Theorem fourierdlem70 46534

Description: A piecewise continuous function is bounded. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem70.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem70.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem70.aleb	⊢ (𝜑 → 𝐴 ≤ 𝐵)
fourierdlem70.f	⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
fourierdlem70.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem70.q	⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
fourierdlem70.q0	⊢ (𝜑 → (𝑄‘0) = 𝐴)
fourierdlem70.qm	⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
fourierdlem70.qlt	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
fourierdlem70.fcn	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
fourierdlem70.r	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖)))
fourierdlem70.l	⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1))))
fourierdlem70.i	⊢ 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))

Assertion

Ref	Expression
fourierdlem70	⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑠 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑠)) ≤ 𝑥)

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

Allowed substitution hints:   𝐴(𝑥,𝑠)   𝐵(𝑥,𝑠)   𝑅(𝑥,𝑖)   𝐿(𝑥,𝑖)   𝑀(𝑥)

Proof of Theorem fourierdlem70

Dummy variables 𝑡 𝑣 𝑦 𝑤 𝑏 𝑧 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prfi 9236	. . 3 ⊢ {ran 𝑄, ∪ ran 𝐼} ∈ Fin
2	1	a1i 11	. 2 ⊢ (𝜑 → {ran 𝑄, ∪ ran 𝐼} ∈ Fin)
3		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼})
4		fourierdlem70.q	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
5		ovex 7401	. . . . . . . . . . 11 ⊢ (0...𝑀) ∈ V
6		fex 7182	. . . . . . . . . . 11 ⊢ ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ V) → 𝑄 ∈ V)
7	4, 5, 6	sylancl 587	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ V)
8		rnexg 7854	. . . . . . . . . 10 ⊢ (𝑄 ∈ V → ran 𝑄 ∈ V)
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑄 ∈ V)
10		fzofi 13909	. . . . . . . . . . . 12 ⊢ (0..^𝑀) ∈ Fin
11		fourierdlem70.i	. . . . . . . . . . . . 13 ⊢ 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
12	11	rnmptfi 45530	. . . . . . . . . . . 12 ⊢ ((0..^𝑀) ∈ Fin → ran 𝐼 ∈ Fin)
13	10, 12	ax-mp 5	. . . . . . . . . . 11 ⊢ ran 𝐼 ∈ Fin
14	13	elexi 3465	. . . . . . . . . 10 ⊢ ran 𝐼 ∈ V
15	14	uniex 7696	. . . . . . . . 9 ⊢ ∪ ran 𝐼 ∈ V
16		uniprg 4881	. . . . . . . . 9 ⊢ ((ran 𝑄 ∈ V ∧ ∪ ran 𝐼 ∈ V) → ∪ {ran 𝑄, ∪ ran 𝐼} = (ran 𝑄 ∪ ∪ ran 𝐼))
17	9, 15, 16	sylancl 587	. . . . . . . 8 ⊢ (𝜑 → ∪ {ran 𝑄, ∪ ran 𝐼} = (ran 𝑄 ∪ ∪ ran 𝐼))
18	17	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → ∪ {ran 𝑄, ∪ ran 𝐼} = (ran 𝑄 ∪ ∪ ran 𝐼))
19	3, 18	eleqtrd 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → 𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼))
20		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ ↦ {𝑣 ∈ (ℝ ↑m (0...𝑦)) ∣ (((𝑣‘0) = 𝐴 ∧ (𝑣‘𝑦) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑦)(𝑣‘𝑖) < (𝑣‘(𝑖 + 1)))}) = (𝑦 ∈ ℕ ↦ {𝑣 ∈ (ℝ ↑m (0...𝑦)) ∣ (((𝑣‘0) = 𝐴 ∧ (𝑣‘𝑦) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑦)(𝑣‘𝑖) < (𝑣‘(𝑖 + 1)))})
21		fourierdlem70.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ)
22		reex 11129	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
23	22, 5	elmap 8821	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ)
24	4, 23	sylibr 234	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑m (0...𝑀)))
25		fourierdlem70.q0	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
26		fourierdlem70.qm	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
27	25, 26	jca 511	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
28		fourierdlem70.qlt	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
29	28	ralrimiva 3130	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
30	24, 27, 29	jca32 515	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
31	20	fourierdlem2 46467	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ ((𝑦 ∈ ℕ ↦ {𝑣 ∈ (ℝ ↑m (0...𝑦)) ∣ (((𝑣‘0) = 𝐴 ∧ (𝑣‘𝑦) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑦)(𝑣‘𝑖) < (𝑣‘(𝑖 + 1)))})‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
32	21, 31	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 ∈ ((𝑦 ∈ ℕ ↦ {𝑣 ∈ (ℝ ↑m (0...𝑦)) ∣ (((𝑣‘0) = 𝐴 ∧ (𝑣‘𝑦) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑦)(𝑣‘𝑖) < (𝑣‘(𝑖 + 1)))})‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
33	30, 32	mpbird 257	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ ((𝑦 ∈ ℕ ↦ {𝑣 ∈ (ℝ ↑m (0...𝑦)) ∣ (((𝑣‘0) = 𝐴 ∧ (𝑣‘𝑦) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑦)(𝑣‘𝑖) < (𝑣‘(𝑖 + 1)))})‘𝑀))
34	20, 21, 33	fourierdlem15 46480	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
35	34	frnd 6678	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
36	35	sselda 3935	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
37	36	adantlr 716	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼)) ∧ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
