Theorem itg2mono 24823

Description: The Monotone Convergence Theorem for nonnegative functions. If {(𝐹‘𝑛):𝑛 ∈ ℕ} is a monotone increasing sequence of positive, measurable, real-valued functions, and 𝐺 is the pointwise limit of the sequence, then (∫₂‘𝐺) is the limit of the sequence {(∫₂‘(𝐹‘𝑛)):𝑛 ∈ ℕ}. (Contributed by Mario Carneiro, 16-Aug-2014.)

Hypotheses

Ref	Expression
itg2mono.1	⊢ 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
itg2mono.2	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn)
itg2mono.3	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞))
itg2mono.4	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))
itg2mono.5	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
itg2mono.6	⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < )

Assertion

Ref	Expression
itg2mono	⊢ (𝜑 → (∫2‘𝐺) = 𝑆)

Distinct variable groups:   𝑥,𝑛,𝑦,𝐺   𝑛,𝐹,𝑥,𝑦   𝜑,𝑛,𝑥,𝑦   𝑆,𝑛,𝑥,𝑦

Proof of Theorem itg2mono

Dummy variables 𝑓 𝑚 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		itg2mono.3	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞))
2		rge0ssre 13117	. . . . . . . . . . . 12 ⊢ (0[,)+∞) ⊆ ℝ
3		fss 6601	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹‘𝑛):ℝ⟶ℝ)
4	1, 2, 3	sylancl 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶ℝ)
5	4	ffvelrnda 6943	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
6	5	an32s 648	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
7	6	fmpttd 6971	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)):ℕ⟶ℝ)
8	7	frnd 6592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ⊆ ℝ)
9		1nn 11914	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
10		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))
11	10, 6	dmmptd 6562	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = ℕ)
12	9, 11	eleqtrrid 2846	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
13	12	ne0d 4266	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅)
14		dm0rn0 5823	. . . . . . . . 9 ⊢ (dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = ∅)
15	14	necon3bii 2995	. . . . . . . 8 ⊢ (dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅)
16	13, 15	sylib 217	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅)
17		itg2mono.5	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
18	7	ffnd 6585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ)
19		breq1 5073	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) → (𝑧 ≤ 𝑦 ↔ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
20	19	ralrn 6946	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
21	18, 20	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
22		fveq2 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
23	22	fveq1d 6758	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛)‘𝑥) = ((𝐹‘𝑚)‘𝑥))
24		fvex 6769	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑚)‘𝑥) ∈ V
25	23, 10, 24	fvmpt 6857	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥))
26	25	breq1d 5080	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦))
27	26	ralbiia 3089	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦)
28	23	breq1d 5080	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦))
29	28	cbvralvw 3372	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦)
30	27, 29	bitr4i 277	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
31	21, 30	bitrdi 286	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦))
32	31	rexbidv 3225	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦))
33	17, 32	mpbird 256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦)
34	8, 16, 33	suprcld 11868	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
35	34	rexrd 10956	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ ℝ*)
36		0red 10909	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
37		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1))
38	37	feq1d 6569	. . . . . . . . . 10 ⊢ (𝑛 = 1 → ((𝐹‘𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘1):ℝ⟶(0[,)+∞)))
39	1	ralrimiva 3107	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛):ℝ⟶(0[,)+∞))
40	9	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℕ)
41	38, 39, 40	rspcdva 3554	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘1):ℝ⟶(0[,)+∞))
42	41	ffvelrnda 6943	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ (0[,)+∞))
43		elrege0 13115	. . . . . . . 8 ⊢ (((𝐹‘1)‘𝑥) ∈ (0[,)+∞) ↔ (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
44	42, 43	sylib 217	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
45	44	simpld 494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ℝ)
46	44	simprd 495	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ ((𝐹‘1)‘𝑥))
47	37	fveq1d 6758	. . . . . . . . . 10 ⊢ (𝑛 = 1 → ((𝐹‘𝑛)‘𝑥) = ((𝐹‘1)‘𝑥))
48		fvex 6769	. . . . . . . . . 10 ⊢ ((𝐹‘1)‘𝑥) ∈ V
49	47, 10, 48	fvmpt 6857	. . . . . . . . 9 ⊢ (1 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘1) = ((𝐹‘1)‘𝑥))
50	9, 49	ax-mp 5	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘1) = ((𝐹‘1)‘𝑥)
51		fnfvelrn 6940	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ ∧ 1 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
52	18, 9, 51	sylancl 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
53	50, 52	eqeltrrid 2844	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
