Theorem fourierdlem71 42456

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 8787	. . . 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 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → 𝐹:dom 𝐹⟶ℝ)
5		simpl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → 𝜑)
6		simpr 487	. . . . . . . 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 7183	. . . . . . . . . . . . 13 ⊢ (0...𝑀) ∈ V
9	8	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝑀) ∈ V)
10		fex 6983	. . . . . . . . . . . 12 ⊢ ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ V) → 𝑄 ∈ V)
11	7, 9, 10	syl2anc 586	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ V)
12		rnexg 7608	. . . . . . . . . . 11 ⊢ (𝑄 ∈ V → ran 𝑄 ∈ V)
13		inex1g 5215	. . . . . . . . . . 11 ⊢ (ran 𝑄 ∈ V → (ran 𝑄 ∩ dom 𝐹) ∈ V)
14	11, 12, 13	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (ran 𝑄 ∩ dom 𝐹) ∈ V)
15	14	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → (ran 𝑄 ∩ dom 𝐹) ∈ V)
16		fourierdlem71.i	. . . . . . . . . . . . . 14 ⊢ 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
17		ovex 7183	. . . . . . . . . . . . . . 15 ⊢ (0..^𝑀) ∈ V
18	17	mptex 6980	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^𝑀) ↦ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ V
19	16, 18	eqeltri 2909	. . . . . . . . . . . . 13 ⊢ 𝐼 ∈ V
20	19	rnex 7611	. . . . . . . . . . . 12 ⊢ ran 𝐼 ∈ V
21	20	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐼 ∈ V)
22	21	uniexd 7462	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran 𝐼 ∈ V)
23	22	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → ∪ ran 𝐼 ∈ V)
24		uniprg 4845	. . . . . . . . 9 ⊢ (((ran 𝑄 ∩ dom 𝐹) ∈ V ∧ ∪ ran 𝐼 ∈ V) → ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
25	15, 23, 24	syl2anc 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
26	6, 25	eleqtrd 2915	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
27		elinel2 4172	. . . . . . . . 9 ⊢ (𝑥 ∈ (ran 𝑄 ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
28	27	adantl 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼)) ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
29		simpll 765	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝜑)
30		elunnel1 4125	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ ∪ ran 𝐼)
31	30	adantll 712	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ ∪ ran 𝐼)
32	16	funmpt2 6388	. . . . . . . . . . . . 13 ⊢ Fun 𝐼
33		elunirn 7004	. . . . . . . . . . . . 13 ⊢ (Fun 𝐼 → (𝑥 ∈ ∪ ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖)))
34	32, 33	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖))
35	34	biimpi 218	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ ran 𝐼 → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖))
36	35	adantl 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖))
37		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ dom 𝐼 → 𝑖 ∈ dom 𝐼)
38		ovex 7183	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V
39	38, 16	dmmpti 6486	. . . . . . . . . . . . . . . . . . 19 ⊢ dom 𝐼 = (0..^𝑀)
40	37, 39	eleqtrdi 2923	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ dom 𝐼 → 𝑖 ∈ (0..^𝑀))
41	40	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → 𝑖 ∈ (0..^𝑀))
42	38	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V)
43	16	fvmpt2 6773	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
44	41, 42, 43	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
45		fourierdlem71.fcn	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
46		cncff 23495	. . . . . . . . . . . . . . . . . . 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)
47		fdm 6516	. . . . . . . . . . . . . . . . . . 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))))
48	45, 46, 47	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
49	40, 48	sylan2 594	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
50		ssdmres 5870	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
51	49, 50	sylibr 236	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ dom 𝐹)
52	44, 51	eqsstrd 4004	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → (𝐼‘𝑖) ⊆ dom 𝐹)
53	52	3adant3 1128	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)) → (𝐼‘𝑖) ⊆ dom 𝐹)
54		simp3 1134	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)) → 𝑥 ∈ (𝐼‘𝑖))
55	53, 54	sseldd 3967	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)) → 𝑥 ∈ dom 𝐹)
56	55	3exp 1115	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑖 ∈ dom 𝐼 → (𝑥 ∈ (𝐼‘𝑖) → 𝑥 ∈ dom 𝐹)))
57	56	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → (𝑖 ∈ dom 𝐼 → (𝑥 ∈ (𝐼‘𝑖) → 𝑥 ∈ dom 𝐹)))
