Theorem itg2gt0 25752

Description: If the function 𝐹 is strictly positive on a set of positive measure, then the integral of the function is positive. (Contributed by Mario Carneiro, 30-Aug-2014.)

Hypotheses

Ref	Expression
itg2gt0.1	⊢ (𝜑 → 𝐴 ∈ dom vol)
itg2gt0.2	⊢ (𝜑 → 0 < (vol‘𝐴))
itg2gt0.3	⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
itg2gt0.4	⊢ (𝜑 → 𝐹 ∈ MblFn)
itg2gt0.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 < (𝐹‘𝑥))

Assertion

Ref	Expression
itg2gt0	⊢ (𝜑 → 0 < (∫2‘𝐹))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝜑,𝑥

Proof of Theorem itg2gt0

Dummy variables 𝑘 𝑛 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		itg2gt0.2	. 2 ⊢ (𝜑 → 0 < (vol‘𝐴))
2		itg2gt0.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ dom vol)
3		iccssxr 13381	. . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ*
4		volf 25521	. . . . . . . . 9 ⊢ vol:dom vol⟶(0[,]+∞)
5	4	ffvelcdmi 7031	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) ∈ (0[,]+∞))
6	3, 5	sselid 3920	. . . . . . 7 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) ∈ ℝ*)
7	2, 6	syl 17	. . . . . 6 ⊢ (𝜑 → (vol‘𝐴) ∈ ℝ*)
8	7	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → (vol‘𝐴) ∈ ℝ*)
9		itg2gt0.4	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 ∈ MblFn)
10	9	elexd 3456	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∈ V)
11		cnvexg 7871	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
12	10, 11	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ◡𝐹 ∈ V)
13		imaexg 7860	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) ∈ V)
14	12, 13	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) ∈ V)
15	14	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) ∈ V)
16	15	fmpttd 7063	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))):ℕ⟶V)
17	16	ffnd 6663	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) Fn ℕ)
18		fniunfv 7198	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) Fn ℕ → ∪ 𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) = ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))))
19	17, 18	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) = ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))))
20		itg2gt0.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))
21		rge0ssre 13407	. . . . . . . . . . . . . . . 16 ⊢ (0[,)+∞) ⊆ ℝ
22		fss 6678	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐹:ℝ⟶ℝ)
23	20, 21, 22	sylancl 592	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
24		mbfima 25622	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ) → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) ∈ dom vol)
25	9, 23, 24	syl2anc 590	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) ∈ dom vol)
26	25	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) ∈ dom vol)
27	26	fmpttd 7063	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))):ℕ⟶dom vol)
28	27	ffvelcdmda 7032	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ∈ dom vol)
29	28	ralrimiva 3132	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ∈ dom vol)
30		iunmbl 25545	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ∈ dom vol → ∪ 𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ∈ dom vol)
31	29, 30	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ∈ dom vol)
32	19, 31	eqeltrrd 2841	. . . . . . . 8 ⊢ (𝜑 → ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ∈ dom vol)
33		mblss 25523	. . . . . . . 8 ⊢ (∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ∈ dom vol → ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ⊆ ℝ)
34	32, 33	syl 17	. . . . . . 7 ⊢ (𝜑 → ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ⊆ ℝ)
35		ovolcl 25470	. . . . . . 7 ⊢ (∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ⊆ ℝ → (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) ∈ ℝ*)
36	34, 35	syl 17	. . . . . 6 ⊢ (𝜑 → (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) ∈ ℝ*)
37	36	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) ∈ ℝ*)
38		0xr 11190	. . . . . 6 ⊢ 0 ∈ ℝ*
39	38	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → 0 ∈ ℝ*)
40		mblvol 25522	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) = (vol*‘𝐴))
41	2, 40	syl 17	. . . . . . 7 ⊢ (𝜑 → (vol‘𝐴) = (vol*‘𝐴))
42		mblss 25523	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
43	2, 42	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
44	43	sselda 3922	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
45	20	ffvelcdmda 7032	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ (0[,)+∞))
46		elrege0 13405	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
47	45, 46	sylib 219	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
48	47	simpld 495	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
49	44, 48	syldan 597	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ)
