Theorem itg2mono 23326

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.1	. . . . . . . 8 ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
2		itg2mono.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn)
3	2	adantlr 747	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn)
4		itg2mono.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞))
5	4	adantlr 747	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞))
6		itg2mono.4	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1)))
7	6	adantlr 747	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘𝑟 ≤ (𝐹‘(𝑛 + 1)))
8		itg2mono.5	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
9	8	adantlr 747	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
10		itg2mono.6	. . . . . . . 8 ⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < )
11		simprll 798	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) → 𝑓 ∈ dom ∫1)
12		simprlr 799	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) → 𝑓 ∘𝑟 ≤ 𝐺)
13		simprr 792	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) → ¬ (∫1‘𝑓) ≤ 𝑆)
14	1, 3, 5, 7, 9, 10, 11, 12, 13	itg2monolem3 23325	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐺) ∧ ¬ (∫1‘𝑓) ≤ 𝑆)) → (∫1‘𝑓) ≤ 𝑆)
15	14	expr 641	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐺)) → (¬ (∫1‘𝑓) ≤ 𝑆 → (∫1‘𝑓) ≤ 𝑆))
16	15	pm2.18d 123	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑓 ∘𝑟 ≤ 𝐺)) → (∫1‘𝑓) ≤ 𝑆)
17	16	expr 641	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ dom ∫1) → (𝑓 ∘𝑟 ≤ 𝐺 → (∫1‘𝑓) ≤ 𝑆))
18	17	ralrimiva 2949	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐺 → (∫1‘𝑓) ≤ 𝑆))
19		rge0ssre 12151	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ ℝ
20		fss 5969	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → (𝐹‘𝑛):ℝ⟶ℝ)
21	4, 19, 20	sylancl 693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶ℝ)
22	21	ffvelrnda 6267	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
23	22	an32s 842	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ)
24		eqid 2610	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))
25	23, 24	fmptd 6292	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)):ℕ⟶ℝ)
26		frn 5966	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)):ℕ⟶ℝ → ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ⊆ ℝ)
27	25, 26	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ⊆ ℝ)
28		1nn 10908	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ
29	24, 23	dmmptd 5937	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = ℕ)
30	28, 29	syl5eleqr 2695	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
31		ne0i 3880	. . . . . . . . . 10 ⊢ (1 ∈ dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) → dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅)
32	30, 31	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅)
33		dm0rn0 5263	. . . . . . . . . 10 ⊢ (dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) = ∅)
34	33	necon3bii 2834	. . . . . . . . 9 ⊢ (dom (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅ ↔ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅)
35	32, 34	sylib 207	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅)
36		ffn 5958	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)):ℕ⟶ℝ → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ)
37	25, 36	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ)
38		breq1 4586	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) → (𝑧 ≤ 𝑦 ↔ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
39	38	ralrn 6270	. . . . . . . . . . . 12 ⊢ ((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))
40	37, 39	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦))
41		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
42	41	fveq1d 6105	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛)‘𝑥) = ((𝐹‘𝑚)‘𝑥))
43		fvex 6113	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑚)‘𝑥) ∈ V
44	42, 24, 43	fvmpt 6191	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥))
45	44	breq1d 4593	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦))
46	45	ralbiia 2962	. . . . . . . . . . . 12 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦)
47	42	breq1d 4593	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦))
48	47	cbvralv 3147	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑚 ∈ ℕ ((𝐹‘𝑚)‘𝑥) ≤ 𝑦)
49	46, 48	bitr4i 266	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
50	40, 49	syl6bb 275	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦 ↔ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦))
51	50	rexbidv 3034	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦))
52	8, 51	mpbird 246	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦)
53		suprcl 10862	. . . . . . . 8 ⊢ ((ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))𝑧 ≤ 𝑦) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
54	27, 35, 52, 53	syl3anc 1318	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ ℝ)
55	54	rexrd 9968	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ ℝ*)
56		0red 9920	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
57	4	ralrimiva 2949	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛):ℝ⟶(0[,)+∞))
58		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1))
