Theorem itg2mono 26104

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 13614	. . . . . . . . . . . 12 ⊢ (0[,)+∞) ⊆ ℝ
3		fss 6748	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹‘𝑛):ℝ⟶ℝ)
4	1, 2, 3	sylancl 598	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶ℝ)
5	4	ffvelcdmda 7108	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
6	5	an32s 665	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
7	6	fmpttd 7139	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)):ℕ⟶ℝ)
8	7	frnd 6740	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ⊆ ℝ)
9		1nn 12373	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
10		eqid 2763	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))
11	10, 6	dmmptd 6706	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = ℕ)
12	9, 11	eleqtrrid 2871	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
13	12	ne0d 4291	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅)
14		dm0rn0 5913	. . . . . . . . 9 ⊢ (dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = ∅)
15	14	necon3bii 3011	. . . . . . . 8 ⊢ (dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅)
16	13, 15	sylib 221	. . . . . . 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 6732	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ)
19		breq1 5109	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) → (𝑧 ≤ 𝑦 ↔ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
20	19	ralrn 7112	. . . . . . . . . . 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 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
22		fveq2 6908	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
23	22	fveq1d 6910	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛)‘𝑥) = ((𝐹‘𝑚)‘𝑥))
24		fvex 6921	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑚)‘𝑥) ∈ V
25	23, 10, 24	fvmpt 7016	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥))
26	25	breq1d 5116	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦))
27	26	ralbiia 3110	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦)
28	23	breq1d 5116	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦))
29	28	cbvralvw 3244	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦)
30	27, 29	bitr4i 281	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
31	21, 30	bitrdi 290	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦))
32	31	rexbidv 3190	. . . . . . . 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 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦)
34	8, 16, 33	suprcld 12307	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
35	34	rexrd 11386	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ ℝ*)
36		0red 11338	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
37		fveq2 6908	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1))
38	37	feq1d 6713	. . . . . . . . . 10 ⊢ (𝑛 = 1 → ((𝐹‘𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘1):ℝ⟶(0[,)+∞)))
39	1	ralrimiva 3158	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛):ℝ⟶(0[,)+∞))
40	9	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℕ)
41	38, 39, 40	rspcdva 3581	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘1):ℝ⟶(0[,)+∞))
42	41	ffvelcdmda 7108	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ (0[,)+∞))
43		elrege0 13612	. . . . . . . 8 ⊢ (((𝐹‘1)‘𝑥) ∈ (0[,)+∞) ↔ (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
44	42, 43	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
45	44	simpld 500	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ℝ)
46	44	simprd 501	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ ((𝐹‘1)‘𝑥))
47	37	fveq1d 6910	. . . . . . . . . 10 ⊢ (𝑛 = 1 → ((𝐹‘𝑛)‘𝑥) = ((𝐹‘1)‘𝑥))
48		fvex 6921	. . . . . . . . . 10 ⊢ ((𝐹‘1)‘𝑥) ∈ V
49	47, 10, 48	fvmpt 7016	. . . . . . . . 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 7104	. . . . . . . . 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 598	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
53	50, 52	eqeltrrid 2869	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
54	8, 16, 33, 53	suprubd 12306	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
55	36, 45, 34, 46, 54	letrd 11494	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
56		elxrge0 13615	. . . . 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 595	. . . 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 7138	. . 3 ⊢ (𝜑 → 𝐺:ℝ⟶(0[,]+∞))
60		itg2cl 26083	. . 3 ⊢ (𝐺:ℝ⟶(0[,]+∞) → (∫2‘𝐺) ∈ ℝ*)
61	59, 60	syl 18	. 2 ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ*)
62		itg2mono.6	. . 3 ⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < )
63		icossicc 13594	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
64		fss 6748	. . . . . . . 8 ⊢ (((𝐹‘𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹‘𝑛):ℝ⟶(0[,]+∞))
65	1, 63, 64	sylancl 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,]+∞))
66		itg2cl 26083	. . . . . . 7 ⊢ ((𝐹‘𝑛):ℝ⟶(0[,]+∞) → (∫2‘(𝐹‘𝑛)) ∈ ℝ*)
67	65, 66	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∫2‘(𝐹‘𝑛)) ∈ ℝ*)
68	67	fmpttd 7139	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))):ℕ⟶ℝ*)
69	68	frnd 6740	. . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) ⊆ ℝ*)
70		supxrcl 13472	. . . 4 ⊢ (ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) ⊆ ℝ* → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ∈ ℝ*)
71	69, 70	syl 18	. . 3 ⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ∈ ℝ*)
72	62, 71	eqeltrid 2868	. 2 ⊢ (𝜑 → 𝑆 ∈ ℝ*)
73		itg2mono.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn)
74	73	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn)
75	1	adantlr 728	. . . . . . . 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 728	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))
78	17	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
79		simprll 791	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) → 𝑓 ∈ dom ∫1)
80		simprlr 792	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) → 𝑓 ∘r ≤ 𝐺)
81		simprr 785	. . . . . . . 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 26103	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) → (∫1‘𝑓) ≤ 𝑆)
83	82	expr 462	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺)) → (¬ (∫1‘𝑓) ≤ 𝑆 → (∫1‘𝑓) ≤ 𝑆))
84	83	pm2.18d 128	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘r ≤ 𝐺)) → (∫1‘𝑓) ≤ 𝑆)
85	84	expr 462	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ dom ∫1) → (𝑓 ∘r ≤ 𝐺 → (∫1‘𝑓) ≤ 𝑆))
86	85	ralrimiva 3158	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐺 → (∫1‘𝑓) ≤ 𝑆))
87		itg2leub 26085	. . . 4 ⊢ ((𝐺:ℝ⟶(0[,]+∞) ∧ 𝑆 ∈ ℝ*) → ((∫2‘𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐺 → (∫1‘𝑓) ≤ 𝑆)))
88	59, 72, 87	syl2anc 596	. . 3 ⊢ (𝜑 → ((∫2‘𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓 ∘r ≤ 𝐺 → (∫1‘𝑓) ≤ 𝑆)))
89	86, 88	mpbird 260	. 2 ⊢ (𝜑 → (∫2‘𝐺) ≤ 𝑆)
90	22	feq1d 6713	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘𝑚):ℝ⟶(0[,)+∞)))
91	90	cbvralvw 3244	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ ℕ (𝐹‘𝑛):ℝ⟶(0[,)+∞) ↔ ∀𝑚 ∈ ℕ (𝐹‘𝑚):ℝ⟶(0[,)+∞))
92	39, 91	sylib 221	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝐹‘𝑚):ℝ⟶(0[,)+∞))
93	92	r19.21bi 3258	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚):ℝ⟶(0[,)+∞))
94		fss 6748	. . . . . . . 8 ⊢ (((𝐹‘𝑚):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹‘𝑚):ℝ⟶(0[,]+∞))
95	93, 63, 94	sylancl 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚):ℝ⟶(0[,]+∞))
96	59	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐺:ℝ⟶(0[,]+∞))
97	8, 16, 33	3jca 1146	. . . . . . . . . . . . 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 728	. . . . . . . . . . . 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 740	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥))
100	18	adantlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ)
101		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑚 ∈ ℕ)
102		fnfvelrn 7104	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
104	99, 103	eqeltrrd 2865	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
105		suprub 12305	. . . . . . . . . . . 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 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
107		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
108		ltso 11417	. . . . . . . . . . . . 13 ⊢ < Or ℝ
109	108	supex 9483	. . . . . . . . . . . 12 ⊢ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ V
110	58	fvmpt2 7028	. . . . . . . . . . . 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 598	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
112	106, 111	breqtrrd 5136	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥))
113	112	ralrimiva 3158	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑥 ∈ ℝ ((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥))
114		fveq2 6908	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑧))
115		fveq2 6908	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
116	114, 115	breq12d 5119	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥) ↔ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧)))
117	116	cbvralvw 3244	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℝ ((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥) ↔ ∀𝑧 ∈ ℝ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧))