50		itg2gt0.5	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 < (𝐹‘𝑥))
51		nnrecl 12433	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) ∈ ℝ ∧ 0 < (𝐹‘𝑥)) → ∃𝑘 ∈ ℕ (1 / 𝑘) < (𝐹‘𝑥))
52	49, 50, 51	syl2anc 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑘 ∈ ℕ (1 / 𝑘) < (𝐹‘𝑥))
53	20	ffnd 6663	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 Fn ℝ)
54	53	ad2antrr 732	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → 𝐹 Fn ℝ)
55		elpreima 7006	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 Fn ℝ → (𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ((1 / 𝑘)(,)+∞))))
56	54, 55	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ((1 / 𝑘)(,)+∞))))
57	44	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
58	57	biantrurd 537	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥) ∈ ((1 / 𝑘)(,)+∞) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ((1 / 𝑘)(,)+∞))))
59		nnrecre 12217	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
60	59	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
61	60	rexrd 11193	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ*)
62	61	adantlr 721	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ*)
63		elioopnf 13394	. . . . . . . . . . . . . . . 16 ⊢ ((1 / 𝑘) ∈ ℝ* → ((𝐹‘𝑥) ∈ ((1 / 𝑘)(,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ (1 / 𝑘) < (𝐹‘𝑥))))
64	62, 63	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥) ∈ ((1 / 𝑘)(,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ (1 / 𝑘) < (𝐹‘𝑥))))
65	56, 58, 64	3bitr2d 308	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ (1 / 𝑘) < (𝐹‘𝑥))))
66		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ)
67		imaexg 7860	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ V)
68	12, 67	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ V)
69	68	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ V)
70		oveq2 7371	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘))
71	70	oveq1d 7378	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → ((1 / 𝑛)(,)+∞) = ((1 / 𝑘)(,)+∞))
72	71	imaeq2d 6019	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) = (◡𝐹 “ ((1 / 𝑘)(,)+∞)))
73		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) = (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))
74	72, 73	fvmptg 6940	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ ∧ (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ V) → ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) = (◡𝐹 “ ((1 / 𝑘)(,)+∞)))
75	66, 69, 74	syl2anr 603	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) = (◡𝐹 “ ((1 / 𝑘)(,)+∞)))
76	75	eleq2d 2826	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ↔ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))))
77	49	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑥) ∈ ℝ)
78	77	biantrurd 537	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → ((1 / 𝑘) < (𝐹‘𝑥) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ (1 / 𝑘) < (𝐹‘𝑥))))
79	65, 76, 78	3bitr4rd 313	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → ((1 / 𝑘) < (𝐹‘𝑥) ↔ 𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
80	79	rexbidva 3162	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑘 ∈ ℕ (1 / 𝑘) < (𝐹‘𝑥) ↔ ∃𝑘 ∈ ℕ 𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
81	52, 80	mpbid 233	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑘 ∈ ℕ 𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘))
82	81	ex 413	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ∃𝑘 ∈ ℕ 𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
83		eluni2 4849	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ↔ ∃𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))𝑥 ∈ 𝑧)
84		eleq2 2829	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
85	84	rexrn 7035	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) Fn ℕ → (∃𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))𝑥 ∈ 𝑧 ↔ ∃𝑘 ∈ ℕ 𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
86	17, 85	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))𝑥 ∈ 𝑧 ↔ ∃𝑘 ∈ ℕ 𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
87	83, 86	bitrid 284	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ↔ ∃𝑘 ∈ ℕ 𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
88	82, 87	sylibrd 260	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))))
89	88	ssrdv 3928	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))))
90		ovolss 25477	. . . . . . . 8 ⊢ ((𝐴 ⊆ ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ∧ ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ⊆ ℝ) → (vol*‘𝐴) ≤ (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))))
91	89, 34, 90	syl2anc 590	. . . . . . 7 ⊢ (𝜑 → (vol*‘𝐴) ≤ (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))))
92	41, 91	eqbrtrd 5101	. . . . . 6 ⊢ (𝜑 → (vol‘𝐴) ≤ (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))))
