Theorem fourierdlem70 43392

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 8946	. . 3 ⊢ {ran 𝑄, ∪ ran 𝐼} ∈ Fin
2	1	a1i 11	. 2 ⊢ (𝜑 → {ran 𝑄, ∪ ran 𝐼} ∈ Fin)
3		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼})
4		fourierdlem70.q	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
5		ovex 7246	. . . . . . . . . . 11 ⊢ (0...𝑀) ∈ V
6		fex 7042	. . . . . . . . . . 11 ⊢ ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ V) → 𝑄 ∈ V)
7	4, 5, 6	sylancl 589	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ V)
8		rnexg 7682	. . . . . . . . . 10 ⊢ (𝑄 ∈ V → ran 𝑄 ∈ V)
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑄 ∈ V)
10		fzofi 13547	. . . . . . . . . . . 12 ⊢ (0..^𝑀) ∈ Fin
11		fourierdlem70.i	. . . . . . . . . . . . 13 ⊢ 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
12	11	rnmptfi 42380	. . . . . . . . . . . 12 ⊢ ((0..^𝑀) ∈ Fin → ran 𝐼 ∈ Fin)
13	10, 12	ax-mp 5	. . . . . . . . . . 11 ⊢ ran 𝐼 ∈ Fin
14	13	elexi 3427	. . . . . . . . . 10 ⊢ ran 𝐼 ∈ V
15	14	uniex 7529	. . . . . . . . 9 ⊢ ∪ ran 𝐼 ∈ V
16		uniprg 4836	. . . . . . . . 9 ⊢ ((ran 𝑄 ∈ V ∧ ∪ ran 𝐼 ∈ V) → ∪ {ran 𝑄, ∪ ran 𝐼} = (ran 𝑄 ∪ ∪ ran 𝐼))
17	9, 15, 16	sylancl 589	. . . . . . . 8 ⊢ (𝜑 → ∪ {ran 𝑄, ∪ ran 𝐼} = (ran 𝑄 ∪ ∪ ran 𝐼))
18	17	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → ∪ {ran 𝑄, ∪ ran 𝐼} = (ran 𝑄 ∪ ∪ ran 𝐼))
19	3, 18	eleqtrd 2840	. . . . . 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 10820	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
23	22, 5	elmap 8552	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ)
24	4, 23	sylibr 237	. . . . . . . . . . . . 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 515	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
28		fourierdlem70.qlt	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
29	28	ralrimiva 3105	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
30	24, 27, 29	jca32 519	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
31	20	fourierdlem2 43325	. . . . . . . . . . . . 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 260	. . . . . . . . . . 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 43338	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
35	34	frnd 6553	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
36	35	sselda 3901	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
37	36	adantlr 715	. . . . . . 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 4064	. . . . . . . . 9 ⊢ ((𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ ∪ ran 𝐼)
40	39	adantll 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ ∪ ran 𝐼)
41		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran 𝐼) → 𝑠 ∈ ∪ ran 𝐼)
42	11	funmpt2 6419	. . . . . . . . . . 11 ⊢ Fun 𝐼
43		elunirn 7064	. . . . . . . . . . 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 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran 𝐼) → ∃𝑖 ∈ dom 𝐼 𝑠 ∈ (𝐼‘𝑖))
46		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ dom 𝐼 → 𝑖 ∈ dom 𝐼)
47		ovex 7246	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V
48	47, 11	dmmpti 6522	. . . . . . . . . . . . . . . . . 18 ⊢ dom 𝐼 = (0..^𝑀)
49	46, 48	eleqtrdi 2848	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ dom 𝐼 → 𝑖 ∈ (0..^𝑀))
50	11	fvmpt2 6829	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
51	49, 47, 50	sylancl 589	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ dom 𝐼 → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
52	51	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
53		ioossicc 13021	. . . . . . . . . . . . . . . 16 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
54		fourierdlem70.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ ℝ)
55	54	rexrd 10883	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
56	55	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → 𝐴 ∈ ℝ*)
57		fourierdlem70.2	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 ∈ ℝ)
58	57	rexrd 10883	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
59	58	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → 𝐵 ∈ ℝ*)
