Theorem fourierdlem70 46948

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 9290	. . 3 ⊢ {ran 𝑄, ∪ ran 𝐼} ∈ Fin
2	1	a1i 11	. 2 ⊢ (𝜑 → {ran 𝑄, ∪ ran 𝐼} ∈ Fin)
3		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼})
4		fourierdlem70.q	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
5		ovex 7452	. . . . . . . . . . 11 ⊢ (0...𝑀) ∈ V
6		fex 7231	. . . . . . . . . . 11 ⊢ ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ V) → 𝑄 ∈ V)
7	4, 5, 6	sylancl 598	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ V)
8		rnexg 7905	. . . . . . . . . 10 ⊢ (𝑄 ∈ V → ran 𝑄 ∈ V)
9	7, 8	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑄 ∈ V)
10		fzofi 14028	. . . . . . . . . . . 12 ⊢ (0..^𝑀) ∈ Fin
11		fourierdlem70.i	. . . . . . . . . . . . 13 ⊢ 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
12	11	rnmptfi 45947	. . . . . . . . . . . 12 ⊢ ((0..^𝑀) ∈ Fin → ran 𝐼 ∈ Fin)
13	10, 12	ax-mp 5	. . . . . . . . . . 11 ⊢ ran 𝐼 ∈ Fin
14	13	elexi 3479	. . . . . . . . . 10 ⊢ ran 𝐼 ∈ V
15	14	uniex 7749	. . . . . . . . 9 ⊢ ∪ ran 𝐼 ∈ V
16		uniprg 4890	. . . . . . . . 9 ⊢ ((ran 𝑄 ∈ V ∧ ∪ ran 𝐼 ∈ V) → ∪ {ran 𝑄, ∪ ran 𝐼} = (ran 𝑄 ∪ ∪ ran 𝐼))
17	9, 15, 16	sylancl 598	. . . . . . . 8 ⊢ (𝜑 → ∪ {ran 𝑄, ∪ ran 𝐼} = (ran 𝑄 ∪ ∪ ran 𝐼))
18	17	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → ∪ {ran 𝑄, ∪ ran 𝐼} = (ran 𝑄 ∪ ∪ ran 𝐼))
19	3, 18	eleqtrd 2867	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → 𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼))
20		eqid 2765	. . . . . . . . . . 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 11206	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
23	22, 5	elmap 8875	. . . . . . . . . . . . . 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 521	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
28		fourierdlem70.qlt	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
29	28	ralrimiva 3159	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
30	24, 27, 29	jca32 525	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
31	20	fourierdlem2 46881	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 46894	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
35	34	frnd 6718	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
36	35	sselda 3938	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
37	36	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼)) ∧ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
38		simpll 779	. . . . . . . 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 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ ∪ ran 𝐼)
41		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran 𝐼) → 𝑠 ∈ ∪ ran 𝐼)
42	11	funmpt2 6579	. . . . . . . . . . 11 ⊢ Fun 𝐼
43		elunirn 7254	. . . . . . . . . . 11 ⊢ (Fun 𝐼 → (𝑠 ∈ ∪ ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑠 ∈ (𝐼‘𝑖)))
44	42, 43	mp1i 14	. . . . . . . . . 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 23	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ dom 𝐼 → 𝑖 ∈ dom 𝐼)
47		ovex 7452	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V
48	47, 11	dmmpti 6683	. . . . . . . . . . . . . . . . . 18 ⊢ dom 𝐼 = (0..^𝑀)
49	46, 48	eleqtrdi 2875	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ dom 𝐼 → 𝑖 ∈ (0..^𝑀))
50	11	fvmpt2 7005	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
51	49, 47, 50	sylancl 598	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ dom 𝐼 → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
52	51	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
53		ioossicc 13476	. . . . . . . . . . . . . . . 16 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
54		fourierdlem70.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ ℝ)
