Theorem fourierdlem70 43607

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 9019	. . 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 7288	. . . . . . . . . . 11 ⊢ (0...𝑀) ∈ V
6		fex 7084	. . . . . . . . . . 11 ⊢ ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ V) → 𝑄 ∈ V)
7	4, 5, 6	sylancl 585	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ V)
8		rnexg 7725	. . . . . . . . . 10 ⊢ (𝑄 ∈ V → ran 𝑄 ∈ V)
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑄 ∈ V)
10		fzofi 13622	. . . . . . . . . . . 12 ⊢ (0..^𝑀) ∈ Fin
11		fourierdlem70.i	. . . . . . . . . . . . 13 ⊢ 𝐼 = (𝑖 ∈ (0..^𝑀) ↦ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
12	11	rnmptfi 42596	. . . . . . . . . . . 12 ⊢ ((0..^𝑀) ∈ Fin → ran 𝐼 ∈ Fin)
13	10, 12	ax-mp 5	. . . . . . . . . . 11 ⊢ ran 𝐼 ∈ Fin
14	13	elexi 3441	. . . . . . . . . 10 ⊢ ran 𝐼 ∈ V
15	14	uniex 7572	. . . . . . . . 9 ⊢ ∪ ran 𝐼 ∈ V
16		uniprg 4853	. . . . . . . . 9 ⊢ ((ran 𝑄 ∈ V ∧ ∪ ran 𝐼 ∈ V) → ∪ {ran 𝑄, ∪ ran 𝐼} = (ran 𝑄 ∪ ∪ ran 𝐼))
17	9, 15, 16	sylancl 585	. . . . . . . 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 2841	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → 𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼))
20		eqid 2738	. . . . . . . . . . 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 10893	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
23	22, 5	elmap 8617	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ)
24	4, 23	sylibr 233	. . . . . . . . . . . . 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 3107	. . . . . . . . . . . . 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 43540	. . . . . . . . . . . . 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 256	. . . . . . . . . . 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 43553	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
35	34	frnd 6592	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
36	35	sselda 3917	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
37	36	adantlr 711	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼)) ∧ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
38		simpll 763	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝜑)
39		elunnel1 4080	. . . . . . . . 9 ⊢ ((𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ ∪ ran 𝐼)
40	39	adantll 710	. . . . . . . 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 6457	. . . . . . . . . . 11 ⊢ Fun 𝐼
43		elunirn 7106	. . . . . . . . . . 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 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ ran 𝐼) → ∃𝑖 ∈ dom 𝐼 𝑠 ∈ (𝐼‘𝑖))
46		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ dom 𝐼 → 𝑖 ∈ dom 𝐼)
47		ovex 7288	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V
48	47, 11	dmmpti 6561	. . . . . . . . . . . . . . . . . 18 ⊢ dom 𝐼 = (0..^𝑀)
49	46, 48	eleqtrdi 2849	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ dom 𝐼 → 𝑖 ∈ (0..^𝑀))
50	11	fvmpt2 6868	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ∈ V) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
51	49, 47, 50	sylancl 585	. . . . . . . . . . . . . . . 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 13094	. . . . . . . . . . . . . . . 16 ⊢ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1)))
54		fourierdlem70.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ ℝ)
55	54	rexrd 10956	. . . . . . . . . . . . . . . . . 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 10956	. . . . . . . . . . . . . . . . . 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 43546	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → ((𝑄‘𝑖)[,](𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