38		simpll 767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝜑)
39		elunnel1 4108	. . . . . . . . 9 ⊢ ((𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ ∪ ran 𝐼)
40	39	adantll 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ ∪ ran 𝐼)
41		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran 𝐼) → 𝑠 ∈ ∪ ran 𝐼)
42	11	funmpt2 6539	. . . . . . . . . . 11 ⊢ Fun 𝐼
43		elunirn 7207	. . . . . . . . . . 11 ⊢ (Fun 𝐼 → (𝑠 ∈ ∪ ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑠 ∈ (𝐼‘𝑖)))
44	42, 43	mp1i 13	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran 𝐼) → (𝑠 ∈ ∪ ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑠 ∈ (𝐼‘𝑖)))
45	41, 44	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran 𝐼) → ∃𝑖 ∈ dom 𝐼 𝑠 ∈ (𝐼‘𝑖))
46		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ dom 𝐼 → 𝑖 ∈ dom 𝐼)
47		ovex 7401	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V
48	47, 11	dmmpti 6644	. . . . . . . . . . . . . . . . . 18 ⊢ dom 𝐼 = (0..^𝑀)
49	46, 48	eleqtrdi 2847	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ dom 𝐼 → 𝑖 ∈ (0..^𝑀))
50	11	fvmpt2 6961	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
51	49, 47, 50	sylancl 587	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ dom 𝐼 → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
52	51	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
53		ioossicc 13361	. . . . . . . . . . . . . . . 16 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
54		fourierdlem70.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ ℝ)
55	54	rexrd 11194	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
56	55	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → 𝐴 ∈ ℝ*)
57		fourierdlem70.2	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 ∈ ℝ)
58	57	rexrd 11194	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
59	58	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → 𝐵 ∈ ℝ*)
60	34	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
61	49	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → 𝑖 ∈ (0..^𝑀))
62	56, 59, 60, 61	fourierdlem8 46473	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
63	53, 62	sstrid 3947	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
64	52, 63	eqsstrd 3970	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → (𝐼‘𝑖) ⊆ (𝐴[,]𝐵))
65	64	3adant3 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑠 ∈ (𝐼‘𝑖)) → (𝐼‘𝑖) ⊆ (𝐴[,]𝐵))
66		simp3 1139	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑠 ∈ (𝐼‘𝑖)) → 𝑠 ∈ (𝐼‘𝑖))
67	65, 66	sseldd 3936	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑠 ∈ (𝐼‘𝑖)) → 𝑠 ∈ (𝐴[,]𝐵))
68	67	3exp 1120	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑖 ∈ dom 𝐼 → (𝑠 ∈ (𝐼‘𝑖) → 𝑠 ∈ (𝐴[,]𝐵))))
69	68	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran 𝐼) → (𝑖 ∈ dom 𝐼 → (𝑠 ∈ (𝐼‘𝑖) → 𝑠 ∈ (𝐴[,]𝐵))))
70	69	rexlimdv 3137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran 𝐼) → (∃𝑖 ∈ dom 𝐼 𝑠 ∈ (𝐼‘𝑖) → 𝑠 ∈ (𝐴[,]𝐵)))
71	45, 70	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran 𝐼) → 𝑠 ∈ (𝐴[,]𝐵))
72	38, 40, 71	syl2anc 585	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
73	37, 72	pm2.61dan 813	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼)) → 𝑠 ∈ (𝐴[,]𝐵))
74	19, 73	syldan 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → 𝑠 ∈ (𝐴[,]𝐵))
75		fourierdlem70.f	. . . . . 6 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
76	75	ffvelcdmda 7038	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑠) ∈ ℝ)
77	74, 76	syldan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → (𝐹‘𝑠) ∈ ℝ)
78	77	recnd 11172	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → (𝐹‘𝑠) ∈ ℂ)
79	78	abscld 15374	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → (abs‘(𝐹‘𝑠)) ∈ ℝ)
80		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → 𝑤 = ran 𝑄)
81	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
82		fzfid 13908	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → (0...𝑀) ∈ Fin)
83		rnffi 45534	. . . . . . 7 ⊢ ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
84	81, 82, 83	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → ran 𝑄 ∈ Fin)
85	80, 84	eqeltrd 2837	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → 𝑤 ∈ Fin)
86	85	adantlr 716	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ 𝑤 = ran 𝑄) → 𝑤 ∈ Fin)
87	75	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
88		simpll 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → 𝜑)
89		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑤 = ran 𝑄 ∧ 𝑠 ∈ 𝑤) → 𝑠 ∈ 𝑤)
90		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑤 = ran 𝑄 ∧ 𝑠 ∈ 𝑤) → 𝑤 = ran 𝑄)
91	89, 90	eleqtrd 2839	. . . . . . . . . . 11 ⊢ ((𝑤 = ran 𝑄 ∧ 𝑠 ∈ 𝑤) → 𝑠 ∈ ran 𝑄)
92	91	adantll 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → 𝑠 ∈ ran 𝑄)
93	88, 92, 36	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → 𝑠 ∈ (𝐴[,]𝐵))
94	87, 93	ffvelcdmd 7039	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → (𝐹‘𝑠) ∈ ℝ)