54	8, 16, 33, 53	suprubd 11867	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
55	36, 45, 34, 46, 54	letrd 11062	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
56		elxrge0 13118	. . . . 5 ⊢ (sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ (0[,]+∞) ↔ (sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ ℝ* ∧ 0 ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )))
57	35, 55, 56	sylanbrc 582	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ (0[,]+∞))
58		itg2mono.1	. . . 4 ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
59	57, 58	fmptd 6970	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶(0[,]+∞))
60		itg2cl 24802	. . 3 ⊢ (𝐺:ℝ⟶(0[,]+∞) → (∫2‘𝐺) ∈ ℝ*)
61	59, 60	syl 17	. 2 ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ*)
62		itg2mono.6	. . 3 ⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < )
63		icossicc 13097	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
64		fss 6601	. . . . . . . 8 ⊢ (((𝐹‘𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹‘𝑛):ℝ⟶(0[,]+∞))
65	1, 63, 64	sylancl 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,]+∞))
66		itg2cl 24802	. . . . . . 7 ⊢ ((𝐹‘𝑛):ℝ⟶(0[,]+∞) → (∫2‘(𝐹‘𝑛)) ∈ ℝ*)
67	65, 66	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∫2‘(𝐹‘𝑛)) ∈ ℝ*)
68	67	fmpttd 6971	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))):ℕ⟶ℝ*)
69	68	frnd 6592	. . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) ⊆ ℝ*)
70		supxrcl 12978	. . . 4 ⊢ (ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) ⊆ ℝ* → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ∈ ℝ*)
71	69, 70	syl 17	. . 3 ⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ∈ ℝ*)
72	62, 71	eqeltrid 2843	. 2 ⊢ (𝜑 → 𝑆 ∈ ℝ*)
73		itg2mono.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn)
74	73	adantlr 711	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn)
75	1	adantlr 711	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞))
76		itg2mono.4	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))
77	76	adantlr 711	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))
78	17	adantlr 711	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
79		simprll 775	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) → 𝑓 ∈ dom ∫1)
80		simprlr 776	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) → 𝑓 ∘r ≤ 𝐺)
81		simprr 769	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) → ¬ (∫1‘𝑓) ≤ 𝑆)
82	58, 74, 75, 77, 78, 62, 79, 80, 81	itg2monolem3 24822	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) → (∫1‘𝑓) ≤ 𝑆)
83	82	expr 456	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺)) → (¬ (∫1‘𝑓) ≤ 𝑆 → (∫1‘𝑓) ≤ 𝑆))
84	83	pm2.18d 127	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺)) → (∫1‘𝑓) ≤ 𝑆)
85	84	expr 456	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ dom ∫1) → (𝑓 ∘r ≤ 𝐺 → (∫1‘𝑓) ≤ 𝑆))
86	85	ralrimiva 3107	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐺 → (∫1‘𝑓) ≤ 𝑆))
87		itg2leub 24804	. . . 4 ⊢ ((𝐺:ℝ⟶(0[,]+∞) ∧ 𝑆 ∈ ℝ*) → ((∫2‘𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐺 → (∫1‘𝑓) ≤ 𝑆)))
88	59, 72, 87	syl2anc 583	. . 3 ⊢ (𝜑 → ((∫2‘𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐺 → (∫1‘𝑓) ≤ 𝑆)))
89	86, 88	mpbird 256	. 2 ⊢ (𝜑 → (∫2‘𝐺) ≤ 𝑆)
90	22	feq1d 6569	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘𝑚):ℝ⟶(0[,)+∞)))
91	90	cbvralvw 3372	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ ℕ (𝐹‘𝑛):ℝ⟶(0[,)+∞) ↔ ∀𝑚 ∈ ℕ (𝐹‘𝑚):ℝ⟶(0[,)+∞))
92	39, 91	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝐹‘𝑚):ℝ⟶(0[,)+∞))
93	92	r19.21bi 3132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚):ℝ⟶(0[,)+∞))
94		fss 6601	. . . . . . . 8 ⊢ (((𝐹‘𝑚):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹‘𝑚):ℝ⟶(0[,]+∞))
95	93, 63, 94	sylancl 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚):ℝ⟶(0[,]+∞))
96	59	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐺:ℝ⟶(0[,]+∞))
97	8, 16, 33	3jca 1126	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦))
98	97	adantlr 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦))
99	25	ad2antlr 723	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥))
100	18	adantlr 711	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ)
101		simplr 765	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑚 ∈ ℕ)
102		fnfvelrn 6940	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
103	100, 101, 102	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
104	99, 103	eqeltrrd 2840	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
105		suprub 11866	. . . . . . . . . . . 12 ⊢ (((ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦) ∧ ((𝐹‘𝑚)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))) → ((𝐹‘𝑚)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
106	98, 104, 105	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
107		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
108		ltso 10986	. . . . . . . . . . . . 13 ⊢ < Or ℝ
109	108	supex 9152	. . . . . . . . . . . 12 ⊢ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ V
110	58	fvmpt2 6868	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ V) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
111	107, 109, 110	sylancl 585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
112	106, 111	breqtrrd 5098	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥))
