Theorem itg2mono 25784

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 13446	. . . . . . . . . . . 12 ⊢ (0[,)+∞) ⊆ ℝ
3		fss 6693	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹‘𝑛):ℝ⟶ℝ)
4	1, 2, 3	sylancl 594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶ℝ)
5	4	ffvelcdmda 7050	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
6	5	an32s 660	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
7	6	fmpttd 7081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)):ℕ⟶ℝ)
8	7	frnd 6685	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ⊆ ℝ)
9		1nn 12207	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
10		eqid 2752	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))
11	10, 6	dmmptd 6651	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = ℕ)
12	9, 11	eleqtrrid 2859	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
13	12	ne0d 4285	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅)
14		dm0rn0 5889	. . . . . . . . 9 ⊢ (dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = ∅)
15	14	necon3bii 2999	. . . . . . . 8 ⊢ (dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅)
16	13, 15	sylib 220	. . . . . . 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 6677	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ)
19		breq1 5093	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) → (𝑧 ≤ 𝑦 ↔ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
20	19	ralrn 7054	. . . . . . . . . . 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 6852	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
23	22	fveq1d 6854	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛)‘𝑥) = ((𝐹‘𝑚)‘𝑥))
24		fvex 6865	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑚)‘𝑥) ∈ V
25	23, 10, 24	fvmpt 6960	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥))
26	25	breq1d 5100	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦))
27	26	ralbiia 3096	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦)
28	23	breq1d 5100	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦))
29	28	cbvralvw 3230	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦)
30	27, 29	bitr4i 280	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
31	21, 30	bitrdi 289	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦))
32	31	rexbidv 3176	. . . . . . . 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 259	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦)
34	8, 16, 33	suprcld 12141	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
35	34	rexrd 11218	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ ℝ*)
36		0red 11170	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
37		fveq2 6852	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1))
38	37	feq1d 6658	. . . . . . . . . 10 ⊢ (𝑛 = 1 → ((𝐹‘𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘1):ℝ⟶(0[,)+∞)))
39	1	ralrimiva 3144	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛):ℝ⟶(0[,)+∞))
40	9	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℕ)
41	38, 39, 40	rspcdva 3573	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘1):ℝ⟶(0[,)+∞))
42	41	ffvelcdmda 7050	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ (0[,)+∞))
43		elrege0 13444	. . . . . . . 8 ⊢ (((𝐹‘1)‘𝑥) ∈ (0[,)+∞) ↔ (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
44	42, 43	sylib 220	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
45	44	simpld 497	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ℝ)
46	44	simprd 498	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ ((𝐹‘1)‘𝑥))
47	37	fveq1d 6854	. . . . . . . . . 10 ⊢ (𝑛 = 1 → ((𝐹‘𝑛)‘𝑥) = ((𝐹‘1)‘𝑥))
48		fvex 6865	. . . . . . . . . 10 ⊢ ((𝐹‘1)‘𝑥) ∈ V
49	47, 10, 48	fvmpt 6960	. . . . . . . . 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 7046	. . . . . . . . 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 594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
53	50, 52	eqeltrrid 2857	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
54	8, 16, 33, 53	suprubd 12140	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
55	36, 45, 34, 46, 54	letrd 11326	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
56		elxrge0 13447	. . . . 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 591	. . . 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 7080	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶(0[,]+∞))
60		itg2cl 25763	. . 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 13426	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
