ dom 𝐼)
176	175	ad2antrl 724	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑖 ∈ dom 𝐼)
177		simprr 769	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
178	125	eqcomd 2744	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼‘𝑖))
179	178	ad2antrl 724	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼‘𝑖))
180	177, 179	eleqtrd 2841	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑥 ∈ (𝐼‘𝑖))
181	176, 180	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)))
182	173, 181	impbida 797	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)) ↔ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))))
183	182	rexbidv2 3223	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
184	183	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
185	168, 184	mpbird 256	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖))
186	185, 33	sylibr 233	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ∪ ran 𝐼)
187		elun2 4107	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ ran 𝐼 → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
188	186, 187	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
189	148, 188	pm2.61dan 809	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
190	189	ralrimiva 3107	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
191		dfss3 3905	. . . . 5 ⊢ (((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼) ↔ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
192	190, 191	sylibr 233	. . . 4 ⊢ (𝜑 → ((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
193	13, 21, 23	syl2anc 583	. . . 4 ⊢ (𝜑 → ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
194	192, 193	sseqtrrd 3958	. . 3 ⊢ (𝜑 → ((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼})
195	2, 64, 142, 194	ssfiunibd 42738	. 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦)
196		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑥𝜑
197		nfra1 3142	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦
198	196, 197	nfan 1903	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦)
199		fourierdlem71.dmf	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ)
200	199	sselda 3917	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ℝ)
201		fourierdlem71.b	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 ∈ ℝ)
202	201	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝐵 ∈ ℝ)
203	202, 200	resubcld 11333	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐵 − 𝑥) ∈ ℝ)
204		fourierdlem71.t	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑇 = (𝐵 − 𝐴)
205		fourierdlem71.a	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐴 ∈ ℝ)
206	201, 205	resubcld 11333	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
207	204, 206	eqeltrid 2843	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇 ∈ ℝ)
208	207	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝑇 ∈ ℝ)
209		fourierdlem71.altb	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐴 < 𝐵)
210	205, 201	posdifd 11492	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
211	209, 210	mpbid 231	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
212	211, 204	breqtrrdi 5112	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 < 𝑇)
213	212	gt0ne0d 11469	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇 ≠ 0)
214	213	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝑇 ≠ 0)
215	203, 208, 214	redivcld 11733	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → ((𝐵 − 𝑥) / 𝑇) ∈ ℝ)
216	215	flcld 13446	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ)
217	216	zred 12355	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℝ)
218	217, 208	remulcld 10936	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) ∈ ℝ)
219	200, 218	readdcld 10935	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ ℝ)
220		fourierdlem71.e	. . . . . . . . . . . . 13 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
221	220	fvmpt2 6868	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ ℝ) → (𝐸‘𝑥) = (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
222	200, 219, 221	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) = (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
223	222	fveq2d 6760	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘(𝐸‘𝑥)) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))))
224		fvex 6769	. . . . . . . . . . . 12 ⊢ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ V
225		eleq1 2826	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (𝑘 ∈ ℤ ↔ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ))
226	225	anbi2d 628	. . . . . . . . . . . . 13 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) ↔ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ)))
227		oveq1 7262	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (𝑘 · 𝑇) = ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))
228	227	oveq2d 7271	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (𝑥 + (𝑘 · 𝑇)) = (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
229	228	fveq2d 6760	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))))
230	229	eqeq1d 2740	. . . . . . . . . . . . 13 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → ((𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥) ↔ (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝐹‘𝑥)))
231	226, 230	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → ((((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥)) ↔ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))))
232		fourierdlem71.fxpt	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥))
233	224, 231, 232	vtocl 3488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))
234	216, 233	mpdan 683	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))
235	223, 234	eqtr2d 2779	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = (𝐹‘(𝐸‘𝑥)))
236	235	fveq2d 6760	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘(𝐸‘𝑥))))