58	57	rexlimdv 3283	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖) → 𝑥 ∈ dom 𝐹))
59	36, 58	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → 𝑥 ∈ dom 𝐹)
60	29, 31, 59	syl2anc 586	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
61	28, 60	pm2.61dan 811	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼)) → 𝑥 ∈ dom 𝐹)
62	5, 26, 61	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → 𝑥 ∈ dom 𝐹)
63	4, 62	ffvelrnd 6846	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → (𝐹‘𝑥) ∈ ℝ)
64	63	recnd 10663	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → (𝐹‘𝑥) ∈ ℂ)
65	64	abscld 14790	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
66		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = (ran 𝑄 ∩ dom 𝐹))
67		fzfid 13335	. . . . . . . . . 10 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
68		rnffi 41424	. . . . . . . . . 10 ⊢ ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
69	7, 67, 68	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑄 ∈ Fin)
70		infi 8736	. . . . . . . . 9 ⊢ (ran 𝑄 ∈ Fin → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
71	69, 70	syl 17	. . . . . . . 8 ⊢ (𝜑 → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
72	71	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
73	66, 72	eqeltrd 2913	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 ∈ Fin)
74		simpll 765	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥 ∈ 𝑤) → 𝜑)
75		simpr 487	. . . . . . . . . 10 ⊢ ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥 ∈ 𝑤) → 𝑥 ∈ 𝑤)
76		simpl 485	. . . . . . . . . 10 ⊢ ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥 ∈ 𝑤) → 𝑤 = (ran 𝑄 ∩ dom 𝐹))
77	75, 76	eleqtrd 2915	. . . . . . . . 9 ⊢ ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥 ∈ 𝑤) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
78	77	adantll 712	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥 ∈ 𝑤) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
79	3	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝐹:dom 𝐹⟶ℝ)
80	27	adantl 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
81	79, 80	ffvelrnd 6846	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (𝐹‘𝑥) ∈ ℝ)
82	81	recnd 10663	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (𝐹‘𝑥) ∈ ℂ)
83	82	abscld 14790	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
84	74, 78, 83	syl2anc 586	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥 ∈ 𝑤) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
85	84	ralrimiva 3182	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ∈ ℝ)
86		fimaxre3 11581	. . . . . 6 ⊢ ((𝑤 ∈ Fin ∧ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦)
87	73, 85, 86	syl2anc 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦)
88	87	adantlr 713	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦)
89		simpll 765	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝜑)
90		neqne 3024	. . . . . . 7 ⊢ (¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹) → 𝑤 ≠ (ran 𝑄 ∩ dom 𝐹))
91		elprn1 41907	. . . . . . 7 ⊢ ((𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} ∧ 𝑤 ≠ (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ∪ ran 𝐼)
92	90, 91	sylan2 594	. . . . . 6 ⊢ ((𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ∪ ran 𝐼)
93	92	adantll 712	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ∪ ran 𝐼)
94		fzofi 13336	. . . . . . . 8 ⊢ (0..^𝑀) ∈ Fin
95	16	rnmptfi 41420	. . . . . . . 8 ⊢ ((0..^𝑀) ∈ Fin → ran 𝐼 ∈ Fin)
96	94, 95	ax-mp 5	. . . . . . 7 ⊢ ran 𝐼 ∈ Fin
97	96	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → ran 𝐼 ∈ Fin)
98	3	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → 𝐹:dom 𝐹⟶ℝ)
99	98, 59	ffvelrnd 6846	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → (𝐹‘𝑥) ∈ ℝ)
100	99	recnd 10663	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → (𝐹‘𝑥) ∈ ℂ)
101	100	adantlr 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑥 ∈ ∪ ran 𝐼) → (𝐹‘𝑥) ∈ ℂ)
102	101	abscld 14790	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑥 ∈ ∪ ran 𝐼) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
103	38, 16	fnmpti 6485	. . . . . . . . . . 11 ⊢ 𝐼 Fn (0..^𝑀)
104		fvelrnb 6720	. . . . . . . . . . 11 ⊢ (𝐼 Fn (0..^𝑀) → (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡))
105	103, 104	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡)
106	105	biimpi 218	. . . . . . . . 9 ⊢ (𝑡 ∈ ran 𝐼 → ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡)
107	106	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡)
108	7	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
