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Theorem itg2mono 25263

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 13430	. . . . . . . . . . . 12 ⊢ (0[,)+∞) ⊆ ℝ
3		fss 6732	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹‘𝑛):ℝ⟶ℝ)
4	1, 2, 3	sylancl 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶ℝ)
5	4	ffvelcdmda 7084	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
6	5	an32s 651	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
7	6	fmpttd 7112	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)):ℕ⟶ℝ)
8	7	frnd 6723	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ⊆ ℝ)
9		1nn 12220	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
10		eqid 2733	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))
11	10, 6	dmmptd 6693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = ℕ)
12	9, 11	eleqtrrid 2841	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
13	12	ne0d 4335	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅)
14		dm0rn0 5923	. . . . . . . . 9 ⊢ (dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = ∅)
15	14	necon3bii 2994	. . . . . . . 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 6716	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ)
19		breq1 5151	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) → (𝑧 ≤ 𝑦 ↔ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
20	19	ralrn 7087	. . . . . . . . . . 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 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
23	22	fveq1d 6891	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛)‘𝑥) = ((𝐹‘𝑚)‘𝑥))
24		fvex 6902	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑚)‘𝑥) ∈ V
25	23, 10, 24	fvmpt 6996	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥))
26	25	breq1d 5158	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦))
27	26	ralbiia 3092	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦)
28	23	breq1d 5158	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦))
29	28	cbvralvw 3235	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦)
30	27, 29	bitr4i 278	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
31	21, 30	bitrdi 287	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦))
32	31	rexbidv 3179	. . . . . . . 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 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦)
34	8, 16, 33	suprcld 12174	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
35	34	rexrd 11261	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ ℝ^*)
36		0red 11214	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
37		fveq2 6889	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1))
38	37	feq1d 6700	. . . . . . . . . 10 ⊢ (𝑛 = 1 → ((𝐹‘𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘1):ℝ⟶(0[,)+∞)))
39	1	ralrimiva 3147	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛):ℝ⟶(0[,)+∞))
40	9	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℕ)
41	38, 39, 40	rspcdva 3614	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘1):ℝ⟶(0[,)+∞))
42	41	ffvelcdmda 7084	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ (0[,)+∞))
43		elrege0 13428	. . . . . . . 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 496	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ℝ)
46	44	simprd 497	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ ((𝐹‘1)‘𝑥))
47	37	fveq1d 6891	. . . . . . . . . 10 ⊢ (𝑛 = 1 → ((𝐹‘𝑛)‘𝑥) = ((𝐹‘1)‘𝑥))
48		fvex 6902	. . . . . . . . . 10 ⊢ ((𝐹‘1)‘𝑥) ∈ V
49	47, 10, 48	fvmpt 6996	. . . . . . . . 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 7080	. . . . . . . . 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 587	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
53	50, 52	eqeltrrid 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
54	8, 16, 33, 53	suprubd 12173	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
55	36, 45, 34, 46, 54	letrd 11368	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
56		elxrge0 13431	. . . . 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 584	. . . 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 7111	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶(0[,]+∞))