118	113, 117	sylib 221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑧 ∈ ℝ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧))
119	93	ffnd 6732	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) Fn ℝ)
120	34, 58	fmptd 7138	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
121	120	ffnd 6732	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn ℝ)
122	121	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐺 Fn ℝ)
123		reex 11318	. . . . . . . . . 10 ⊢ ℝ ∈ V
124	123	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ℝ ∈ V)
125		inidm 4126	. . . . . . . . 9 ⊢ (ℝ ∩ ℝ) = ℝ
126		eqidd 2764	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑚)‘𝑧) = ((𝐹‘𝑚)‘𝑧))
127		eqidd 2764	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧))
128	119, 122, 124, 124, 125, 126, 127	ofrfval 7727	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐹‘𝑚) ∘r ≤ 𝐺 ↔ ∀𝑧 ∈ ℝ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧)))
129	118, 128	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∘r ≤ 𝐺)
130		itg2le 26090	. . . . . . 7 ⊢ (((𝐹‘𝑚):ℝ⟶(0[,]+∞) ∧ 𝐺:ℝ⟶(0[,]+∞) ∧ (𝐹‘𝑚) ∘r ≤ 𝐺) → (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))
131	95, 96, 129, 130	syl3anc 1399	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))
132	131	ralrimiva 3158	. . . . 5 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))
133	68	ffnd 6732	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) Fn ℕ)
134		breq1 5109	. . . . . . . 8 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) → (𝑧 ≤ (∫2‘𝐺) ↔ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺)))
135	134	ralrn 7112	. . . . . . 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 18	. . . . . 6 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺) ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺)))
137		2fveq3 6913	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (∫2‘(𝐹‘𝑛)) = (∫2‘(𝐹‘𝑚)))
138		eqid 2763	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))
139		fvex 6921	. . . . . . . . 9 ⊢ (∫2‘(𝐹‘𝑚)) ∈ V
140	137, 138, 139	fvmpt 7016	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) = (∫2‘(𝐹‘𝑚)))
141	140	breq1d 5116	. . . . . . 7 ⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺) ↔ (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺)))
142	141	ralbiia 3110	. . . . . 6 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))
143	136, 142	bitrdi 290	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺)))
144	132, 143	mpbird 260	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺))
145		supxrleub 13483	. . . . 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 596	. . . 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 260	. . 3 ⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ≤ (∫2‘𝐺))
148	62, 147	eqbrtrid 5143	. 2 ⊢ (𝜑 → 𝑆 ≤ (∫2‘𝐺))
149	61, 72, 89, 148	xrletrid 13311	1 ⊢ (𝜑 → (∫2‘𝐺) = 𝑆)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  ∅c0 4281   class class class wbr 5106   ↦ cmpt 5189  dom cdm 5655  ran crn 5656   Fn wfn 6553  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443   ∘_r cofr 7716  supcsup 9459  ℝcr 11226  0cc0 11227  1c1 11228   + caddc 11230  +∞cpnf 11367  ℝ^*cxr 11369   < clt 11370   ≤ cle 11371  ℕcn 12362  [,)cico 13505  [,]cicc 13506  MblFncmbf 25965  ∫₁citg1 25966  ∫₂citg2 25967

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-inf2 9669  ax-cc 10540  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304  ax-pre-sup 11305

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-disj 5074  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7717  df-ofr 7718  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-oadd 8508  df-omul 8509  df-er 8745  df-map 8877  df-pm 8878  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-fi 9430  df-sup 9461  df-inf 9462  df-oi 9531  df-dju 10009  df-card 10047  df-acn 10050  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-div 12001  df-nn 12363  df-2 12432  df-3 12433  df-n0 12634  df-z 12721  df-uz 12993  df-q 13103  df-rp 13148  df-xneg 13268  df-xadd 13269  df-xmul 13270  df-ioo 13507  df-ioc 13508  df-ico 13509  df-icc 13510  df-fz 13667  df-fzo 13816  df-fl 13959  df-seq 14172  df-exp 14232  df-hash 14502  df-cj 15295  df-re 15296  df-im 15297  df-sqrt 15431  df-abs 15432  df-clim 15684  df-rlim 15685  df-sum 15883  df-rest 17622  df-topgen 17643  df-psmet 21700  df-xmet 21701  df-met 21702  df-bl 21703  df-mopn 21704  df-top 23242  df-topon 23259  df-bases 23294  df-cmp 23735  df-ovol 25815  df-vol 25816  df-mbf 25970  df-itg1 25971  df-itg2 25972

This theorem is used by:  itg2i1fseq  26106  itg2cnlem1  26112