Theorem fourierdlem71 44408

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 9266	. . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → 𝐹:dom 𝐹⟶ℝ)
5		simpl 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → 𝜑)
6		simpr 485	. . . . . . . 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 7390	. . . . . . . . . . . . 13 ⊢ (0...𝑀) ∈ V
9	8	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝑀) ∈ V)
10	7, 9	fexd 7177	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ V)
11		rnexg 7841	. . . . . . . . . . 11 ⊢ (𝑄 ∈ V → ran 𝑄 ∈ V)
12		inex1g 5276	. . . . . . . . . . 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 481	. . . . . . . . 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 7390	. . . . . . . . . . . . . . 15 ⊢ (0..^𝑀) ∈ V
17	16	mptex 7173	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0..^𝑀) ↦ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ V
18	15, 17	eqeltri 2834	. . . . . . . . . . . . 13 ⊢ 𝐼 ∈ V
19	18	rnex 7849	. . . . . . . . . . . 12 ⊢ ran 𝐼 ∈ V
20	19	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝐼 ∈ V)
21	20	uniexd 7679	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran 𝐼 ∈ V)
22	21	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → ∪ ran 𝐼 ∈ V)
23		uniprg 4882	. . . . . . . . 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 584	. . . . . . . 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 2840	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
26		elinel2 4156	. . . . . . . . 9 ⊢ (𝑥 ∈ (ran 𝑄 ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
27	26	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼)) ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
28		simpll 765	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝜑)
29		elunnel1 4109	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ ∪ ran 𝐼)
30	29	adantll 712	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ ∪ ran 𝐼)
31	15	funmpt2 6540	. . . . . . . . . . . . 13 ⊢ Fun 𝐼
32		elunirn 7198	. . . . . . . . . . . . 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 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → ∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖))
36		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ dom 𝐼 → 𝑖 ∈ dom 𝐼)
37		ovex 7390	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V
38	37, 15	dmmpti 6645	. . . . . . . . . . . . . . . . . . 19 ⊢ dom 𝐼 = (0..^𝑀)
39	36, 38	eleqtrdi 2848	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ dom 𝐼 → 𝑖 ∈ (0..^𝑀))
40	39	adantl 482	. . . . . . . . . . . . . . . . 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 6959	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
43	40, 41, 42	syl2anc 584	. . . . . . . . . . . . . . . 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 24256	. . . . . . . . . . . . . . . . . . 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 6677	. . . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → dom (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
49		ssdmres 5960	. . . . . . . . . . . . . . . . 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 3982	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → (𝐼‘𝑖) ⊆ dom 𝐹)
52	51	3adant3 1132	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)) → (𝐼‘𝑖) ⊆ dom 𝐹)
53		simp3 1138	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)) → 𝑥 ∈ (𝐼‘𝑖))
54	52, 53	sseldd 3945	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)) → 𝑥 ∈ dom 𝐹)
55	54	3exp 1119	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑖 ∈ dom 𝐼 → (𝑥 ∈ (𝐼‘𝑖) → 𝑥 ∈ dom 𝐹)))
56	55	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → (𝑖 ∈ dom 𝐼 → (𝑥 ∈ (𝐼‘𝑖) → 𝑥 ∈ dom 𝐹)))
57	56	rexlimdv 3150	. . . . . . . . . 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 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼)) ∧ ¬ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
60	27, 59	pm2.61dan 811	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼)) → 𝑥 ∈ dom 𝐹)
61	5, 25, 60	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → 𝑥 ∈ dom 𝐹)
62	4, 61	ffvelcdmd 7036	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → (𝐹‘𝑥) ∈ ℝ)
63	62	recnd 11183	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → (𝐹‘𝑥) ∈ ℂ)
64	63	abscld 15321	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
65		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = (ran 𝑄 ∩ dom 𝐹))
66		fzfid 13878	. . . . . . . . . 10 ⊢ (𝜑 → (0...𝑀) ∈ Fin)
67		rnffi 43382	. . . . . . . . . 10 ⊢ ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