59	58	feq1d 5943	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → ((𝐹‘𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘1):ℝ⟶(0[,)+∞)))
60	59	rspcv 3278	. . . . . . . . . . 11 ⊢ (1 ∈ ℕ → (∀𝑛 ∈ ℕ (𝐹‘𝑛):ℝ⟶(0[,)+∞) → (𝐹‘1):ℝ⟶(0[,)+∞)))
61	28, 57, 60	mpsyl 66	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘1):ℝ⟶(0[,)+∞))
62	61	ffvelrnda 6267	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ (0[,)+∞))
63		elrege0 12149	. . . . . . . . 9 ⊢ (((𝐹‘1)‘𝑥) ∈ (0[,)+∞) ↔ (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
64	62, 63	sylib 207	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (((𝐹‘1)‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝐹‘1)‘𝑥)))
65	64	simpld 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ℝ)
66	64	simprd 478	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ ((𝐹‘1)‘𝑥))
67	58	fveq1d 6105	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → ((𝐹‘𝑛)‘𝑥) = ((𝐹‘1)‘𝑥))
68		fvex 6113	. . . . . . . . . . 11 ⊢ ((𝐹‘1)‘𝑥) ∈ V
69	67, 24, 68	fvmpt 6191	. . . . . . . . . 10 ⊢ (1 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘1) = ((𝐹‘1)‘𝑥))
70	28, 69	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘1) = ((𝐹‘1)‘𝑥)
71		fnfvelrn 6264	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ ∧ 1 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
72	37, 28, 71	sylancl 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘1) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
73	70, 72	syl5eqelr 2693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
74		suprub 10863	. . . . . . . 8 ⊢ (((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‘1)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))) → ((𝐹‘1)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
75	27, 35, 52, 73, 74	syl31anc 1321	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘1)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
76	56, 65, 54, 66, 75	letrd 10073	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
77		elxrge0 12152	. . . . . 6 ⊢ (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, < )))
78	55, 76, 77	sylanbrc 695	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ (0[,]+∞))
79	78, 1	fmptd 6292	. . . 4 ⊢ (𝜑 → 𝐺:ℝ⟶(0[,]+∞))
80		icossicc 12131	. . . . . . . . . 10 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
81		fss 5969	. . . . . . . . . 10 ⊢ (((𝐹‘𝑛):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹‘𝑛):ℝ⟶(0[,]+∞))
82	4, 80, 81	sylancl 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,]+∞))
83		itg2cl 23305	. . . . . . . . 9 ⊢ ((𝐹‘𝑛):ℝ⟶(0[,]+∞) → (∫2‘(𝐹‘𝑛)) ∈ ℝ*)
84	82, 83	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∫2‘(𝐹‘𝑛)) ∈ ℝ*)
85		eqid 2610	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))
86	84, 85	fmptd 6292	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))):ℕ⟶ℝ*)
87		frn 5966	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))):ℕ⟶ℝ* → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) ⊆ ℝ*)
88	86, 87	syl 17	. . . . . 6 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) ⊆ ℝ*)
89		supxrcl 12017	. . . . . 6 ⊢ (ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) ⊆ ℝ* → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ∈ ℝ*)
90	88, 89	syl 17	. . . . 5 ⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ∈ ℝ*)
91	10, 90	syl5eqel 2692	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℝ*)
92		itg2leub 23307	. . . 4 ⊢ ((𝐺:ℝ⟶(0[,]+∞) ∧ 𝑆 ∈ ℝ*) → ((∫2‘𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐺 → (∫1‘𝑓) ≤ 𝑆)))
93	79, 91, 92	syl2anc 691	. . 3 ⊢ (𝜑 → ((∫2‘𝐺) ≤ 𝑆 ↔ ∀𝑓 ∈ dom ∫1(𝑓 ∘𝑟 ≤ 𝐺 → (∫1‘𝑓) ≤ 𝑆)))
94	18, 93	mpbird 246	. 2 ⊢ (𝜑 → (∫2‘𝐺) ≤ 𝑆)
95	41	feq1d 5943	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛):ℝ⟶(0[,)+∞) ↔ (𝐹‘𝑚):ℝ⟶(0[,)+∞)))
96	95	cbvralv 3147	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ ℕ (𝐹‘𝑛):ℝ⟶(0[,)+∞) ↔ ∀𝑚 ∈ ℕ (𝐹‘𝑚):ℝ⟶(0[,)+∞))
97	57, 96	sylib 207	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (𝐹‘𝑚):ℝ⟶(0[,)+∞))
98	97	r19.21bi 2916	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚):ℝ⟶(0[,)+∞))
99		fss 5969	. . . . . . . 8 ⊢ (((𝐹‘𝑚):ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → (𝐹‘𝑚):ℝ⟶(0[,]+∞))
100	98, 80, 99	sylancl 693	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚):ℝ⟶(0[,]+∞))
101	79	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐺:ℝ⟶(0[,]+∞))
102	27, 35, 52	3jca 1235	. . . . . . . . . . . . 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))
103	102	adantlr 747	. . . . . . . . . . . 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))
104	44	ad2antlr 759	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥))
105	37	adantlr 747	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ)
106		simplr 788	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑚 ∈ ℕ)
107		fnfvelrn 6264	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)) Fn ℕ ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
108	105, 106, 107	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
109	104, 108	eqeltrrd 2689	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ∈ ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)))
110		suprub 10863	. . . . . . . . . . . 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, < ))
111	103, 109, 110	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
112		simpr 476	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