109		elfzofz 13047	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
110	109	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
111	108, 110	ffvelrnd 6846	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
112		fzofzp1 13128	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
113	112	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
114	108, 113	ffvelrnd 6846	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
115		fourierdlem71.l	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1))))
116		fourierdlem71.r	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖)))
117	111, 114, 45, 115, 116	cncfioobd 42173	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏)
118	117	3adant3 1128	. . . . . . . . . . . 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)
119		fvres 6683	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥) = (𝐹‘𝑥))
120	119	fveq2d 6668	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) = (abs‘(𝐹‘𝑥)))
121	120	breq1d 5068	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑥)) ≤ 𝑏))
122	121	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑥)) ≤ 𝑏))
123	122	ralbidva 3196	. . . . . . . . . . . . . . 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))
124	123	rexbidv 3297	. . . . . . . . . . . . . 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))
125	124	3adant3 1128	. . . . . . . . . . . . 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))
126	38, 43	mpan2 689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0..^𝑀) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
127		id 22	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼‘𝑖) = 𝑡 → (𝐼‘𝑖) = 𝑡)
128	126, 127	sylan9req 2877	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
129	128	3adant1 1126	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
130	129	raleqdv 3415	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → (∀𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑥)) ≤ 𝑏 ↔ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏))
131	130	rexbidv 3297	. . . . . . . . . . . . 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))
132	125, 131	bitrd 281	. . . . . . . . . . . 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))
133	118, 132	mpbid 234	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏)
134	133	3exp 1115	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) → ((𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏)))
135	134	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → (𝑖 ∈ (0..^𝑀) → ((𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏)))
136	135	rexlimdv 3283	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → (∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏))
137	107, 136	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏)
138	137	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏)
139		eqimss 4022	. . . . . . 7 ⊢ (𝑤 = ∪ ran 𝐼 → 𝑤 ⊆ ∪ ran 𝐼)
140	139	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → 𝑤 ⊆ ∪ ran 𝐼)
141	97, 102, 138, 140	ssfiunibd 41569	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦)
142	89, 93, 141	syl2anc 586	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦)
143	88, 142	pm2.61dan 811	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦)
144		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ran 𝑄)
145		elinel2 4172	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
146	145	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ dom 𝐹)
147	144, 146	elind 4170	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
148		elun1 4151	. . . . . . . 8 ⊢ (𝑥 ∈ (ran 𝑄 ∩ dom 𝐹) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
149	147, 148	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
150		fourierdlem71.7	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℕ)
151	150	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑀 ∈ ℕ)
152	7	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
153		elinel1 4171	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹) → 𝑥 ∈ (𝐴[,]𝐵))
154	153	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ (𝐴[,]𝐵))
155		fourierdlem71.q0	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
156	155	eqcomd 2827	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 = (𝑄‘0))
157	156	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝐴 = (𝑄‘0))
158		fourierdlem71.10	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
159	158	eqcomd 2827	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 = (𝑄‘𝑀))
160	159	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝐵 = (𝑄‘𝑀))