68	7, 66, 67	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑄 ∈ Fin)
69		infi 9212	. . . . . . . . 9 ⊢ (ran 𝑄 ∈ Fin → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
70	68, 69	syl 17	. . . . . . . 8 ⊢ (𝜑 → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
71	70	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → (ran 𝑄 ∩ dom 𝐹) ∈ Fin)
72	65, 71	eqeltrd 2838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 ∈ Fin)
73		simpll 765	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥 ∈ 𝑤) → 𝜑)
74		simpr 485	. . . . . . . . . 10 ⊢ ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥 ∈ 𝑤) → 𝑥 ∈ 𝑤)
75		simpl 483	. . . . . . . . . 10 ⊢ ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥 ∈ 𝑤) → 𝑤 = (ran 𝑄 ∩ dom 𝐹))
76	74, 75	eleqtrd 2840	. . . . . . . . 9 ⊢ ((𝑤 = (ran 𝑄 ∩ dom 𝐹) ∧ 𝑥 ∈ 𝑤) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
77	76	adantll 712	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥 ∈ 𝑤) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
78	3	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝐹:dom 𝐹⟶ℝ)
79	26	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → 𝑥 ∈ dom 𝐹)
80	78, 79	ffvelcdmd 7036	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (𝐹‘𝑥) ∈ ℝ)
81	80	recnd 11183	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (𝐹‘𝑥) ∈ ℂ)
82	81	abscld 15321	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹)) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
83	73, 77, 82	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) ∧ 𝑥 ∈ 𝑤) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
84	83	ralrimiva 3143	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ∈ ℝ)
85		fimaxre3 12101	. . . . . 6 ⊢ ((𝑤 ∈ Fin ∧ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦)
86	72, 84, 85	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦)
87	86	adantlr 713	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) ∧ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦)
88		simpll 765	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝜑)
89		neqne 2951	. . . . . . 7 ⊢ (¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹) → 𝑤 ≠ (ran 𝑄 ∩ dom 𝐹))
90		elprn1 43864	. . . . . . 7 ⊢ ((𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} ∧ 𝑤 ≠ (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ∪ ran 𝐼)
91	89, 90	sylan2 593	. . . . . 6 ⊢ ((𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ∪ ran 𝐼)
92	91	adantll 712	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) ∧ ¬ 𝑤 = (ran 𝑄 ∩ dom 𝐹)) → 𝑤 = ∪ ran 𝐼)
93		fzofi 13879	. . . . . . . 8 ⊢ (0..^𝑀) ∈ Fin
94	15	rnmptfi 43378	. . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → 𝐹:dom 𝐹⟶ℝ)
98	97, 58	ffvelcdmd 7036	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → (𝐹‘𝑥) ∈ ℝ)
99	98	recnd 11183	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ ran 𝐼) → (𝐹‘𝑥) ∈ ℂ)
100	99	adantlr 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑥 ∈ ∪ ran 𝐼) → (𝐹‘𝑥) ∈ ℂ)
101	100	abscld 15321	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑥 ∈ ∪ ran 𝐼) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
102	37, 15	fnmpti 6644	. . . . . . . . . . 11 ⊢ 𝐼 Fn (0..^𝑀)
103		fvelrnb 6903	. . . . . . . . . . 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 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡)
107	7	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
108		elfzofz 13588	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
109	108	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
110	107, 109	ffvelcdmd 7036	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
111		fzofzp1 13669	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
112	111	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
113	107, 112	ffvelcdmd 7036	. . . . . . . . . . . . . 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 44128	. . . . . . . . . . . . 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 1132	. . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥) = (𝐹‘𝑥))
119	118	fveq2d 6846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) = (abs‘(𝐹‘𝑥)))
120	119	breq1d 5115	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑥)) ≤ 𝑏))
121	120	adantl 482	. . . . . . . . . . . . . . . 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 3172	. . . . . . . . . . . . . . 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 3175	. . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . 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 689	. . . . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