113		ltso 9997	. . . . . . . . . . . . 13 ⊢ < Or ℝ
114	113	supex 8252	. . . . . . . . . . . 12 ⊢ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ V
115	1	fvmpt2 6200	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ V) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
116	112, 114, 115	sylancl 693	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐺‘𝑥) = sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
117	111, 116	breqtrrd 4611	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥))
118	117	ralrimiva 2949	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑥 ∈ ℝ ((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥))
119		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑧))
120		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
121	119, 120	breq12d 4596	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥) ↔ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧)))
122	121	cbvralv 3147	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ℝ ((𝐹‘𝑚)‘𝑥) ≤ (𝐺‘𝑥) ↔ ∀𝑧 ∈ ℝ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧))
123	118, 122	sylib 207	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∀𝑧 ∈ ℝ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧))
124		ffn 5958	. . . . . . . . . 10 ⊢ ((𝐹‘𝑚):ℝ⟶(0[,]+∞) → (𝐹‘𝑚) Fn ℝ)
125	100, 124	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) Fn ℝ)
126	54, 1	fmptd 6292	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
127		ffn 5958	. . . . . . . . . . 11 ⊢ (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
128	126, 127	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn ℝ)
129	128	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐺 Fn ℝ)
130		reex 9906	. . . . . . . . . 10 ⊢ ℝ ∈ V
131	130	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ℝ ∈ V)
132		inidm 3784	. . . . . . . . 9 ⊢ (ℝ ∩ ℝ) = ℝ
133		eqidd 2611	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑚)‘𝑧) = ((𝐹‘𝑚)‘𝑧))
134		eqidd 2611	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑧 ∈ ℝ) → (𝐺‘𝑧) = (𝐺‘𝑧))
135	125, 129, 131, 131, 132, 133, 134	ofrfval 6803	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐹‘𝑚) ∘𝑟 ≤ 𝐺 ↔ ∀𝑧 ∈ ℝ ((𝐹‘𝑚)‘𝑧) ≤ (𝐺‘𝑧)))
136	123, 135	mpbird 246	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∘𝑟 ≤ 𝐺)
137		itg2le 23312	. . . . . . 7 ⊢ (((𝐹‘𝑚):ℝ⟶(0[,]+∞) ∧ 𝐺:ℝ⟶(0[,]+∞) ∧ (𝐹‘𝑚) ∘𝑟 ≤ 𝐺) → (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))
138	100, 101, 136, 137	syl3anc 1318	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))
139	138	ralrimiva 2949	. . . . 5 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))
140		ffn 5958	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))):ℕ⟶ℝ* → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) Fn ℕ)
141	86, 140	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))) Fn ℕ)
142		breq1 4586	. . . . . . . 8 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) → (𝑧 ≤ (∫2‘𝐺) ↔ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺)))
143	142	ralrn 6270	. . . . . . 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)))
144	141, 143	syl 17	. . . . . 6 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺) ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺)))
145	41	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (∫2‘(𝐹‘𝑛)) = (∫2‘(𝐹‘𝑚)))
146		fvex 6113	. . . . . . . . 9 ⊢ (∫2‘(𝐹‘𝑚)) ∈ V
147	145, 85, 146	fvmpt 6191	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) = (∫2‘(𝐹‘𝑚)))
148	147	breq1d 4593	. . . . . . 7 ⊢ (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺) ↔ (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺)))
149	148	ralbiia 2962	. . . . . 6 ⊢ (∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))‘𝑚) ≤ (∫2‘𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺))
150	144, 149	syl6bb 275	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺) ↔ ∀𝑚 ∈ ℕ (∫2‘(𝐹‘𝑚)) ≤ (∫2‘𝐺)))
151	139, 150	mpbird 246	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺))
152		itg2cl 23305	. . . . . 6 ⊢ (𝐺:ℝ⟶(0[,]+∞) → (∫2‘𝐺) ∈ ℝ*)
153	79, 152	syl 17	. . . . 5 ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ*)
154		supxrleub 12028	. . . . 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)))
155	88, 153, 154	syl2anc 691	. . . 4 ⊢ (𝜑 → (sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ≤ (∫2‘𝐺) ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛)))𝑧 ≤ (∫2‘𝐺)))
156	151, 155	mpbird 246	. . 3 ⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < ) ≤ (∫2‘𝐺))
157	10, 156	syl5eqbr 4618	. 2 ⊢ (𝜑 → 𝑆 ≤ (∫2‘𝐺))
158		xrletri3 11861	. . 3 ⊢ (((∫2‘𝐺) ∈ ℝ* ∧ 𝑆 ∈ ℝ*) → ((∫2‘𝐺) = 𝑆 ↔ ((∫2‘𝐺) ≤ 𝑆 ∧ 𝑆 ≤ (∫2‘𝐺))))
159	153, 91, 158	syl2anc 691	. 2 ⊢ (𝜑 → ((∫2‘𝐺) = 𝑆 ↔ ((∫2‘𝐺) ≤ 𝑆 ∧ 𝑆 ≤ (∫2‘𝐺))))
160	94, 157, 159	mpbir2and 959	1 ⊢ (𝜑 → (∫2‘𝐺) = 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘_𝑟 cofr 6794  supcsup 8229  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818  +∞cpnf 9950  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954  ℕcn 10897  [,)cico 12048  [,]cicc 12049  MblFncmbf 23189  ∫₁citg1 23190  ∫₂citg2 23191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cc 9140  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-mbf 23194  df-itg1 23195  df-itg2 23196

This theorem is referenced by:  itg2i1fseq  23328  itg2cnlem1  23334