60	34	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
61	49	adantl 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → 𝑖 ∈ (0..^𝑀))
62	56, 59, 60, 61	fourierdlem8 43331	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
63	53, 62	sstrid 3912	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
64	52, 63	eqsstrd 3939	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → (𝐼‘𝑖) ⊆ (𝐴[,]𝐵))
65	64	3adant3 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑠 ∈ (𝐼‘𝑖)) → (𝐼‘𝑖) ⊆ (𝐴[,]𝐵))
66		simp3 1140	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑠 ∈ (𝐼‘𝑖)) → 𝑠 ∈ (𝐼‘𝑖))
67	65, 66	sseldd 3902	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑠 ∈ (𝐼‘𝑖)) → 𝑠 ∈ (𝐴[,]𝐵))
68	67	3exp 1121	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑖 ∈ dom 𝐼 → (𝑠 ∈ (𝐼‘𝑖) → 𝑠 ∈ (𝐴[,]𝐵))))
69	68	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran 𝐼) → (𝑖 ∈ dom 𝐼 → (𝑠 ∈ (𝐼‘𝑖) → 𝑠 ∈ (𝐴[,]𝐵))))
70	69	rexlimdv 3202	. . . . . . . . 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 587	. . . . . . 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 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → 𝑠 ∈ (𝐴[,]𝐵))
75		fourierdlem70.f	. . . . . 6 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
76	75	ffvelrnda 6904	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑠) ∈ ℝ)
77	74, 76	syldan 594	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → (𝐹‘𝑠) ∈ ℝ)
78	77	recnd 10861	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → (𝐹‘𝑠) ∈ ℂ)
79	78	abscld 15000	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → (abs‘(𝐹‘𝑠)) ∈ ℝ)
80		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → 𝑤 = ran 𝑄)
81	4	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
82		fzfid 13546	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → (0...𝑀) ∈ Fin)
83		rnffi 42384	. . . . . . 7 ⊢ ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
84	81, 82, 83	syl2anc 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → ran 𝑄 ∈ Fin)
85	80, 84	eqeltrd 2838	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → 𝑤 ∈ Fin)
86	85	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ 𝑤 = ran 𝑄) → 𝑤 ∈ Fin)
87	75	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
88		simpll 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → 𝜑)
89		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝑤 = ran 𝑄 ∧ 𝑠 ∈ 𝑤) → 𝑠 ∈ 𝑤)
90		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝑤 = ran 𝑄 ∧ 𝑠 ∈ 𝑤) → 𝑤 = ran 𝑄)
91	89, 90	eleqtrd 2840	. . . . . . . . . . 11 ⊢ ((𝑤 = ran 𝑄 ∧ 𝑠 ∈ 𝑤) → 𝑠 ∈ ran 𝑄)
92	91	adantll 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → 𝑠 ∈ ran 𝑄)
93	88, 92, 36	syl2anc 587	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → 𝑠 ∈ (𝐴[,]𝐵))
94	87, 93	ffvelrnd 6905	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → (𝐹‘𝑠) ∈ ℝ)
95	94	recnd 10861	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → (𝐹‘𝑠) ∈ ℂ)
96	95	abscld 15000	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → (abs‘(𝐹‘𝑠)) ∈ ℝ)
97	96	ralrimiva 3105	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ∈ ℝ)
98	97	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ 𝑤 = ran 𝑄) → ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ∈ ℝ)
99		fimaxre3 11778	. . . 4 ⊢ ((𝑤 ∈ Fin ∧ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ≤ 𝑧)
100	86, 98, 99	syl2anc 587	. . 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 2948	. . . . . 6 ⊢ (¬ 𝑤 = ran 𝑄 → 𝑤 ≠ ran 𝑄)
103		elprn1 42849	. . . . . 6 ⊢ ((𝑤 ∈ {ran 𝑄, ∪ ran 𝐼} ∧ 𝑤 ≠ ran 𝑄) → 𝑤 = ∪ ran 𝐼)
104	102, 103	sylan2 596	. . . . 5 ⊢ ((𝑤 ∈ {ran 𝑄, ∪ ran 𝐼} ∧ ¬ 𝑤 = ran 𝑄) → 𝑤 = ∪ ran 𝐼)
105	104	adantll 714	. . . 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 10786	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
108	107	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℂ)
109	75, 108	fssd 6563	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
110	109	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑠 ∈ ∪ ran 𝐼) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
111	71	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑠 ∈ ∪ ran 𝐼) → 𝑠 ∈ (𝐴[,]𝐵))