95	94	recnd 11172	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → (𝐹‘𝑠) ∈ ℂ)
96	95	abscld 15374	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → (abs‘(𝐹‘𝑠)) ∈ ℝ)
97	96	ralrimiva 3130	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ∈ ℝ)
98	97	adantlr 716	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ 𝑤 = ran 𝑄) → ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ∈ ℝ)
99		fimaxre3 12100	. . . 4 ⊢ ((𝑤 ∈ Fin ∧ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ≤ 𝑧)
100	86, 98, 99	syl2anc 585	. . 3 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ 𝑤 = ran 𝑄) → ∃𝑧 ∈ ℝ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ≤ 𝑧)
101		simpll 767	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → 𝜑)
102		neqne 2941	. . . . . 6 ⊢ (¬ 𝑤 = ran 𝑄 → 𝑤 ≠ ran 𝑄)
103		elprn1 4610	. . . . . 6 ⊢ ((𝑤 ∈ {ran 𝑄, ∪ ran 𝐼} ∧ 𝑤 ≠ ran 𝑄) → 𝑤 = ∪ ran 𝐼)
104	102, 103	sylan2 594	. . . . 5 ⊢ ((𝑤 ∈ {ran 𝑄, ∪ ran 𝐼} ∧ ¬ 𝑤 = ran 𝑄) → 𝑤 = ∪ ran 𝐼)
105	104	adantll 715	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → 𝑤 = ∪ ran 𝐼)
106	10, 12	mp1i 13	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → ran 𝐼 ∈ Fin)
107		ax-resscn 11095	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
108	107	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℂ)
109	75, 108	fssd 6687	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
110	109	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑠 ∈ ∪ ran 𝐼) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
111	71	adantlr 716	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑠 ∈ ∪ ran 𝐼) → 𝑠 ∈ (𝐴[,]𝐵))
112	110, 111	ffvelcdmd 7039	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑠 ∈ ∪ ran 𝐼) → (𝐹‘𝑠) ∈ ℂ)
113	112	abscld 15374	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑠 ∈ ∪ ran 𝐼) → (abs‘(𝐹‘𝑠)) ∈ ℝ)
114	47, 11	fnmpti 6643	. . . . . . . . . 10 ⊢ 𝐼 Fn (0..^𝑀)
115		fvelrnb 6902	. . . . . . . . . 10 ⊢ (𝐼 Fn (0..^𝑀) → (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡))
116	114, 115	ax-mp 5	. . . . . . . . 9 ⊢ (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡)
117	116	biimpi 216	. . . . . . . 8 ⊢ (𝑡 ∈ ran 𝐼 → ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡)
118	117	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡)
119	4	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
120		elfzofz 13603	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
121	120	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
122	119, 121	ffvelcdmd 7039	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
123		fzofzp1 13692	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
124	123	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
125	119, 124	ffvelcdmd 7039	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
126		fourierdlem70.fcn	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
127		fourierdlem70.l	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1))))
128		fourierdlem70.r	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖)))
129	122, 125, 126, 127, 128	cncfioobd 46255	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏)
130		fvres 6861	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠) = (𝐹‘𝑠))
131	130	fveq2d 6846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) = (abs‘(𝐹‘𝑠)))
132	131	breq1d 5110	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑠)) ≤ 𝑏))
133	132	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑠)) ≤ 𝑏))
134	133	ralbidva 3159	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏 ↔ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑠)) ≤ 𝑏))
135	134	rexbidv 3162	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑠)) ≤ 𝑏))
136	129, 135	mpbid 232	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑠)) ≤ 𝑏)
137	136	3adant3 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑠)) ≤ 𝑏)
138	47, 50	mpan2 692	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0..^𝑀) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
139	138	eqcomd 2743	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼‘𝑖))
140	139	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼‘𝑖))
141		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → (𝐼‘𝑖) = 𝑡)
142	140, 141	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
143	142	raleqdv 3298	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → (∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑠)) ≤ 𝑏 ↔ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏))
144	143	rexbidv 3162	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑠)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏))
145	144	3adant1 1131	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → (∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑠)) ≤ 𝑏 ↔ ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏))
146	137, 145	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏)
147	146	3exp 1120	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) → ((𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏)))