93	92	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → (vol‘𝐴) ≤ (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))))
94		mblvol 25522	. . . . . . . . 9 ⊢ (∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ∈ dom vol → (vol‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) = (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))))
95	32, 94	syl 17	. . . . . . . 8 ⊢ (𝜑 → (vol‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) = (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))))
96		peano2nn 12184	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
97	96	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
98		nnrecre 12217	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 1) ∈ ℕ → (1 / (𝑘 + 1)) ∈ ℝ)
99	97, 98	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℝ)
100	99	rexrd 11193	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℝ*)
101		nnre 12179	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
102	101	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
103	102	lep1d 12085	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ (𝑘 + 1))
104		nngt0 12206	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 0 < 𝑘)
105	104	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < 𝑘)
106	97	nnred 12187	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℝ)
107	97	nngt0d 12224	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < (𝑘 + 1))
108		lerec 12037	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ℝ ∧ 0 < 𝑘) ∧ ((𝑘 + 1) ∈ ℝ ∧ 0 < (𝑘 + 1))) → (𝑘 ≤ (𝑘 + 1) ↔ (1 / (𝑘 + 1)) ≤ (1 / 𝑘)))
109	102, 105, 106, 107, 108	syl22anc 844	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 ≤ (𝑘 + 1) ↔ (1 / (𝑘 + 1)) ≤ (1 / 𝑘)))
110	103, 109	mpbid 233	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ≤ (1 / 𝑘))
111		iooss1 13331	. . . . . . . . . . . . 13 ⊢ (((1 / (𝑘 + 1)) ∈ ℝ* ∧ (1 / (𝑘 + 1)) ≤ (1 / 𝑘)) → ((1 / 𝑘)(,)+∞) ⊆ ((1 / (𝑘 + 1))(,)+∞))
112	100, 110, 111	syl2anc 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1 / 𝑘)(,)+∞) ⊆ ((1 / (𝑘 + 1))(,)+∞))
113		imass2 6061	. . . . . . . . . . . 12 ⊢ (((1 / 𝑘)(,)+∞) ⊆ ((1 / (𝑘 + 1))(,)+∞) → (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ⊆ (◡𝐹 “ ((1 / (𝑘 + 1))(,)+∞)))
114	112, 113	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ⊆ (◡𝐹 “ ((1 / (𝑘 + 1))(,)+∞)))
115	66, 68, 74	syl2anr 603	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) = (◡𝐹 “ ((1 / 𝑘)(,)+∞)))
116		imaexg 7860	. . . . . . . . . . . . 13 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ ((1 / (𝑘 + 1))(,)+∞)) ∈ V)
117	12, 116	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (◡𝐹 “ ((1 / (𝑘 + 1))(,)+∞)) ∈ V)
118		oveq2 7371	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑘 + 1) → (1 / 𝑛) = (1 / (𝑘 + 1)))
119	118	oveq1d 7378	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑘 + 1) → ((1 / 𝑛)(,)+∞) = ((1 / (𝑘 + 1))(,)+∞))
120	119	imaeq2d 6019	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑘 + 1) → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) = (◡𝐹 “ ((1 / (𝑘 + 1))(,)+∞)))
121	120, 73	fvmptg 6940	. . . . . . . . . . . 12 ⊢ (((𝑘 + 1) ∈ ℕ ∧ (◡𝐹 “ ((1 / (𝑘 + 1))(,)+∞)) ∈ V) → ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘(𝑘 + 1)) = (◡𝐹 “ ((1 / (𝑘 + 1))(,)+∞)))
122	96, 117, 121	syl2anr 603	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘(𝑘 + 1)) = (◡𝐹 “ ((1 / (𝑘 + 1))(,)+∞)))
123	114, 115, 122	3sstr4d 3977	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ⊆ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘(𝑘 + 1)))
124	123	ralrimiva 3132	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ⊆ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘(𝑘 + 1)))
125		volsup 25548	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))):ℕ⟶dom vol ∧ ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ⊆ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘(𝑘 + 1))) → (vol‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) = sup((vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))), ℝ*, < ))
126	27, 124, 125	syl2anc 590	. . . . . . . 8 ⊢ (𝜑 → (vol‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) = sup((vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))), ℝ*, < ))
127	95, 126	eqtr3d 2777	. . . . . . 7 ⊢ (𝜑 → (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) = sup((vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))), ℝ*, < ))
128	127	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) = sup((vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))), ℝ*, < ))
129	68	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ V)
130	66, 129, 74	syl2anr 603	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) = (◡𝐹 “ ((1 / 𝑘)(,)+∞)))
131	130	fveq2d 6838	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) ∧ 𝑘 ∈ ℕ) → (vol‘((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)) = (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))