113	112	ralrimiva 3107	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑥 ∈ ℝ ((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥))
114		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑧))
115		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
116	114, 115	breq12d 5083	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥) ↔ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧)))
117	116	cbvralvw 3372	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℝ ((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥) ↔ ∀𝑧 ∈ ℝ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧))
118	113, 117	sylib 217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑧 ∈ ℝ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧))
119	93	ffnd 6585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) Fn ℝ)
120	34, 58	fmptd 6970	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
121	120	ffnd 6585	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn ℝ)
122	121	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐺 Fn ℝ)
123		reex 10893	. . . . . . . . . 10 ⊢ ℝ ∈ V
124	123	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ℝ ∈ V)
125		inidm 4149	. . . . . . . . 9 ⊢ (ℝ ∩ ℝ) = ℝ
126		eqidd 2739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑚)‘𝑧) = ((𝐹‘𝑚)‘𝑧))
127		eqidd 2739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧))
128	119, 122, 124, 124, 125, 126, 127	ofrfval 7521	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐹‘𝑚) ∘r ≤ 𝐺 ↔ ∀𝑧 ∈ ℝ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧)))
129	118, 128	mpbird 256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∘r ≤ 𝐺)
130		itg2le 24809	. . . . . . 7 ⊢ (((𝐹‘𝑚):ℝ⟶(0[,]+∞) ∧ 𝐺:ℝ⟶(0[,]+∞) ∧ (𝐹‘𝑚) ∘r ≤ 𝐺) → (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))
131	95, 96, 129, 130	syl3anc 1369	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))
132	131	ralrimiva 3107	. . . . 5 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))
133	68	ffnd 6585	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) Fn ℕ)
134		breq1 5073	. . . . . . . 8 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) → (𝑧 ≤ (∫2‘𝐺) ↔ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺)))
135	134	ralrn 6946	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺) ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺)))
136	133, 135	syl 17	. . . . . 6 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺) ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺)))
137		2fveq3 6761	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (∫2‘(𝐹‘𝑛)) = (∫2‘(𝐹‘𝑚)))
138		eqid 2738	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))
139		fvex 6769	. . . . . . . . 9 ⊢ (∫2‘(𝐹‘𝑚)) ∈ V
140	137, 138, 139	fvmpt 6857	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) = (∫2‘(𝐹‘𝑚)))
141	140	breq1d 5080	. . . . . . 7 ⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺) ↔ (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺)))
142	141	ralbiia 3089	. . . . . 6 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))
143	136, 142	bitrdi 286	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺)))
144	132, 143	mpbird 256	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺))
145		supxrleub 12989	. . . . 5 ⊢ ((ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) ⊆ ℝ* ∧ (∫2‘𝐺) ∈ ℝ*) → (sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ≤ (∫2‘𝐺) ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺)))
146	69, 61, 145	syl2anc 583	. . . 4 ⊢ (𝜑 → (sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ≤ (∫2‘𝐺) ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺)))
147	144, 146	mpbird 256	. . 3 ⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ≤ (∫2‘𝐺))
148	62, 147	eqbrtrid 5105	. 2 ⊢ (𝜑 → 𝑆 ≤ (∫2‘𝐺))
149	61, 72, 89, 148	xrletrid 12818	1 ⊢ (𝜑 → (∫2‘𝐺) = 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  ∅c0 4253   class class class wbr 5070   ↦ cmpt 5153  dom cdm 5580  ran crn 5581   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ∘_r cofr 7510  supcsup 9129  ℝcr 10801  0cc0 10802  1c1 10803   + caddc 10805  +∞cpnf 10937  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941  ℕcn 11903  [,)cico 13010  [,]cicc 13011  MblFncmbf 24683  ∫₁citg1 24684  ∫₂citg2 24685

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-inf2 9329  ax-cc 10122  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

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-disj 5036  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-ofr 7512  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-omul 8272  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fi 9100  df-sup 9131  df-inf 9132  df-oi 9199  df-dju 9590  df-card 9628  df-acn 9631  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-n0 12164  df-z 12250  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-fl 13440  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-clim 15125  df-rlim 15126  df-sum 15326  df-rest 17050  df-topgen 17071  df-psmet 20502  df-xmet 20503  df-met 20504  df-bl 20505  df-mopn 20506  df-top 21951  df-topon 21968  df-bases 22004  df-cmp 22446  df-ovol 24533  df-vol 24534  df-mbf 24688  df-itg1 24689  df-itg2 24690

This theorem is referenced by:  itg2i1fseq  24825  itg2cnlem1  24831