161	157, 160	oveq12d 7168	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄‘𝑀)))
162	154, 161	eleqtrd 2915	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
163	162	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
164		simpr 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ¬ 𝑥 ∈ ran 𝑄)
165		fveq2 6664	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝑄‘𝑘) = (𝑄‘𝑗))
166	165	breq1d 5068	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → ((𝑄‘𝑘) < 𝑥 ↔ (𝑄‘𝑗) < 𝑥))
167	166	cbvrabv 3491	. . . . . . . . . . . 12 ⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝑥} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < 𝑥}
168	167	supeq1i 8905	. . . . . . . . . . 11 ⊢ sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝑥}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < 𝑥}, ℝ, < )
169	151, 152, 163, 164, 168	fourierdlem25 42411	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
170	40	ad2antrl 726	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) → 𝑖 ∈ (0..^𝑀))
171		simprr 771	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) → 𝑥 ∈ (𝐼‘𝑖))
172	170, 126	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
173	171, 172	eleqtrd 2915	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) → 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
174	170, 173	jca 514	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) → (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
175		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0..^𝑀))
176	175, 39	eleqtrrdi 2924	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ dom 𝐼)
177	176	ad2antrl 726	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑖 ∈ dom 𝐼)
178		simprr 771	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
179	126	eqcomd 2827	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼‘𝑖))
180	179	ad2antrl 726	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼‘𝑖))
181	178, 180	eleqtrd 2915	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑥 ∈ (𝐼‘𝑖))
182	177, 181	jca 514	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)))
183	174, 182	impbida 799	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)) ↔ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))))
184	183	rexbidv2 3295	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
185	184	ad2antrr 724	. . . . . . . . . 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)))))
186	169, 185	mpbird 259	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖))
187	186, 34	sylibr 236	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ∪ ran 𝐼)
188		elun2 4152	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ ran 𝐼 → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
189	187, 188	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
190	149, 189	pm2.61dan 811	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
191	190	ralrimiva 3182	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
192		dfss3 3955	. . . . 5 ⊢ (((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼) ↔ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
193	191, 192	sylibr 236	. . . 4 ⊢ (𝜑 → ((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
194	14, 22, 24	syl2anc 586	. . . 4 ⊢ (𝜑 → ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
195	193, 194	sseqtrrd 4007	. . 3 ⊢ (𝜑 → ((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼})
196	2, 65, 143, 195	ssfiunibd 41569	. 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦)
197		nfv 1911	. . . . . 6 ⊢ Ⅎ𝑥𝜑
198		nfra1 3219	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦
199	197, 198	nfan 1896	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦)
200		fourierdlem71.dmf	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ)
201	200	sselda 3966	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ℝ)
202		fourierdlem71.b	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 ∈ ℝ)
203	202	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝐵 ∈ ℝ)
204	203, 201	resubcld 11062	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐵 − 𝑥) ∈ ℝ)
205		fourierdlem71.t	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑇 = (𝐵 − 𝐴)
206		fourierdlem71.a	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐴 ∈ ℝ)
207	202, 206	resubcld 11062	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
208	205, 207	eqeltrid 2917	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇 ∈ ℝ)
209	208	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝑇 ∈ ℝ)
210		fourierdlem71.altb	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐴 < 𝐵)
211	206, 202	posdifd 11221	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
212	210, 211	mpbid 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
213	212, 205	breqtrrdi 5100	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 < 𝑇)
214	213	gt0ne0d 11198	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇 ≠ 0)
215	214	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝑇 ≠ 0)
216	204, 209, 215	redivcld 11462	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → ((𝐵 − 𝑥) / 𝑇) ∈ ℝ)
217	216	flcld 13162	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ)
218	217	zred 12081	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℝ)
219	218, 209	remulcld 10665	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) ∈ ℝ)
220	201, 219	readdcld 10664	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ ℝ)
221		fourierdlem71.e	. . . . . . . . . . . . 13 ⊢ 𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
222	221	fvmpt2 6773	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ ℝ) → (𝐸‘𝑥) = (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
223	201, 220, 222	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) = (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
224	223	fveq2d 6668	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘(𝐸‘𝑥)) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))))
225		fvex 6677	. . . . . . . . . . . 12 ⊢ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ V
226		eleq1 2900	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (𝑘 ∈ ℤ ↔ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ))
227	226	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) ↔ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ)))
228		oveq1 7157	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (𝑘 · 𝑇) = ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))
229	228	oveq2d 7166	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (𝑥 + (𝑘 · 𝑇)) = (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
230	229	fveq2d 6668	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))))
231	230	eqeq1d 2823	. . . . . . . . . . . . 13 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → ((𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥) ↔ (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝐹‘𝑥)))
232	227, 231	imbi12d 347	. . . . . . . . . . . 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))))
233		fourierdlem71.fxpt	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘𝑥))
234	225, 232, 233	vtocl 3559	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))
235	217, 234	mpdan 685	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))
236	224, 235	eqtr2d 2857	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = (𝐹‘(𝐸‘𝑥)))
237	236	fveq2d 6668	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘(𝐸‘𝑥))))
238	237	adantlr 713	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘(𝐸‘𝑥))))
239		fveq2 6664	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
240	239	fveq2d 6668	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘𝑤)))
241	240	breq1d 5068	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((abs‘(𝐹‘𝑥)) ≤ 𝑦 ↔ (abs‘(𝐹‘𝑤)) ≤ 𝑦))
242	241	cbvralvw 3449	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦 ↔ ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑤)) ≤ 𝑦)
243	242	biimpi 218	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦 → ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑤)) ≤ 𝑦)
244	243	ad2antlr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → ∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑤)) ≤ 𝑦)
245		iocssicc 12819	. . . . . . . . . . 11 ⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
246	206	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝐴 ∈ ℝ)
247	210	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝐴 < 𝐵)
248		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
249		oveq2 7158	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝐵 − 𝑥) = (𝐵 − 𝑦))
250	249	oveq1d 7165	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − 𝑦) / 𝑇))
251	250	fveq2d 6668	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − 𝑦) / 𝑇)))
252	251	oveq1d 7165	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − 𝑦) / 𝑇)) · 𝑇))
253	248, 252	oveq12d 7168	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = (𝑦 + ((⌊‘((𝐵 − 𝑦) / 𝑇)) · 𝑇)))
254	253	cbvmptv 5161	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵 − 𝑦) / 𝑇)) · 𝑇)))
255	221, 254	eqtri 2844	. . . . . . . . . . . . 13 ⊢ 𝐸 = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵 − 𝑦) / 𝑇)) · 𝑇)))