64		fss 6693	. . . . . . . 8 ⊢ (((𝐹‘𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹‘𝑛):ℝ⟶(0[,]+∞))
65	1, 63, 64	sylancl 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,]+∞))
66		itg2cl 25763	. . . . . . 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 7081	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))):ℕ⟶ℝ*)
69	68	frnd 6685	. . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) ⊆ ℝ*)
70		supxrcl 13304	. . . 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 2856	. 2 ⊢ (𝜑 → 𝑆 ∈ ℝ*)
73		itg2mono.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn)
74	73	adantlr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn)
75	1	adantlr 723	. . . . . . . 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 723	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))
78	17	adantlr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
79		simprll 786	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) → 𝑓 ∈ dom ∫1)
80		simprlr 787	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) → 𝑓 ∘r ≤ 𝐺)
81		simprr 780	. . . . . . . 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 25783	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) → (∫1‘𝑓) ≤ 𝑆)
83	82	expr 459	. . . . . 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 459	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ dom ∫1) → (𝑓 ∘r ≤ 𝐺 → (∫1‘𝑓) ≤ 𝑆))
86	85	ralrimiva 3144	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐺 → (∫1‘𝑓) ≤ 𝑆))
87		itg2leub 25765	. . . 4 ⊢ ((𝐺:ℝ⟶(0[,]+∞) ∧ 𝑆 ∈ ℝ*) → ((∫2‘𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐺 → (∫1‘𝑓) ≤ 𝑆)))
88	59, 72, 87	syl2anc 592	. . 3 ⊢ (𝜑 → ((∫2‘𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐺 → (∫1‘𝑓) ≤ 𝑆)))
89	86, 88	mpbird 259	. 2 ⊢ (𝜑 → (∫2‘𝐺) ≤ 𝑆)
90	22	feq1d 6658	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘𝑚):ℝ⟶(0[,)+∞)))
91	90	cbvralvw 3230	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ ℕ (𝐹‘𝑛):ℝ⟶(0[,)+∞) ↔ ∀𝑚 ∈ ℕ (𝐹‘𝑚):ℝ⟶(0[,)+∞))
92	39, 91	sylib 220	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝐹‘𝑚):ℝ⟶(0[,)+∞))
93	92	r19.21bi 3244	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚):ℝ⟶(0[,)+∞))
94		fss 6693	. . . . . . . 8 ⊢ (((𝐹‘𝑚):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹‘𝑚):ℝ⟶(0[,]+∞))
95	93, 63, 94	sylancl 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚):ℝ⟶(0[,]+∞))
96	59	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐺:ℝ⟶(0[,]+∞))
97	8, 16, 33	3jca 1137	. . . . . . . . . . . . 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 723	. . . . . . . . . . . 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 735	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥))
100	18	adantlr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ)
101		simplr 776	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑚 ∈ ℕ)
102		fnfvelrn 7046	. . . . . . . . . . . . . 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 592	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
104	99, 103	eqeltrrd 2853	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
105		suprub 12139	. . . . . . . . . . . 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 592	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
107		simpr 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
108		ltso 11249	. . . . . . . . . . . . 13 ⊢ < Or ℝ
109	108	supex 9396	. . . . . . . . . . . 12 ⊢ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ V
110	58	fvmpt2 6972	. . . . . . . . . . . 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 594	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
112	106, 111	breqtrrd 5118	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥))
113	112	ralrimiva 3144	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑥 ∈ ℝ ((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥))
114		fveq2 6852	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑧))
115		fveq2 6852	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
116	114, 115	breq12d 5103	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥) ↔ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧)))
117	116	cbvralvw 3230	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℝ ((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥) ↔ ∀𝑧 ∈ ℝ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧))
118	113, 117	sylib 220	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑧 ∈ ℝ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧))
119	93	ffnd 6677	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) Fn ℝ)