55	54	rexrd 11274	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
56	55	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → 𝐴 ∈ ℝ*)
57		fourierdlem70.2	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 ∈ ℝ)
58	57	rexrd 11274	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
59	58	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → 𝐵 ∈ ℝ*)
60	34	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
61	49	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → 𝑖 ∈ (0..^𝑀))
62	56, 59, 60, 61	fourierdlem8 46887	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
63	53, 62	sstrid 3949	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
64	52, 63	eqsstrd 3972	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → (𝐼‘𝑖) ⊆ (𝐴[,]𝐵))
65	64	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑠 ∈ (𝐼‘𝑖)) → (𝐼‘𝑖) ⊆ (𝐴[,]𝐵))
66		simp3 1156	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑠 ∈ (𝐼‘𝑖)) → 𝑠 ∈ (𝐼‘𝑖))
67	65, 66	sseldd 3939	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑠 ∈ (𝐼‘𝑖)) → 𝑠 ∈ (𝐴[,]𝐵))
68	67	3exp 1137	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑖 ∈ dom 𝐼 → (𝑠 ∈ (𝐼‘𝑖) → 𝑠 ∈ (𝐴[,]𝐵))))
69	68	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran 𝐼) → (𝑖 ∈ dom 𝐼 → (𝑠 ∈ (𝐼‘𝑖) → 𝑠 ∈ (𝐴[,]𝐵))))
70	69	rexlimdv 3166	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran 𝐼) → (∃𝑖 ∈ dom 𝐼 𝑠 ∈ (𝐼‘𝑖) → 𝑠 ∈ (𝐴[,]𝐵)))
71	45, 70	mpd 16	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran 𝐼) → 𝑠 ∈ (𝐴[,]𝐵))
72	38, 40, 71	syl2anc 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
73	37, 72	pm2.61dan 825	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼)) → 𝑠 ∈ (𝐴[,]𝐵))
74	19, 73	syldan 603	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → 𝑠 ∈ (𝐴[,]𝐵))
75		fourierdlem70.f	. . . . . 6 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
76	75	ffvelcdmda 7083	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑠) ∈ ℝ)
77	74, 76	syldan 603	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → (𝐹‘𝑠) ∈ ℝ)
78	77	recnd 11252	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → (𝐹‘𝑠) ∈ ℂ)
79	78	abscld 15514	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → (abs‘(𝐹‘𝑠)) ∈ ℝ)
80		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → 𝑤 = ran 𝑄)
81	4	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
82		fzfid 14027	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → (0...𝑀) ∈ Fin)
83		rnffi 45951	. . . . . . 7 ⊢ ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
84	81, 82, 83	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → ran 𝑄 ∈ Fin)
85	80, 84	eqeltrd 2865	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → 𝑤 ∈ Fin)
86	85	adantlr 728	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ 𝑤 = ran 𝑄) → 𝑤 ∈ Fin)
87	75	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
88		simpll 779	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → 𝜑)
89		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑤 = ran 𝑄 ∧ 𝑠 ∈ 𝑤) → 𝑠 ∈ 𝑤)
90		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑤 = ran 𝑄 ∧ 𝑠 ∈ 𝑤) → 𝑤 = ran 𝑄)
91	89, 90	eleqtrd 2867	. . . . . . . . . . 11 ⊢ ((𝑤 = ran 𝑄 ∧ 𝑠 ∈ 𝑤) → 𝑠 ∈ ran 𝑄)
92	91	adantll 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → 𝑠 ∈ ran 𝑄)
93	88, 92, 36	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → 𝑠 ∈ (𝐴[,]𝐵))
94	87, 93	ffvelcdmd 7084	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → (𝐹‘𝑠) ∈ ℝ)
95	94	recnd 11252	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → (𝐹‘𝑠) ∈ ℂ)
96	95	abscld 15514	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → (abs‘(𝐹‘𝑠)) ∈ ℝ)
97	96	ralrimiva 3159	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ∈ ℝ)
98	97	adantlr 728	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ 𝑤 = ran 𝑄) → ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ∈ ℝ)