63	53, 62	sstrid 3928	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ (𝐴[,]𝐵))
64	52, 63	eqsstrd 3955	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼) → (𝐼‘𝑖) ⊆ (𝐴[,]𝐵))
65	64	3adant3 1130	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑠 ∈ (𝐼‘𝑖)) → (𝐼‘𝑖) ⊆ (𝐴[,]𝐵))
66		simp3 1136	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑠 ∈ (𝐼‘𝑖)) → 𝑠 ∈ (𝐼‘𝑖))
67	65, 66	sseldd 3918	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ dom 𝐼 ∧ 𝑠 ∈ (𝐼‘𝑖)) → 𝑠 ∈ (𝐴[,]𝐵))
68	67	3exp 1117	. . . . . . . . . . 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 3211	. . . . . . . . 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 583	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼)) ∧ ¬ 𝑠 ∈ ran 𝑄) → 𝑠 ∈ (𝐴[,]𝐵))
73	37, 72	pm2.61dan 809	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (ran 𝑄 ∪ ∪ ran 𝐼)) → 𝑠 ∈ (𝐴[,]𝐵))
74	19, 73	syldan 590	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → 𝑠 ∈ (𝐴[,]𝐵))
75		fourierdlem70.f	. . . . . 6 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
76	75	ffvelrnda 6943	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑠) ∈ ℝ)
77	74, 76	syldan 590	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → (𝐹‘𝑠) ∈ ℝ)
78	77	recnd 10934	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ ∪ {ran 𝑄, ∪ ran 𝐼}) → (𝐹‘𝑠) ∈ ℂ)
79	78	abscld 15076	. 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 13621	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → (0...𝑀) ∈ Fin)
83		rnffi 42600	. . . . . . 7 ⊢ ((𝑄:(0...𝑀)⟶ℝ ∧ (0...𝑀) ∈ Fin) → ran 𝑄 ∈ Fin)
84	81, 82, 83	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → ran 𝑄 ∈ Fin)
85	80, 84	eqeltrd 2839	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → 𝑤 ∈ Fin)
86	85	adantlr 711	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ 𝑤 = ran 𝑄) → 𝑤 ∈ Fin)
87	75	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
88		simpll 763	. . . . . . . . . 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 2841	. . . . . . . . . . 11 ⊢ ((𝑤 = ran 𝑄 ∧ 𝑠 ∈ 𝑤) → 𝑠 ∈ ran 𝑄)
92	91	adantll 710	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → 𝑠 ∈ ran 𝑄)
93	88, 92, 36	syl2anc 583	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → 𝑠 ∈ (𝐴[,]𝐵))
94	87, 93	ffvelrnd 6944	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → (𝐹‘𝑠) ∈ ℝ)
95	94	recnd 10934	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → (𝐹‘𝑠) ∈ ℂ)
96	95	abscld 15076	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 = ran 𝑄) ∧ 𝑠 ∈ 𝑤) → (abs‘(𝐹‘𝑠)) ∈ ℝ)
97	96	ralrimiva 3107	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 = ran 𝑄) → ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ∈ ℝ)
98	97	adantlr 711	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ 𝑤 = ran 𝑄) → ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ∈ ℝ)
99		fimaxre3 11851	. . . 4 ⊢ ((𝑤 ∈ Fin ∧ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ≤ 𝑧)
100	86, 98, 99	syl2anc 583	. . 3 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ 𝑤 = ran 𝑄) → ∃𝑧 ∈ ℝ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ≤ 𝑧)
101		simpll 763	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → 𝜑)
102		neqne 2950	. . . . . 6 ⊢ (¬ 𝑤 = ran 𝑄 → 𝑤 ≠ ran 𝑄)
103		elprn1 43064	. . . . . 6 ⊢ ((𝑤 ∈ {ran 𝑄, ∪ ran 𝐼} ∧ 𝑤 ≠ ran 𝑄) → 𝑤 = ∪ ran 𝐼)
104	102, 103	sylan2 592	. . . . 5 ⊢ ((𝑤 ∈ {ran 𝑄, ∪ ran 𝐼} ∧ ¬ 𝑤 = ran 𝑄) → 𝑤 = ∪ ran 𝐼)
105	104	adantll 710	. . . 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 10859	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
108	107	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ ℂ)
109	75, 108	fssd 6602	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
110	109	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑠 ∈ ∪ ran 𝐼) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