148	147	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → (𝑖 ∈ (0..^𝑀) → ((𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏)))
149	148	rexlimdv 3137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → (∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏))
150	118, 149	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏)
151	150	adantlr 716	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏)
152		eqimss 3994	. . . . . 6 ⊢ (𝑤 = ∪ ran 𝐼 → 𝑤 ⊆ ∪ ran 𝐼)
153	152	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → 𝑤 ⊆ ∪ ran 𝐼)
154	106, 113, 151, 153	ssfiunibd 45671	. . . 4 ⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → ∃𝑧 ∈ ℝ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ≤ 𝑧)
155	101, 105, 154	syl2anc 585	. . 3 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → ∃𝑧 ∈ ℝ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ≤ 𝑧)
156	100, 155	pm2.61dan 813	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) → ∃𝑧 ∈ ℝ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ≤ 𝑧)
157	21	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑀 ∈ ℕ)
158	4	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
159		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (𝐴[,]𝐵))
160	25	eqcomd 2743	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 = (𝑄‘0))
161	26	eqcomd 2743	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 = (𝑄‘𝑀))
162	160, 161	oveq12d 7386	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄‘𝑀)))
163	162	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄‘𝑀)))
164	159, 163	eleqtrd 2839	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
165	164	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑡 ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
166		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ¬ 𝑡 ∈ ran 𝑄)
167		fveq2 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (𝑄‘𝑘) = (𝑄‘𝑗))
168	167	breq1d 5110	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → ((𝑄‘𝑘) < 𝑡 ↔ (𝑄‘𝑗) < 𝑡))
169	168	cbvrabv 3411	. . . . . . . . . . . . 13 ⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝑡} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < 𝑡}
170	169	supeq1i 9362	. . . . . . . . . . . 12 ⊢ sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝑡}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < 𝑡}, ℝ, < )
171	157, 158, 165, 166, 170	fourierdlem25 46490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
172	138	eleq2d 2823	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑡 ∈ (𝐼‘𝑖) ↔ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
173	172	rexbiia 3083	. . . . . . . . . . 11 ⊢ (∃𝑖 ∈ (0..^𝑀)𝑡 ∈ (𝐼‘𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
174	171, 173	sylibr 234	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ (𝐼‘𝑖))
175	48	eqcomi 2746	. . . . . . . . . . 11 ⊢ (0..^𝑀) = dom 𝐼
176	175	rexeqi 3297	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ (0..^𝑀)𝑡 ∈ (𝐼‘𝑖) ↔ ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼‘𝑖))
177	174, 176	sylib 218	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼‘𝑖))
178		elunirn 7207	. . . . . . . . . 10 ⊢ (Fun 𝐼 → (𝑡 ∈ ∪ ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼‘𝑖)))
179	42, 178	mp1i 13	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → (𝑡 ∈ ∪ ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼‘𝑖)))
180	177, 179	mpbird 257	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑡 ∈ ∪ ran 𝐼)
181	180	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (¬ 𝑡 ∈ ran 𝑄 → 𝑡 ∈ ∪ ran 𝐼))
182	181	orrd 864	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ ran 𝑄 ∨ 𝑡 ∈ ∪ ran 𝐼))
183		elun 4107	. . . . . 6 ⊢ (𝑡 ∈ (ran 𝑄 ∪ ∪ ran 𝐼) ↔ (𝑡 ∈ ran 𝑄 ∨ 𝑡 ∈ ∪ ran 𝐼))
184	182, 183	sylibr 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (ran 𝑄 ∪ ∪ ran 𝐼))
185	184	ralrimiva 3130	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ (𝐴[,]𝐵)𝑡 ∈ (ran 𝑄 ∪ ∪ ran 𝐼))
186		dfss3 3924	. . . 4 ⊢ ((𝐴[,]𝐵) ⊆ (ran 𝑄 ∪ ∪ ran 𝐼) ↔ ∀𝑡 ∈ (𝐴[,]𝐵)𝑡 ∈ (ran 𝑄 ∪ ∪ ran 𝐼))
187	185, 186	sylibr 234	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (ran 𝑄 ∪ ∪ ran 𝐼))
188	187, 17	sseqtrrd 3973	. 2 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ {ran 𝑄, ∪ ran 𝐼})
189	2, 79, 156, 188	ssfiunibd 45671	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑠 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑠)) ≤ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3401  Vcvv 3442   ∪ cun 3901   ⊆ wss 3903  {cpr 4584  ∪ cuni 4865   class class class wbr 5100   ↦ cmpt 5181  dom cdm 5632  ran crn 5633   ↾ cres 5634  Fun wfun 6494   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ↑_m cmap 8775  Fincfn 8895  supcsup 9355  ℂcc 11036  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041  ℝ^*cxr 11177   < clt 11178   ≤ cle 11179  ℕcn 12157  (,)cioo 13273  [,]cicc 13276  ...cfz 13435  ..^cfzo 13582  abscabs 15169  –cn→ccncf 24837   lim_ℂ climc 25831