99		fimaxre3 12176	. . . 4 ⊢ ((𝑤 ∈ Fin ∧ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ≤ 𝑧)
100	86, 98, 99	syl2anc 596	. . 3 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ 𝑤 = ran 𝑄) → ∃𝑧 ∈ ℝ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ≤ 𝑧)
101		simpll 779	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → 𝜑)
102		neqne 2968	. . . . . 6 ⊢ (¬ 𝑤 = ran 𝑄 → 𝑤 ≠ ran 𝑄)
103		elprn1 4619	. . . . . 6 ⊢ ((𝑤 ∈ {ran 𝑄, ∪ ran 𝐼} ∧ 𝑤 ≠ ran 𝑄) → 𝑤 = ∪ ran 𝐼)
104	102, 103	sylan2 605	. . . . 5 ⊢ ((𝑤 ∈ {ran 𝑄, ∪ ran 𝐼} ∧ ¬ 𝑤 = ran 𝑄) → 𝑤 = ∪ ran 𝐼)
105	104	adantll 727	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → 𝑤 = ∪ ran 𝐼)
106	10, 12	mp1i 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → ran 𝐼 ∈ Fin)
107		ax-resscn 11172	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
108	107	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℂ)
109	75, 108	fssd 6727	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
110	109	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑠 ∈ ∪ ran 𝐼) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
111	71	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑠 ∈ ∪ ran 𝐼) → 𝑠 ∈ (𝐴[,]𝐵))
112	110, 111	ffvelcdmd 7084	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑠 ∈ ∪ ran 𝐼) → (𝐹‘𝑠) ∈ ℂ)
113	112	abscld 15514	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑠 ∈ ∪ ran 𝐼) → (abs‘(𝐹‘𝑠)) ∈ ℝ)
114	47, 11	fnmpti 6682	. . . . . . . . 9 ⊢ 𝐼 Fn (0..^𝑀)
115		fvelrnb 6945	. . . . . . . . 9 ⊢ (𝐼 Fn (0..^𝑀) → (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡))
116	114, 115	ax-mp 5	. . . . . . . 8 ⊢ (𝑡 ∈ ran 𝐼 ↔ ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡)
117	116	bilani 510	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → ∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡)
118	4	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
119		elfzofz 13721	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
120	119	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
121	118, 120	ffvelcdmd 7084	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
122		fzofzp1 13810	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
123	122	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
124	118, 123	ffvelcdmd 7084	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) ∈ ℝ)
125		fourierdlem70.fcn	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
126		fourierdlem70.l	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘(𝑖 + 1))))
127		fourierdlem70.r	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) limℂ (𝑄‘𝑖)))
128	121, 124, 125, 126, 127	cncfioobd 46669	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏)
129		fvres 6904	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠) = (𝐹‘𝑠))
130	129	fveq2d 6889	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) = (abs‘(𝐹‘𝑠)))
131	130	breq1d 5121	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑠)) ≤ 𝑏))
132	131	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))) → ((abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏 ↔ (abs‘(𝐹‘𝑠)) ≤ 𝑏))
133	132	ralbidva 3188	. . . . . . . . . . . . . 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))
134	133	rexbidv 3191	. . . . . . . . . . . . 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))
135	128, 134	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑠)) ≤ 𝑏)
136	135	3adant3 1150	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑠)) ≤ 𝑏)
137	47, 50	mpan2 704	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0..^𝑀) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
138	137	eqcomd 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼‘𝑖))
139	138	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = (𝐼‘𝑖))