256	246, 203, 247, 205, 255	fourierdlem4 42390	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝐸:ℝ⟶(𝐴(,]𝐵))
257	256, 201	ffvelrnd 6846	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ (𝐴(,]𝐵))
258	245, 257	sseldi 3964	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ (𝐴[,]𝐵))
259	229	eleq1d 2897	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → ((𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹 ↔ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹))
260	227, 259	imbi12d 347	. . . . . . . . . . . . 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)))
261		fourierdlem71.xpt	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ dom 𝐹)
262	225, 260, 261	vtocl 3559	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹)
263	217, 262	mpdan 685	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹)
264	223, 263	eqeltrd 2913	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ dom 𝐹)
265	258, 264	elind 4170	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹))
266	265	adantlr 713	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹))
267		fveq2 6664	. . . . . . . . . . 11 ⊢ (𝑤 = (𝐸‘𝑥) → (𝐹‘𝑤) = (𝐹‘(𝐸‘𝑥)))
268	267	fveq2d 6668	. . . . . . . . . 10 ⊢ (𝑤 = (𝐸‘𝑥) → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘(𝐸‘𝑥))))
269	268	breq1d 5068	. . . . . . . . 9 ⊢ (𝑤 = (𝐸‘𝑥) → ((abs‘(𝐹‘𝑤)) ≤ 𝑦 ↔ (abs‘(𝐹‘(𝐸‘𝑥))) ≤ 𝑦))
270	269	rspccva 3621	. . . . . . . 8 ⊢ ((∀𝑤 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑤)) ≤ 𝑦 ∧ (𝐸‘𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → (abs‘(𝐹‘(𝐸‘𝑥))) ≤ 𝑦)
271	244, 266, 270	syl2anc 586	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘(𝐸‘𝑥))) ≤ 𝑦)
272	238, 271	eqbrtrd 5080	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘𝑥)) ≤ 𝑦)
273	272	ex 415	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) → (𝑥 ∈ dom 𝐹 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))
274	199, 273	ralrimi 3216	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) → ∀𝑥 ∈ dom 𝐹(abs‘(𝐹‘𝑥)) ≤ 𝑦)
275	274	ex 415	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦 → ∀𝑥 ∈ dom 𝐹(abs‘(𝐹‘𝑥)) ≤ 𝑦))
276	275	reximdv 3273	. 2 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐹(abs‘(𝐹‘𝑥)) ≤ 𝑦))
277	196, 276	mpd 15	1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ dom 𝐹(abs‘(𝐹‘𝑥)) ≤ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  {crab 3142  Vcvv 3494   ∪ cun 3933   ∩ cin 3934   ⊆ wss 3935  {cpr 4562  ∪ cuni 4831   class class class wbr 5058   ↦ cmpt 5138  dom cdm 5549  ran crn 5550   ↾ cres 5551  Fun wfun 6343   Fn wfn 6344  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150  Fincfn 8503  supcsup 8898  ℂcc 10529  ℝcr 10530  0cc0 10531  1c1 10532   + caddc 10534   · cmul 10536   < clt 10669   ≤ cle 10670   − cmin 10864   / cdiv 11291  ℕcn 11632  ℤcz 11975  (,)cioo 12732  (,]cioc 12733  [,]cicc 12735  ...cfz 12886  ..^cfzo 13027  ⌊cfl 13154  abscabs 14587  –cn→ccncf 23478   lim_ℂ climc 24454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609  ax-mulf 10611

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-iin 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-om 7575  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-pm 8403  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fsupp 8828  df-fi 8869  df-sup 8900  df-inf 8901  df-oi 8968  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-q 12343  df-rp 12384  df-xneg 12501  df-xadd 12502  df-xmul 12503  df-ioo 12736  df-ioc 12737  df-ico 12738  df-icc 12739  df-fz 12887  df-fzo 13028  df-fl 13156  df-seq 13364  df-exp 13424  df-hash 13685  df-cj 14452  df-re 14453  df-im 14454  df-sqrt 14588  df-abs 14589  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-mulr 16573  df-starv 16574  df-sca 16575  df-vsca 16576  df-ip 16577  df-tset 16578  df-ple 16579  df-ds 16581  df-unif 16582  df-hom 16583  df-cco 16584  df-rest 16690  df-topn 16691  df-0g 16709  df-gsum 16710  df-topgen 16711  df-pt 16712  df-prds 16715  df-xrs 16769  df-qtop 16774  df-imas 16775  df-xps 16777  df-mre 16851  df-mrc 16852  df-acs 16854  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-submnd 17951  df-mulg 18219  df-cntz 18441  df-cmn 18902  df-psmet 20531  df-xmet 20532  df-met 20533  df-bl 20534  df-mopn 20535  df-cnfld 20540  df-top 21496  df-topon 21513  df-topsp 21535  df-bases 21548  df-cld 21621  df-ntr 21622  df-cls 21623  df-cn 21829  df-cnp 21830  df-cmp 21989  df-tx 22164  df-hmeo 22357  df-xms 22924  df-ms 22925  df-tms 22926  df-cncf 23480  df-limc 24458

This theorem is referenced by:  fourierdlem94  42479  fourierdlem113  42498