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-of 7632  df-om 7819  df-1st 7943  df-2nd 7944  df-supp 8113  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-map 8777  df-pm 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9277  df-fi 9326  df-sup 9357  df-inf 9358  df-oi 9427  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-q 12874  df-rp 12918  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-ioo 13277  df-ioc 13278  df-ico 13279  df-icc 13280  df-fz 13436  df-fzo 13583  df-seq 13937  df-exp 13997  df-hash 14266  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-mulr 17203  df-starv 17204  df-sca 17205  df-vsca 17206  df-ip 17207  df-tset 17208  df-ple 17209  df-ds 17211  df-unif 17212  df-hom 17213  df-cco 17214  df-rest 17354  df-topn 17355  df-0g 17373  df-gsum 17374  df-topgen 17375  df-pt 17376  df-prds 17379  df-xrs 17435  df-qtop 17440  df-imas 17441  df-xps 17443  df-mre 17517  df-mrc 17518  df-acs 17520  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-submnd 18721  df-mulg 19010  df-cntz 19258  df-cmn 19723  df-psmet 21313  df-xmet 21314  df-met 21315  df-bl 21316  df-mopn 21317  df-cnfld 21322  df-top 22850  df-topon 22867  df-topsp 22889  df-bases 22902  df-cld 22975  df-ntr 22976  df-cls 22977  df-cn 23183  df-cnp 23184  df-cmp 23343  df-tx 23518  df-hmeo 23711  df-xms 24276  df-ms 24277  df-tms 24278  df-cncf 24839  df-limc 25835

This theorem is referenced by:  fourierdlem103  46567  fourierdlem104  46568