128	127	3adant1 1130	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
129	128	raleqdv 3313	. . . . . . . . . . . . . 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 3175	. . . . . . . . . . . . 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 1119	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) → ((𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏)))
134	133	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → (𝑖 ∈ (0..^𝑀) → ((𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏)))
135	134	rexlimdv 3150	. . . . . . . 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 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑥 ∈ 𝑡 (abs‘(𝐹‘𝑥)) ≤ 𝑏)
138		eqimss 4000	. . . . . . 7 ⊢ (𝑤 = ∪ ran 𝐼 → 𝑤 ⊆ ∪ ran 𝐼)
139	138	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → 𝑤 ⊆ ∪ ran 𝐼)
140	96, 101, 137, 139	ssfiunibd 43533	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦)
141	88, 92, 140	syl2anc 584	. . . 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 811	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼}) → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝑤 (abs‘(𝐹‘𝑥)) ≤ 𝑦)
143		simpr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ran 𝑄)
144		elinel2 4156	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹) → 𝑥 ∈ dom 𝐹)
145	144	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ dom 𝐹)
146	143, 145	elind 4154	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ (ran 𝑄 ∩ dom 𝐹))
147		elun1 4136	. . . . . . . 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 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑀 ∈ ℕ)
151	7	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
152		elinel1 4155	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹) → 𝑥 ∈ (𝐴[,]𝐵))
153	152	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ (𝐴[,]𝐵))
154		fourierdlem71.q0	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
155	154	eqcomd 2742	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 = (𝑄‘0))
156	155	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝐴 = (𝑄‘0))
157		fourierdlem71.10	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
158	157	eqcomd 2742	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 = (𝑄‘𝑀))
159	158	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝐵 = (𝑄‘𝑀))
160	156, 159	oveq12d 7375	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄‘𝑀)))
161	153, 160	eleqtrd 2840	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
162	161	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → 𝑥 ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
163		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ¬ 𝑥 ∈ ran 𝑄)
164		fveq2 6842	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (𝑄‘𝑘) = (𝑄‘𝑗))
165	164	breq1d 5115	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → ((𝑄‘𝑘) < 𝑥 ↔ (𝑄‘𝑗) < 𝑥))
166	165	cbvrabv 3417	. . . . . . . . . . . 12 ⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝑥} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < 𝑥}
167	166	supeq1i 9383	. . . . . . . . . . 11 ⊢ sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝑥}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < 𝑥}, ℝ, < )
168	150, 151, 162, 163, 167	fourierdlem25 44363	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) ∧ ¬ 𝑥 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
169	39	ad2antrl 726	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) → 𝑖 ∈ (0..^𝑀))
170		simprr 771	. . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖))) → 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
173	169, 172	jca 512	. . . . . . . . . . . . 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 2849	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ dom 𝐼)
176	175	ad2antrl 726	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑖 ∈ dom 𝐼)
177		simprr 771	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
178	125	eqcomd 2742	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼‘𝑖))
179	178	ad2antrl 726	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼‘𝑖))
180	177, 179	eleqtrd 2840	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → 𝑥 ∈ (𝐼‘𝑖))
181	176, 180	jca 512	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))) → (𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)))
182	173, 181	impbida 799	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑖 ∈ dom 𝐼 ∧ 𝑥 ∈ (𝐼‘𝑖)) ↔ (𝑖 ∈ (0..^𝑀) ∧ 𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))))