111	71	adantlr 711	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑠 ∈ ∪ ran 𝐼) → 𝑠 ∈ (𝐴[,]𝐵))
112	110, 111	ffvelrnd 6944	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑠 ∈ ∪ ran 𝐼) → (𝐹‘𝑠) ∈ ℂ)
113	112	abscld 15076	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑠 ∈ ∪ ran 𝐼) → (abs‘(𝐹‘𝑠)) ∈ ℝ)
114	47, 11	fnmpti 6560	. . . . . . . . . 10 ⊢ 𝐼 Fn (0..^𝑀)
115		fvelrnb 6812	. . . . . . . . . 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 215	. . . . . . . 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 13331	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
121	120	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
122	119, 121	ffvelrnd 6944	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) ∈ ℝ)
123		fzofzp1 13412	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
124	123	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
125	119, 124	ffvelrnd 6944	. . . . . . . . . . . . . 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 43328	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) ≤ 𝑏)
130		fvres 6775	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → ((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠) = (𝐹‘𝑠))
131	130	fveq2d 6760	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) → (abs‘((𝐹 ↾ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))‘𝑠)) = (abs‘(𝐹‘𝑠)))
132	131	breq1d 5080	. . . . . . . . . . . . . . . 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 3119	. . . . . . . . . . . . . 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 3225	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑠)) ≤ 𝑏)
137	136	3adant3 1130	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))(abs‘(𝐹‘𝑠)) ≤ 𝑏)
138	47, 50	mpan2 687	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0..^𝑀) → (𝐼‘𝑖) = ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
139	138	eqcomd 2744	. . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))) = 𝑡)
143	142	raleqdv 3339	. . . . . . . . . . . . 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 3225	. . . . . . . . . . . 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 1128	. . . . . . . . . . 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 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀) ∧ (𝐼‘𝑖) = 𝑡) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏)
147	146	3exp 1117	. . . . . . . . 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 3211	. . . . . . 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 711	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 = ∪ ran 𝐼) ∧ 𝑡 ∈ ran 𝐼) → ∃𝑏 ∈ ℝ ∀𝑠 ∈ 𝑡 (abs‘(𝐹‘𝑠)) ≤ 𝑏)
152		eqimss 3973	. . . . . 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 42738	. . . 4 ⊢ ((𝜑 ∧ 𝑤 = ∪ ran 𝐼) → ∃𝑧 ∈ ℝ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ≤ 𝑧)
155	101, 105, 154	syl2anc 583	. . 3 ⊢ (((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) ∧ ¬ 𝑤 = ran 𝑄) → ∃𝑧 ∈ ℝ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ≤ 𝑧)
156	100, 155	pm2.61dan 809	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ {ran 𝑄, ∪ ran 𝐼}) → ∃𝑧 ∈ ℝ ∀𝑠 ∈ 𝑤 (abs‘(𝐹‘𝑠)) ≤ 𝑧)
157	21	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → 𝑀 ∈ ℕ)
158	4	ad2antrr 722	. . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 = (𝑄‘0))
161	26	eqcomd 2744	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 = (𝑄‘𝑀))
162	160, 161	oveq12d 7273	. . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . 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 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (𝑄‘𝑘) = (𝑄‘𝑗))
168	167	breq1d 5080	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → ((𝑄‘𝑘) < 𝑡 ↔ (𝑄‘𝑗) < 𝑡))