112	110, 111	ffvelrnd 6905	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑠 ∈ ∪ ran 𝐼) → (𝐹‘𝑠) ∈ ℂ)
113	112	abscld 15000	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑠 ∈ ∪ ran 𝐼) → (abs‘(𝐹‘𝑠)) ∈ ℝ)
114	47, 11	fnmpti 6521	. . . . . . . . . 10 ⊢ 𝐼 Fn (0..^𝑀)
115		fvelrnb 6773	. . . . . . . . . 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 219	. . . . . . . 8 ⊢ (𝑡 ∈ ran 𝐼 → ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡)
118	117	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡)
119	4	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
120		elfzofz 13258	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
121	120	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
122	119, 121	ffvelrnd 6905	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
123		fzofzp1 13339	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
124	123	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
125	119, 124	ffvelrnd 6905	. . . . . . . . . . . . . 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 43113	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏)
130		fvres 6736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠) = (𝐹‘𝑠))
131	130	fveq2d 6721	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) = (abs‘(𝐹‘𝑠)))
132	131	breq1d 5063	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑠)) ≤ 𝑏))
133	132	adantl 485	. . . . . . . . . . . . . . 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 3117	. . . . . . . . . . . . . 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 3216	. . . . . . . . . . . . 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 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑠)) ≤ 𝑏)
137	136	3adant3 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑠)) ≤ 𝑏)
138	47, 50	mpan2 691	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼‘𝑖))
141		simpr 488	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → (𝐼‘𝑖) = 𝑡)
142	140, 141	eqtrd 2777	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
143	142	raleqdv 3325	. . . . . . . . . . . . 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 3216	. . . . . . . . . . . 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 1132	. . . . . . . . . . 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 235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏)
147	146	3exp 1121	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) → ((𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏)))
148	147	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → (𝑖 ∈ (0..^𝑀) → ((𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏)))
149	148	rexlimdv 3202	. . . . . . 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 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏)
152		eqimss 3957	. . . . . 6 ⊢ (𝑤 = ∪ ran 𝐼 → 𝑤 ⊆ ∪ ran 𝐼)
153	152	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → 𝑤 ⊆ ∪ ran 𝐼)
154	106, 113, 151, 153	ssfiunibd 42521	. . . 4 ⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → ∃𝑧 ∈ ℝ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ≤ 𝑧)
155	101, 105, 154	syl2anc 587	. . 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 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑀 ∈ ℕ)
158	4	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
159		simpr 488	. . . . . . . . . . . . . 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 7231	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄‘𝑀)))
163	162	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄‘𝑀)))
164	159, 163	eleqtrd 2840	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
165	164	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑡 ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
166		simpr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ¬ 𝑡 ∈ ran 𝑄)
167		fveq2 6717	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (𝑄‘𝑘) = (𝑄‘𝑗))
168	167	breq1d 5063	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → ((𝑄‘𝑘) < 𝑡 ↔ (𝑄‘𝑗) < 𝑡))