183	182	rexbidv2 3171	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑖 ∈ dom 𝐼 𝑥 ∈ (𝐼‘𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑥 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
184	183	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)))))
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 4137	. . . . . . . 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 811	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)) → 𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
190	189	ralrimiva 3143	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)𝑥 ∈ ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
191		dfss3 3932	. . . . 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 584	. . . 4 ⊢ (𝜑 → ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼} = ((ran 𝑄 ∩ dom 𝐹) ∪ ∪ ran 𝐼))
194	192, 193	sseqtrrd 3985	. . 3 ⊢ (𝜑 → ((𝐴[,]𝐵) ∩ dom 𝐹) ⊆ ∪ {(ran 𝑄 ∩ dom 𝐹), ∪ ran 𝐼})
195	2, 64, 142, 194	ssfiunibd 43533	. 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦)
196		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑥𝜑
197		nfra1 3267	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦
198	196, 197	nfan 1902	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦)
199		fourierdlem71.dmf	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ)
200	199	sselda 3944	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝑥 ∈ ℝ)
201		fourierdlem71.b	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 ∈ ℝ)
202	201	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝐵 ∈ ℝ)
203	202, 200	resubcld 11583	. . . . . . . . . . . . . . . . 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 11583	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
207	204, 206	eqeltrid 2842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇 ∈ ℝ)
208	207	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝑇 ∈ ℝ)
209		fourierdlem71.altb	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐴 < 𝐵)
210	205, 201	posdifd 11742	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
211	209, 210	mpbid 231	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
212	211, 204	breqtrrdi 5147	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 < 𝑇)
213	212	gt0ne0d 11719	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇 ≠ 0)
214	213	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝑇 ≠ 0)
215	203, 208, 214	redivcld 11983	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → ((𝐵 − 𝑥) / 𝑇) ∈ ℝ)
216	215	flcld 13703	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ)
217	216	zred 12607	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℝ)
218	217, 208	remulcld 11185	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) ∈ ℝ)
219	200, 218	readdcld 11184	. . . . . . . . . . . 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 6959	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ ℝ) → (𝐸‘𝑥) = (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
222	200, 219, 221	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) = (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
223	222	fveq2d 6846	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘(𝐸‘𝑥)) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))))
224		fvex 6855	. . . . . . . . . . . 12 ⊢ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ V
225		eleq1 2825	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (𝑘 ∈ ℤ ↔ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ))
226	225	anbi2d 629	. . . . . . . . . . . . 13 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ 𝑘 ∈ ℤ) ↔ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ)))
227		oveq1 7364	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (𝑘 · 𝑇) = ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))
228	227	oveq2d 7373	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (𝑥 + (𝑘 · 𝑇)) = (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)))
229	228	fveq2d 6846	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (⌊‘((𝐵 − 𝑥) / 𝑇)) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))))
230	229	eqeq1d 2738	. . . . . . . . . . . . 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 3518	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ) → (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))
234	216, 233	mpdan 685	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘(𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝐹‘𝑥))
235	223, 234	eqtr2d 2777	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = (𝐹‘(𝐸‘𝑥)))