169	168	cbvrabv 3416	. . . . . . . . . . . . 13 ⊢ {𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝑡} = {𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < 𝑡}
170	169	supeq1i 9136	. . . . . . . . . . . 12 ⊢ sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄‘𝑘) < 𝑡}, ℝ, < ) = sup({𝑗 ∈ (0..^𝑀) ∣ (𝑄‘𝑗) < 𝑡}, ℝ, < )
171	157, 158, 165, 166, 170	fourierdlem25 43563	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
172	138	eleq2d 2824	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑡 ∈ (𝐼‘𝑖) ↔ 𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1)))))
173	172	rexbiia 3176	. . . . . . . . . . 11 ⊢ (∃𝑖 ∈ (0..^𝑀)𝑡 ∈ (𝐼‘𝑖) ↔ ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ ((𝑄‘𝑖)(,)(𝑄‘(𝑖 + 1))))
174	171, 173	sylibr 233	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ (0..^𝑀)𝑡 ∈ (𝐼‘𝑖))
175	48	eqcomi 2747	. . . . . . . . . . 11 ⊢ (0..^𝑀) = dom 𝐼
176	175	rexeqi 3338	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ (0..^𝑀)𝑡 ∈ (𝐼‘𝑖) ↔ ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼‘𝑖))
177	174, 176	sylib 217	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) ∧ ¬ 𝑡 ∈ ran 𝑄) → ∃𝑖 ∈ dom 𝐼 𝑡 ∈ (𝐼‘𝑖))
178		elunirn 7106	. . . . . . . . . 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 256	. . . . . . . 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 859	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → (𝑡 ∈ ran 𝑄 ∨ 𝑡 ∈ ∪ ran 𝐼))
183		elun 4079	. . . . . 6 ⊢ (𝑡 ∈ (ran 𝑄 ∪ ∪ ran 𝐼) ↔ (𝑡 ∈ ran 𝑄 ∨ 𝑡 ∈ ∪ ran 𝐼))
184	182, 183	sylibr 233	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴[,]𝐵)) → 𝑡 ∈ (ran 𝑄 ∪ ∪ ran 𝐼))
185	184	ralrimiva 3107	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ (𝐴[,]𝐵)𝑡 ∈ (ran 𝑄 ∪ ∪ ran 𝐼))
186		dfss3 3905	. . . 4 ⊢ ((𝐴[,]𝐵) ⊆ (ran 𝑄 ∪ ∪ ran 𝐼) ↔ ∀𝑡 ∈ (𝐴[,]𝐵)𝑡 ∈ (ran 𝑄 ∪ ∪ ran 𝐼))
187	185, 186	sylibr 233	. . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ (ran 𝑄 ∪ ∪ ran 𝐼))
188	187, 17	sseqtrrd 3958	. 2 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ {ran 𝑄, ∪ ran 𝐼})
189	2, 79, 156, 188	ssfiunibd 42738	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑠 ∈ (𝐴[,]𝐵)(abs‘(𝐹‘𝑠)) ≤ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  {cpr 4560  ∪ cuni 4836   class class class wbr 5070   ↦ cmpt 5153  dom cdm 5580  ran crn 5581   ↾ cres 5582  Fun wfun 6412   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  Fincfn 8691  supcsup 9129  ℂcc 10800  ℝcr 10801  0cc0 10802  1c1 10803   + caddc 10805  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941  ℕcn 11903  (,)cioo 13008  [,]cicc 13011  ...cfz 13168  ..^cfzo 13311  abscabs 14873  –cn→ccncf 23945   lim_ℂ climc 24931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880  ax-mulf 10882

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-fi 9100  df-sup 9131  df-inf 9132  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-q 12618  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-ioo 13012  df-ioc 13013  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-starv 16903  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-unif 16911  df-hom 16912  df-cco 16913  df-rest 17050  df-topn 17051  df-0g 17069  df-gsum 17070  df-topgen 17071  df-pt 17072  df-prds 17075  df-xrs 17130  df-qtop 17135  df-imas 17136  df-xps 17138  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-mulg 18616  df-cntz 18838  df-cmn 19303  df-psmet 20502  df-xmet 20503  df-met 20504  df-bl 20505  df-mopn 20506  df-cnfld 20511  df-top 21951  df-topon 21968  df-topsp 21990  df-bases 22004  df-cld 22078  df-ntr 22079  df-cls 22080  df-cn 22286  df-cnp 22287  df-cmp 22446  df-tx 22621  df-hmeo 22814  df-xms 23381  df-ms 23382  df-tms 23383  df-cncf 23947  df-limc 24935

This theorem is referenced by:  fourierdlem103  43640  fourierdlem104  43641