169	168	cbvrabv 3402	. . . . . . . . . . . . 13 ⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝑡} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < 𝑡}
170	169	supeq1i 9063	. . . . . . . . . . . 12 ⊢ sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝑡}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < 𝑡}, ℝ, < )
171	157, 158, 165, 166, 170	fourierdlem25 43348	. . . . . . . . . . 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 3169	. . . . . . . . . . 11 ⊢ (∃𝑖 ∈ (0..^𝑀)𝑡 ∈ (𝐼‘𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
174	171, 173	sylibr 237	. . . . . . . . . 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 3324	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ (0..^𝑀)𝑡 ∈ (𝐼‘𝑖) ↔ ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼‘𝑖))
177	174, 176	sylib 221	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼‘𝑖))
178		elunirn 7064	. . . . . . . . . 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 260	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑡 ∈ ∪ ran 𝐼)
181	180	ex 416	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (¬ 𝑡 ∈ ran 𝑄 → 𝑡 ∈ ∪ ran 𝐼))
182	181	orrd 863	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ ran 𝑄 ∨ 𝑡 ∈ ∪ ran 𝐼))
183		elun 4063	. . . . . 6 ⊢ (𝑡 ∈ (ran 𝑄 ∪ ∪ ran 𝐼) ↔ (𝑡 ∈ ran 𝑄 ∨ 𝑡 ∈ ∪ ran 𝐼))
184	182, 183	sylibr 237	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (ran 𝑄 ∪ ∪ ran 𝐼))
185	184	ralrimiva 3105	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ (𝐴[,]𝐵)𝑡 ∈ (ran 𝑄 ∪ ∪ ran 𝐼))
186		dfss3 3888	. . . 4 ⊢ ((𝐴[,]𝐵) ⊆ (ran 𝑄 ∪ ∪ ran 𝐼) ↔ ∀𝑡 ∈ (𝐴[,]𝐵)𝑡 ∈ (ran 𝑄 ∪ ∪ ran 𝐼))
187	185, 186	sylibr 237	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (ran 𝑄 ∪ ∪ ran 𝐼))
188	187, 17	sseqtrrd 3942	. 2 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ {ran 𝑄, ∪ ran 𝐼})
189	2, 79, 156, 188	ssfiunibd 42521	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑠 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑠)) ≤ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∀wral 3061  ∃wrex 3062  {crab 3065  Vcvv 3408   ∪ cun 3864   ⊆ wss 3866  {cpr 4543  ∪ cuni 4819   class class class wbr 5053   ↦ cmpt 5135  dom cdm 5551  ran crn 5552   ↾ cres 5553  Fun wfun 6374   Fn wfn 6375  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213   ↑_m cmap 8508  Fincfn 8626  supcsup 9056  ℂcc 10727  ℝcr 10728  0cc0 10729  1c1 10730   + caddc 10732  ℝ^*cxr 10866   < clt 10867   ≤ cle 10868  ℕcn 11830  (,)cioo 12935  [,]cicc 12938  ...cfz 13095  ..^cfzo 13238  abscabs 14797  –cn→ccncf 23773   lim_ℂ climc 24759

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807  ax-mulf 10809

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-supp 7904  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-er 8391  df-map 8510  df-pm 8511  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fsupp 8986  df-fi 9027  df-sup 9058  df-inf 9059  df-oi 9126  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-dec 12294  df-uz 12439  df-q 12545  df-rp 12587  df-xneg 12704  df-xadd 12705  df-xmul 12706  df-ioo 12939  df-ioc 12940  df-ico 12941  df-icc 12942  df-fz 13096  df-fzo 13239  df-seq 13575  df-exp 13636  df-hash 13897  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-starv 16817  df-sca 16818  df-vsca 16819  df-ip 16820  df-tset 16821  df-ple 16822  df-ds 16824  df-unif 16825  df-hom 16826  df-cco 16827  df-rest 16927  df-topn 16928  df-0g 16946  df-gsum 16947  df-topgen 16948  df-pt 16949  df-prds 16952  df-xrs 17007  df-qtop 17012  df-imas 17013  df-xps 17015  df-mre 17089  df-mrc 17090  df-acs 17092  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-submnd 18219  df-mulg 18489  df-cntz 18711  df-cmn 19172  df-psmet 20355  df-xmet 20356  df-met 20357  df-bl 20358  df-mopn 20359  df-cnfld 20364  df-top 21791  df-topon 21808  df-topsp 21830  df-bases 21843  df-cld 21916  df-ntr 21917  df-cls 21918  df-cn 22124  df-cnp 22125  df-cmp 22284  df-tx 22459  df-hmeo 22652  df-xms 23218  df-ms 23219  df-tms 23220  df-cncf 23775  df-limc 24763

This theorem is referenced by:  fourierdlem103  43425  fourierdlem104  43426