120	34, 58	fmptd 7080	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
121	120	ffnd 6677	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn ℝ)
122	121	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐺 Fn ℝ)
123		reex 11150	. . . . . . . . . 10 ⊢ ℝ ∈ V
124	123	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ℝ ∈ V)
125		inidm 4169	. . . . . . . . 9 ⊢ (ℝ ∩ ℝ) = ℝ
126		eqidd 2753	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑚)‘𝑧) = ((𝐹‘𝑚)‘𝑧))
127		eqidd 2753	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧))
128	119, 122, 124, 124, 125, 126, 127	ofrfval 7655	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐹‘𝑚) ∘r ≤ 𝐺 ↔ ∀𝑧 ∈ ℝ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧)))
129	118, 128	mpbird 259	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∘r ≤ 𝐺)
130		itg2le 25770	. . . . . . 7 ⊢ (((𝐹‘𝑚):ℝ⟶(0[,]+∞) ∧ 𝐺:ℝ⟶(0[,]+∞) ∧ (𝐹‘𝑚) ∘r ≤ 𝐺) → (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))
131	95, 96, 129, 130	syl3anc 1382	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))
132	131	ralrimiva 3144	. . . . 5 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))
133	68	ffnd 6677	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) Fn ℕ)
134		breq1 5093	. . . . . . . 8 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) → (𝑧 ≤ (∫2‘𝐺) ↔ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺)))
135	134	ralrn 7054	. . . . . . 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 6857	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (∫2‘(𝐹‘𝑛)) = (∫2‘(𝐹‘𝑚)))
138		eqid 2752	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))
139		fvex 6865	. . . . . . . . 9 ⊢ (∫2‘(𝐹‘𝑚)) ∈ V
140	137, 138, 139	fvmpt 6960	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) = (∫2‘(𝐹‘𝑚)))
141	140	breq1d 5100	. . . . . . 7 ⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺) ↔ (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺)))
142	141	ralbiia 3096	. . . . . 6 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))
143	136, 142	bitrdi 289	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺)))
144	132, 143	mpbird 259	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺))
145		supxrleub 13315	. . . . 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 592	. . . 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 259	. . 3 ⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ≤ (∫2‘𝐺))
148	62, 147	eqbrtrid 5125	. 2 ⊢ (𝜑 → 𝑆 ≤ (∫2‘𝐺))
149	61, 72, 89, 148	xrletrid 13143	1 ⊢ (𝜑 → (∫2‘𝐺) = 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132   ≠ wne 2947  ∀wral 3066  ∃wrex 3076  Vcvv 3444   ⊆ wss 3895  ∅c0 4276   class class class wbr 5090   ↦ cmpt 5171  dom cdm 5636  ran crn 5637   Fn wfn 6501  ⟶wf 6502  ‘cfv 6506  (class class class)co 7381   ∘_r cofr 7644  supcsup 9372  ℝcr 11058  0cc0 11059  1c1 11060   + caddc 11062  +∞cpnf 11199  ℝ^*cxr 11201   < clt 11202   ≤ cle 11203  ℕcn 12196  [,)cico 13337  [,]cicc 13338  MblFncmbf 25645  ∫₁citg1 25646  ∫₂citg2 25647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-inf2 9582  ax-cc 10378  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-nel 3052  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-int 4896  df-iun 4941  df-disj 5058  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-se 5590  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-isom 6515  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-of 7645  df-ofr 7646  df-om 7832  df-1st 7955  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-2o 8422  df-oadd 8425  df-omul 8426  df-er 8662  df-map 8794  df-pm 8795  df-en 8913  df-dom 8914  df-sdom 8915  df-fin 8916  df-fi 9343  df-sup 9374  df-inf 9375  df-oi 9444  df-dju 9845  df-card 9883  df-acn 9886  df-pnf 11204  df-mnf 11205  df-xr 11206  df-ltxr 11207  df-le 11208  df-sub 11402  df-neg 11403  df-div 11831  df-nn 12197  df-2 12266  df-3 12267  df-n0 12468  df-z 12555  df-uz 12826  df-q 12936  df-rp 12980  df-xneg 13100  df-xadd 13101  df-xmul 13102  df-ioo 13339  df-ioc 13340  df-ico 13341  df-icc 13342  df-fz 13499  df-fzo 13646  df-fl 13788  df-seq 14001  df-exp 14061  df-hash 14330  df-cj 15098  df-re 15099  df-im 15100  df-sqrt 15234  df-abs 15235  df-clim 15487  df-rlim 15488  df-sum 15686  df-rest 17423  df-topgen 17444  df-psmet 21385  df-xmet 21386  df-met 21387  df-bl 21388  df-mopn 21389  df-top 22923  df-topon 22940  df-bases 22975  df-cmp 23416  df-ovol 25495  df-vol 25496  df-mbf 25650  df-itg1 25651  df-itg2 25652

This theorem is referenced by:  itg2i1fseq  25786  itg2cnlem1  25792