140		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → (𝐼‘𝑖) = 𝑡)
141	139, 140	eqtrd 2800	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
142	141	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))
143	142	rexbidv 3191	. . . . . . . . . . . 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))
144	143	3adant1 1148	. . . . . . . . . . 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))
145	136, 144	mpbid 235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏)
146	145	3exp 1137	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) → ((𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏)))
147	146	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → (𝑖 ∈ (0..^𝑀) → ((𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏)))
148	147	rexlimdv 3166	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → (∃𝑖 ∈ (0..^𝑀)(𝐼‘𝑖) = 𝑡 → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏))
149	117, 148	mpd 16	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏)
150	149	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏)
151		eqimss 3996	. . . . . 6 ⊢ (𝑤 = ∪ ran 𝐼 → 𝑤 ⊆ ∪ ran 𝐼)
152	151	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → 𝑤 ⊆ ∪ ran 𝐼)
153	106, 113, 150, 152	ssfiunibd 46086	. . . 4 ⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → ∃𝑧 ∈ ℝ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ≤ 𝑧)
154	101, 105, 153	syl2anc 596	. . 3 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → ∃𝑧 ∈ ℝ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ≤ 𝑧)
155	100, 154	pm2.61dan 825	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) → ∃𝑧 ∈ ℝ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ≤ 𝑧)
156	21	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑀 ∈ ℕ)
157	4	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑄:(0...𝑀)⟶ℝ)
158		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (𝐴[,]𝐵))
159	25	eqcomd 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 = (𝑄‘0))
160	26	eqcomd 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 = (𝑄‘𝑀))
161	159, 160	oveq12d 7437	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄‘𝑀)))
162	161	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝐴[,]𝐵) = ((𝑄‘0)[,](𝑄‘𝑀)))
163	158, 162	eleqtrd 2867	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
164	163	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑡 ∈ ((𝑄‘0)[,](𝑄‘𝑀)))
165		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ¬ 𝑡 ∈ ran 𝑄)
166		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (𝑄‘𝑘) = (𝑄‘𝑗))
167	166	breq1d 5121	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → ((𝑄‘𝑘) < 𝑡 ↔ (𝑄‘𝑗) < 𝑡))
168	167	cbvrabv 3428	. . . . . . . . . . . . 13 ⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝑡} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < 𝑡}
169	168	supeq1i 9414	. . . . . . . . . . . 12 ⊢ sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝑡}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < 𝑡}, ℝ, < )
170	156, 157, 164, 165, 169	fourierdlem25 46904	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
171	137	eleq2d 2851	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑡 ∈ (𝐼‘𝑖) ↔ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
172	171	rexbiia 3112	. . . . . . . . . . 11 ⊢ (∃𝑖 ∈ (0..^𝑀)𝑡 ∈ (𝐼‘𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
173	170, 172	sylibr 237	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ (𝐼‘𝑖))
174	48	eqcomi 2774	. . . . . . . . . . 11 ⊢ (0..^𝑀) = dom 𝐼
175	174	rexeqi 3324	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ (0..^𝑀)𝑡 ∈ (𝐼‘𝑖) ↔ ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼‘𝑖))
176	173, 175	sylib 221	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼‘𝑖))