236	235	fveq2d 6846	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘(𝐸‘𝑥))))
237	236	adantlr 713	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘(𝐸‘𝑥))))
238		fveq2 6842	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
239	238	fveq2d 6846	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘𝑤)))
240	239	breq1d 5115	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ((abs‘(𝐹‘𝑥)) ≤ 𝑦 ↔ (abs‘(𝐹‘𝑤)) ≤ 𝑦))
241	240	cbvralvw 3225	. . . . . . . . . 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 725	. . . . . . . 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 13354	. . . . . . . . . . 11 ⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
245	205	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝐴 ∈ ℝ)
246	209	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝐴 < 𝐵)
247		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
248		oveq2 7365	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝐵 − 𝑥) = (𝐵 − 𝑦))
249	248	oveq1d 7372	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝐵 − 𝑥) / 𝑇) = ((𝐵 − 𝑦) / 𝑇))
250	249	fveq2d 6846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (⌊‘((𝐵 − 𝑥) / 𝑇)) = (⌊‘((𝐵 − 𝑦) / 𝑇)))
251	250	oveq1d 7372	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇) = ((⌊‘((𝐵 − 𝑦) / 𝑇)) · 𝑇))
252	247, 251	oveq12d 7375	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) = (𝑦 + ((⌊‘((𝐵 − 𝑦) / 𝑇)) · 𝑇)))
253	252	cbvmptv 5218	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇))) = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵 − 𝑦) / 𝑇)) · 𝑇)))
254	220, 253	eqtri 2764	. . . . . . . . . . . . 13 ⊢ 𝐸 = (𝑦 ∈ ℝ ↦ (𝑦 + ((⌊‘((𝐵 − 𝑦) / 𝑇)) · 𝑇)))
255	245, 202, 246, 204, 254	fourierdlem4 44342	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → 𝐸:ℝ⟶(𝐴(,]𝐵))
256	255, 200	ffvelcdmd 7036	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ (𝐴(,]𝐵))
257	244, 256	sselid 3942	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ (𝐴[,]𝐵))
258	228	eleq1d 2822	. . . . . . . . . . . . . 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 3518	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐹) ∧ (⌊‘((𝐵 − 𝑥) / 𝑇)) ∈ ℤ) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹)
262	216, 261	mpdan 685	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝑥 + ((⌊‘((𝐵 − 𝑥) / 𝑇)) · 𝑇)) ∈ dom 𝐹)
263	222, 262	eqeltrd 2838	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ dom 𝐹)
264	257, 263	elind 4154	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹))
265	264	adantlr 713	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (𝐸‘𝑥) ∈ ((𝐴[,]𝐵) ∩ dom 𝐹))
266		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑤 = (𝐸‘𝑥) → (𝐹‘𝑤) = (𝐹‘(𝐸‘𝑥)))
267	266	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝑤 = (𝐸‘𝑥) → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘(𝐸‘𝑥))))
268	267	breq1d 5115	. . . . . . . . 9 ⊢ (𝑤 = (𝐸‘𝑥) → ((abs‘(𝐹‘𝑤)) ≤ 𝑦 ↔ (abs‘(𝐹‘(𝐸‘𝑥))) ≤ 𝑦))
269	268	rspccva 3580	. . . . . . . 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 584	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘(𝐸‘𝑥))) ≤ 𝑦)
271	237, 270	eqbrtrd 5127	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) ∧ 𝑥 ∈ dom 𝐹) → (abs‘(𝐹‘𝑥)) ≤ 𝑦)
272	271	ex 413	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) → (𝑥 ∈ dom 𝐹 → (abs‘(𝐹‘𝑥)) ≤ 𝑦))
273	198, 272	ralrimi 3240	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦) → ∀𝑥 ∈ dom 𝐹(abs‘(𝐹‘𝑥)) ≤ 𝑦)
274	273	ex 413	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ ((𝐴[,]𝐵) ∩ dom 𝐹)(abs‘(𝐹‘𝑥)) ≤ 𝑦 → ∀𝑥 ∈ dom 𝐹(abs‘(𝐹‘𝑥)) ≤ 𝑦))
275	274	reximdv 3167	. 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  {crab 3407  Vcvv 3445   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  {cpr 4588  ∪ cuni 4865   class class class wbr 5105   ↦ cmpt 5188  dom cdm 5633  ran crn 5634   ↾ cres 5635  Fun wfun 6490   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  Fincfn 8883  supcsup 9376  ℂcc 11049  ℝcr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056   < clt 11189   ≤ cle 11190   − cmin 11385   / cdiv 11812  ℕcn 12153  ℤcz 12499  (,)cioo 13264  (,]cioc 13265  [,]cicc 13267  ...cfz 13424  ..^cfzo 13567  ⌊cfl 13695  abscabs 15119  –cn→ccncf 24239   lim_ℂ climc 25226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-mulf 11131

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-cn 22578  df-cnp 22579  df-cmp 22738  df-tx 22913  df-hmeo 23106  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241  df-limc 25230

This theorem is referenced by:  fourierdlem94  44431  fourierdlem113  44450