60		itg2cl 25242	. . 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 13410	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
64		fss 6732	. . . . . . . 8 ⊢ (((𝐹‘𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹‘𝑛):ℝ⟶(0[,]+∞))
65	1, 63, 64	sylancl 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,]+∞))
66		itg2cl 25242	. . . . . . 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 7112	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫₂‘(𝐹‘𝑛))):ℕ⟶ℝ^*)
69	68	frnd 6723	. . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫₂‘(𝐹‘𝑛))) ⊆ ℝ^*)
70		supxrcl 13291	. . . 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 2838	. 2 ⊢ (𝜑 → 𝑆 ∈ ℝ^*)
73		itg2mono.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn)
74	73	adantlr 714	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫₁ ∧ 𝑓 ∘_r ≤ 𝐺) ∧ ¬ (∫₁‘𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn)
75	1	adantlr 714	. . . . . . . 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 714	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫₁ ∧ 𝑓 ∘_r ≤ 𝐺) ∧ ¬ (∫₁‘𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘_r ≤ (𝐹‘(𝑛 + 1)))
78	17	adantlr 714	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫₁ ∧ 𝑓 ∘_r ≤ 𝐺) ∧ ¬ (∫₁‘𝑓) ≤ 𝑆)) ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
79		simprll 778	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫₁ ∧ 𝑓 ∘_r ≤ 𝐺) ∧ ¬ (∫₁‘𝑓) ≤ 𝑆)) → 𝑓 ∈ dom ∫₁)
80		simprlr 779	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫₁ ∧ 𝑓 ∘_r ≤ 𝐺) ∧ ¬ (∫₁‘𝑓) ≤ 𝑆)) → 𝑓 ∘_r ≤ 𝐺)
81		simprr 772	. . . . . . . 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 25262	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫₁ ∧ 𝑓 ∘_r ≤ 𝐺) ∧ ¬ (∫₁‘𝑓) ≤ 𝑆)) → (∫₁‘𝑓) ≤ 𝑆)
83	82	expr 458	. . . . . 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 458	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ dom ∫₁) → (𝑓 ∘_r ≤ 𝐺 → (∫₁‘𝑓) ≤ 𝑆))
86	85	ralrimiva 3147	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ dom ∫₁(𝑓 ∘_r ≤ 𝐺 → (∫₁‘𝑓) ≤ 𝑆))
87		itg2leub 25244	. . . 4 ⊢ ((𝐺:ℝ⟶(0[,]+∞) ∧ 𝑆 ∈ ℝ^*) → ((∫₂‘𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫₁(𝑓 ∘_r ≤ 𝐺 → (∫₁‘𝑓) ≤ 𝑆)))
88	59, 72, 87	syl2anc 585	. . 3 ⊢ (𝜑 → ((∫₂‘𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫₁(𝑓 ∘_r ≤ 𝐺 → (∫₁‘𝑓) ≤ 𝑆)))
89	86, 88	mpbird 257	. 2 ⊢ (𝜑 → (∫₂‘𝐺) ≤ 𝑆)
90	22	feq1d 6700	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘𝑚):ℝ⟶(0[,)+∞)))
91	90	cbvralvw 3235	. . . . . . . . . 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 3249	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚):ℝ⟶(0[,)+∞))
94		fss 6732	. . . . . . . 8 ⊢ (((𝐹‘𝑚):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹‘𝑚):ℝ⟶(0[,]+∞))
95	93, 63, 94	sylancl 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚):ℝ⟶(0[,]+∞))
96	59	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐺:ℝ⟶(0[,]+∞))
97	8, 16, 33	3jca 1129	. . . . . . . . . . . . 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 714	. . . . . . . . . . . 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 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥))
100	18	adantlr 714	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ)
101		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑚 ∈ ℕ)
102		fnfvelrn 7080	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
104	99, 103	eqeltrrd 2835	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
105		suprub 12172	. . . . . . . . . . . 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 585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
107		simpr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
108		ltso 11291	. . . . . . . . . . . . 13 ⊢ < Or ℝ
109	108	supex 9455	. . . . . . . . . . . 12 ⊢ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ V
110	58	fvmpt2 7007	. . . . . . . . . . . 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 587	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
112	106, 111	breqtrrd 5176	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥))
113	112	ralrimiva 3147	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑥 ∈ ℝ ((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥))