237	236	adantlr 711	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘(𝐸‘𝑥))))
238		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
239	238	fveq2d 6760	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘𝑤)))
240	239	breq1d 5080	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((abs‘(𝐹‘𝑥)) ≤ 𝑦 ↔ (abs‘(𝐹‘𝑤)) ≤ 𝑦))
241	240	cbvralvw 3372	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦 ↔ ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑤)) ≤ 𝑦)
242	241	biimpi 215	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦 → ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑤)) ≤ 𝑦)
243	242	ad2antlr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑤)) ≤ 𝑦)
244		iocssicc 13098	. . . . . . . . . . 11 ⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
245	205	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝐴 ∈ ℝ)
246	209	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝐴 < 𝐵)
247		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
248		oveq2 7263	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝐵 − 𝑥) = (𝐵 − 𝑦))
249	248	oveq1d 7270	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − 𝑦) / 𝑇))
250	249	fveq2d 6760	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − 𝑦) / 𝑇)))
251	250	oveq1d 7270	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − 𝑦) / 𝑇)) · 𝑇))
252	247, 251	oveq12d 7273	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = (𝑦 + ((⌊‘((𝐵 − 𝑦) / 𝑇)) · 𝑇)))
253	252	cbvmptv 5183	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵 − 𝑦) / 𝑇)) · 𝑇)))
254	220, 253	eqtri 2766	. . . . . . . . . . . . 13 ⊢ 𝐸 = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵 − 𝑦) / 𝑇)) · 𝑇)))
255	245, 202, 246, 204, 254	fourierdlem4 43542	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝐸:ℝ⟶(𝐴(,]𝐵))
256	255, 200	ffvelrnd 6944	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ (𝐴(,]𝐵))
257	244, 256	sselid 3915	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ (𝐴[,]𝐵))
258	228	eleq1d 2823	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → ((𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹 ↔ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹))
259	226, 258	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → ((((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹) ↔ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹)))
260		fourierdlem71.xpt	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹)
261	224, 259, 260	vtocl 3488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹)
262	216, 261	mpdan 683	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹)
263	222, 262	eqeltrd 2839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ dom 𝐹)
264	257, 263	elind 4124	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹))
265	264	adantlr 711	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹))
266		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑤 = (𝐸‘𝑥) → (𝐹‘𝑤) = (𝐹‘(𝐸‘𝑥)))
267	266	fveq2d 6760	. . . . . . . . . 10 ⊢ (𝑤 = (𝐸‘𝑥) → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘(𝐸‘𝑥))))
268	267	breq1d 5080	. . . . . . . . 9 ⊢ (𝑤 = (𝐸‘𝑥) → ((abs‘(𝐹‘𝑤)) ≤ 𝑦 ↔ (abs‘(𝐹‘(𝐸‘𝑥))) ≤ 𝑦))
269	268	rspccva 3551	. . . . . . . 8 ⊢ ((∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑤)) ≤ 𝑦 ∧ (𝐸‘𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → (abs‘(𝐹‘(𝐸‘𝑥))) ≤ 𝑦)
270	243, 265, 269	syl2anc 583	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘(𝐸‘𝑥))) ≤ 𝑦)
271	237, 270	eqbrtrd 5092	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘𝑥)) ≤ 𝑦)
272	271	ex 412	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) → (𝑥 ∈ dom 𝐹 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))
273	198, 272	ralrimi 3139	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) → ∀𝑥 ∈ dom 𝐹(abs‘(𝐹‘𝑥)) ≤ 𝑦)
274	273	ex 412	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦 → ∀𝑥 ∈ dom 𝐹(abs‘(𝐹‘𝑥)) ≤ 𝑦))
275	274	reximdv 3201	. 2 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐹(abs‘(𝐹‘𝑥)) ≤ 𝑦))
276	195, 275	mpd 15	1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐹(abs‘(𝐹‘𝑥)) ≤ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067  Vcvv 3422   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  {cpr 4560  ∪ cuni 4836   class class class wbr 5070   ↦ cmpt 5153  dom cdm 5580  ran crn 5581   ↾ cres 5582  Fun wfun 6412   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  Fincfn 8691  supcsup 9129  ℂcc 10800  ℝcr 10801  0cc0 10802  1c1 10803   + caddc 10805   · cmul 10807   < clt 10940   ≤ cle 10941   − cmin 11135   / cdiv 11562  ℕcn 11903  ℤcz 12249  (,)cioo 13008  (,]cioc 13009  [,]cicc 13011  ...cfz 13168  ..^cfzo 13311  ⌊cfl 13438  abscabs 14873  –cn→ccncf 23945   lim_ℂ climc 24931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880  ax-mulf 10882

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-fi 9100  df-sup 9131  df-inf 9132  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-q 12618  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-ioo 13012  df-ioc 13013  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-fl 13440  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-starv 16903  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-unif 16911  df-hom 16912  df-cco 16913  df-rest 17050  df-topn 17051  df-0g 17069  df-gsum 17070  df-topgen 17071  df-pt 17072  df-prds 17075  df-xrs 17130  df-qtop 17135  df-imas 17136  df-xps 17138  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-mulg 18616  df-cntz 18838  df-cmn 19303  df-psmet 20502  df-xmet 20503  df-met 20504  df-bl 20505  df-mopn 20506  df-cnfld 20511  df-top 21951  df-topon 21968  df-topsp 21990  df-bases 22004  df-cld 22078  df-ntr 22079  df-cls 22080  df-cn 22286  df-cnp 22287  df-cmp 22446  df-tx 22621  df-hmeo 22814  df-xms 23381  df-ms 23382  df-tms 23383  df-cncf 23947  df-limc 24935

This theorem is referenced by:  fourierdlem94  43631  fourierdlem113  43650