177		elunirn 7254	. . . . . . . . . 10 ⊢ (Fun 𝐼 → (𝑡 ∈ ∪ ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼‘𝑖)))
178	42, 177	mp1i 14	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → (𝑡 ∈ ∪ ran 𝐼 ↔ ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼‘𝑖)))
179	176, 178	mpbird 260	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑡 ∈ ∪ ran 𝐼)
180	179	ex 418	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (¬ 𝑡 ∈ ran 𝑄 → 𝑡 ∈ ∪ ran 𝐼))
181	180	orrd 877	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ ran 𝑄 ∨ 𝑡 ∈ ∪ ran 𝐼))
182		elun 4107	. . . . . 6 ⊢ (𝑡 ∈ (ran 𝑄 ∪ ∪ ran 𝐼) ↔ (𝑡 ∈ ran 𝑄 ∨ 𝑡 ∈ ∪ ran 𝐼))
183	181, 182	sylibr 237	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (ran 𝑄 ∪ ∪ ran 𝐼))
184	183	ralrimiva 3159	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ (𝐴[,]𝐵)𝑡 ∈ (ran 𝑄 ∪ ∪ ran 𝐼))
185		dfss3 3927	. . . 4 ⊢ ((𝐴[,]𝐵) ⊆ (ran 𝑄 ∪ ∪ ran 𝐼) ↔ ∀𝑡 ∈ (𝐴[,]𝐵)𝑡 ∈ (ran 𝑄 ∪ ∪ ran 𝐼))
186	184, 185	sylibr 237	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (ran 𝑄 ∪ ∪ ran 𝐼))
187	186, 17	sseqtrrd 3975	. 2 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ {ran 𝑄, ∪ ran 𝐼})
188	2, 79, 155, 187	ssfiunibd 46086	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑠 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑠)) ≤ 𝑥)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  {crab 3418  Vcvv 3457   ∪ cun 3904   ⊆ wss 3906  {cpr 4593  ∪ cuni 4874   class class class wbr 5111   ↦ cmpt 5194  dom cdm 5663  ran crn 5664   ↾ cres 5665  Fun wfun 6534   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419   ↑_m cmap 8830  Fincfn 8949  supcsup 9407  ℂcc 11113  ℝcr 11114  0cc0 11115  1c1 11116   + caddc 11118  ℝ^*cxr 11257   < clt 11258   ≤ cle 11259  ℕcn 12248  (,)cioo 13388  [,]cicc 13391  ...cfz 13551  ..^cfzo 13699  abscabs 15309  –cn→ccncf 25086   lim_ℂ climc 26072

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192  ax-pre-sup 11193

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-om 7869  df-1st 7992  df-2nd 7993  df-supp 8163  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-map 8832  df-pm 8833  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-fsupp 9329  df-fi 9378  df-sup 9409  df-inf 9410  df-oi 9479  df-card 9941  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-div 11887  df-nn 12249  df-2 12318  df-3 12319  df-4 12320  df-5 12321  df-6 12322  df-7 12323  df-8 12324  df-9 12325  df-n0 12520  df-z 12607  df-dec 12728  df-uz 12879  df-q 12989  df-rp 13033  df-xneg 13153  df-xadd 13154  df-xmul 13155  df-ioo 13392  df-ioc 13393  df-ico 13394  df-icc 13395  df-fz 13552  df-fzo 13700  df-seq 14056  df-exp 14116  df-hash 14385  df-cj 15174  df-re 15175  df-im 15176  df-sqrt 15310  df-abs 15311  df-struct 17229  df-sets 17246  df-slot 17264  df-ndx 17276  df-base 17292  df-ress 17313  df-plusg 17345  df-mulr 17346  df-starv 17347  df-sca 17348  df-vsca 17349  df-ip 17350  df-tset 17351  df-ple 17352  df-ds 17354  df-unif 17355  df-hom 17356  df-cco 17357  df-rest 17497  df-topn 17498  df-0g 17516  df-gsum 17517  df-topgen 17518  df-pt 17519  df-prds 17522  df-xrs 17578  df-qtop 17583  df-imas 17584  df-xps 17586  df-mre 17660  df-mrc 17661  df-acs 17663  df-mgm 18720  df-sgrp 18809  df-mnd 18825  df-submnd 18879  df-mulg 19178  df-cntz 19431  df-cmn 19896  df-psmet 21564  df-xmet 21565  df-met 21566  df-bl 21567  df-mopn 21568  df-cnfld 21573  df-top 23101  df-topon 23118  df-topsp 23140  df-bases 23153  df-cld 23226  df-ntr 23227  df-cls 23228  df-cn 23434  df-cnp 23435  df-cmp 23594  df-tx 23770  df-hmeo 23963  df-xms 24528  df-ms 24529  df-tms 24530  df-cncf 25088  df-limc 26076

This theorem is used by:  fourierdlem103  46981  fourierdlem104  46982