114		fveq2 6889	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑧))
115		fveq2 6889	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
116	114, 115	breq12d 5161	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥) ↔ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧)))
117	116	cbvralvw 3235	. . . . . . . . 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 6716	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) Fn ℝ)
120	34, 58	fmptd 7111	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
121	120	ffnd 6716	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn ℝ)
122	121	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐺 Fn ℝ)
123		reex 11198	. . . . . . . . . 10 ⊢ ℝ ∈ V
124	123	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ℝ ∈ V)
125		inidm 4218	. . . . . . . . 9 ⊢ (ℝ ∩ ℝ) = ℝ
126		eqidd 2734	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑚)‘𝑧) = ((𝐹‘𝑚)‘𝑧))
127		eqidd 2734	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧))
128	119, 122, 124, 124, 125, 126, 127	ofrfval 7677	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐹‘𝑚) ∘_r ≤ 𝐺 ↔ ∀𝑧 ∈ ℝ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧)))
129	118, 128	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∘_r ≤ 𝐺)
130		itg2le 25249	. . . . . . 7 ⊢ (((𝐹‘𝑚):ℝ⟶(0[,]+∞) ∧ 𝐺:ℝ⟶(0[,]+∞) ∧ (𝐹‘𝑚) ∘_r ≤ 𝐺) → (∫₂‘(𝐹‘𝑚)) ≤ (∫₂‘𝐺))
131	95, 96, 129, 130	syl3anc 1372	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∫₂‘(𝐹‘𝑚)) ≤ (∫₂‘𝐺))
132	131	ralrimiva 3147	. . . . 5 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (∫₂‘(𝐹‘𝑚)) ≤ (∫₂‘𝐺))
133	68	ffnd 6716	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫₂‘(𝐹‘𝑛))) Fn ℕ)
134		breq1 5151	. . . . . . . 8 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ (∫₂‘(𝐹‘𝑛)))‘𝑚) → (𝑧 ≤ (∫₂‘𝐺) ↔ ((𝑛 ∈ ℕ ↦ (∫₂‘(𝐹‘𝑛)))‘𝑚) ≤ (∫₂‘𝐺)))
135	134	ralrn 7087	. . . . . . 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 6894	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (∫₂‘(𝐹‘𝑛)) = (∫₂‘(𝐹‘𝑚)))
138		eqid 2733	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (∫₂‘(𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ (∫₂‘(𝐹‘𝑛)))
139		fvex 6902	. . . . . . . . 9 ⊢ (∫₂‘(𝐹‘𝑚)) ∈ V
140	137, 138, 139	fvmpt 6996	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫₂‘(𝐹‘𝑛)))‘𝑚) = (∫₂‘(𝐹‘𝑚)))
141	140	breq1d 5158	. . . . . . 7 ⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫₂‘(𝐹‘𝑛)))‘𝑚) ≤ (∫₂‘𝐺) ↔ (∫₂‘(𝐹‘𝑚)) ≤ (∫₂‘𝐺)))
142	141	ralbiia 3092	. . . . . 6 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫₂‘(𝐹‘𝑛)))‘𝑚) ≤ (∫₂‘𝐺) ↔ ∀𝑚 ∈ ℕ (∫₂‘(𝐹‘𝑚)) ≤ (∫₂‘𝐺))
143	136, 142	bitrdi 287	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫₂‘(𝐹‘𝑛)))𝑧 ≤ (∫₂‘𝐺) ↔ ∀𝑚 ∈ ℕ (∫₂‘(𝐹‘𝑚)) ≤ (∫₂‘𝐺)))
144	132, 143	mpbird 257	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫₂‘(𝐹‘𝑛)))𝑧 ≤ (∫₂‘𝐺))
145		supxrleub 13302	. . . . 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 585	. . . 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 257	. . 3 ⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫₂‘(𝐹‘𝑛))), ℝ^*, < ) ≤ (∫₂‘𝐺))
148	62, 147	eqbrtrid 5183	. 2 ⊢ (𝜑 → 𝑆 ≤ (∫₂‘𝐺))
149	61, 72, 89, 148	xrletrid 13131	1 ⊢ (𝜑 → (∫₂‘𝐺) = 𝑆)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ⊆ wss 3948 ∅c0 4322 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ran crn 5677 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ∘_r cofr 7666 supcsup 9432 ℝcr 11106 0cc0 11107 1c1 11108 + caddc 11110 +∞cpnf 11242 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246 ℕcn 12209 [,)cico 13323 [,]cicc 13324 MblFncmbf 25123 ∫₁citg1 25124 ∫₂citg2 25125

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cc 10427 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-oadd 8467 df-omul 8468 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-dju 9893 df-card 9931 df-acn 9934 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-rlim 15430 df-sum 15630 df-rest 17365 df-topgen 17386 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-top 22388 df-topon 22405 df-bases 22441 df-cmp 22883 df-ovol 24973 df-vol 24974 df-mbf 25128 df-itg1 25129 df-itg2 25130

This theorem is referenced by: itg2i1fseq 25265 itg2cnlem1 25271

W3C validator