132	38	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → 0 ∈ ℝ*)
133		nnrecgt0 12218	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ ℕ → 0 < (1 / 𝑘))
134	133	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < (1 / 𝑘))
135		0re 11144	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ ℝ
136		ltle 11232	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((0 ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) → (0 < (1 / 𝑘) → 0 ≤ (1 / 𝑘)))
137	135, 60, 136	sylancr 593	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (0 < (1 / 𝑘) → 0 ≤ (1 / 𝑘)))
138	134, 137	mpd 15	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (1 / 𝑘))
139		elxrge0 13408	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 / 𝑘) ∈ (0[,]+∞) ↔ ((1 / 𝑘) ∈ ℝ* ∧ 0 ≤ (1 / 𝑘)))
140	61, 138, 139	sylanbrc 589	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ (0[,]+∞))
141		0e0iccpnf 13410	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ (0[,]+∞)
142		ifcl 4507	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1 / 𝑘) ∈ (0[,]+∞) ∧ 0 ∈ (0[,]+∞)) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ∈ (0[,]+∞))
143	140, 141, 142	sylancl 592	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ∈ (0[,]+∞))
144	143	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ∈ (0[,]+∞))
145	144	fmpttd 7063	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)):ℝ⟶(0[,]+∞))
146	145	adantrr 723	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)):ℝ⟶(0[,]+∞))
147		itg2cl 25724	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)):ℝ⟶(0[,]+∞) → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))) ∈ ℝ*)
148	146, 147	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))) ∈ ℝ*)
149		icossicc 13387	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
150		fss 6678	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℝ⟶(0[,]+∞))
151	20, 149, 150	sylancl 592	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,]+∞))
152		itg2cl 25724	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) ∈ ℝ*)
153	151, 152	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ*)
154	153	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (∫2‘𝐹) ∈ ℝ*)
155		0nrp 12977	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 0 ∈ ℝ+
156		simpr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) ∧ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)))) → 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))))
157	115, 28	eqeltrrd 2841	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ dom vol)
158	157	adantrr 723	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ dom vol)
159	158	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) ∧ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)))) → (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ dom vol)
160	156, 135	eqeltrrdi 2849	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) ∧ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)))) → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))) ∈ ℝ)
161	60, 134	elrpd 12981	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ+)
162	161	adantrr 723	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (1 / 𝑘) ∈ ℝ+)
163	162	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) ∧ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)))) → (1 / 𝑘) ∈ ℝ+)
164		itg2const2 25733	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ dom vol ∧ (1 / 𝑘) ∈ ℝ+) → ((vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ∈ ℝ ↔ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))) ∈ ℝ))
165	159, 163, 164	syl2anc 590	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) ∧ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)))) → ((vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ∈ ℝ ↔ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))) ∈ ℝ))
166	160, 165	mpbird 258	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) ∧ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)))) → (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ∈ ℝ)
167		elrege0 13405	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((1 / 𝑘) ∈ (0[,)+∞) ↔ ((1 / 𝑘) ∈ ℝ ∧ 0 ≤ (1 / 𝑘)))
168	60, 138, 167	sylanbrc 589	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ (0[,)+∞))
169	168	adantrr 723	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (1 / 𝑘) ∈ (0[,)+∞))
170	169	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) ∧ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)))) → (1 / 𝑘) ∈ (0[,)+∞))
171		itg2const 25732	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ dom vol ∧ (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ∈ ℝ ∧ (1 / 𝑘) ∈ (0[,)+∞)) → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))) = ((1 / 𝑘) · (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞)))))
172	159, 166, 170, 171	syl3anc 1379	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) ∧ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)))) → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))) = ((1 / 𝑘) · (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞)))))
173	156, 172	eqtrd 2775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) ∧ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)))) → 0 = ((1 / 𝑘) · (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞)))))
174		simplrr 783	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) ∧ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)))) → 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))
175	166, 174	elrpd 12981	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) ∧ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)))) → (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ∈ ℝ+)
176	163, 175	rpmulcld 13000	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) ∧ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)))) → ((1 / 𝑘) · (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞)))) ∈ ℝ+)
177	173, 176	eqeltrd 2840	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) ∧ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)))) → 0 ∈ ℝ+)
178	177	ex 413	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))) → 0 ∈ ℝ+))
179	155, 178	mtoi 200	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → ¬ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))))
180		itg2ge0 25727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)):ℝ⟶(0[,]+∞) → 0 ≤ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))))
181	146, 180	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → 0 ≤ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))))
182		xrleloe 13093	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 ∈ ℝ* ∧ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))) ∈ ℝ*) → (0 ≤ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))) ↔ (0 < (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))) ∨ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))))))
183	38, 148, 182	sylancr 593	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (0 ≤ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))) ↔ (0 < (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))) ∨ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))))))
184	181, 183	mpbid 233	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (0 < (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))) ∨ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)))))
185	184	ord 870	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (¬ 0 < (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))) → 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)))))
186	179, 185	mt3d 148	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → 0 < (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))))
187	151	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → 𝐹:ℝ⟶(0[,]+∞))
188	60	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → (1 / 𝑘) ∈ ℝ)
189	53	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹 Fn ℝ)
190	189, 55	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ((1 / 𝑘)(,)+∞))))
191	190	biimpa 477	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ((1 / 𝑘)(,)+∞)))
192	191	simpld 495	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → 𝑥 ∈ ℝ)
193	48	adantlr 721	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
194	192, 193	syldan 597	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → (𝐹‘𝑥) ∈ ℝ)
195	61	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → (1 / 𝑘) ∈ ℝ*)
196	191	simprd 496	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → (𝐹‘𝑥) ∈ ((1 / 𝑘)(,)+∞))
197		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐹‘𝑥) ∈ ℝ ∧ (1 / 𝑘) < (𝐹‘𝑥)) → (1 / 𝑘) < (𝐹‘𝑥))
198	63, 197	biimtrdi 254	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((1 / 𝑘) ∈ ℝ* → ((𝐹‘𝑥) ∈ ((1 / 𝑘)(,)+∞) → (1 / 𝑘) < (𝐹‘𝑥)))
199	195, 196, 198	sylc 65	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → (1 / 𝑘) < (𝐹‘𝑥))
200	188, 194, 199	ltled 11292	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → (1 / 𝑘) ≤ (𝐹‘𝑥))
201	47	simprd 496	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ (𝐹‘𝑥))
202	201	adantlr 721	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ (𝐹‘𝑥))
203	192, 202	syldan 597	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → 0 ≤ (𝐹‘𝑥))
204		breq1 5082	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((1 / 𝑘) = if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) → ((1 / 𝑘) ≤ (𝐹‘𝑥) ↔ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥)))
205		breq1 5082	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 = if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) → (0 ≤ (𝐹‘𝑥) ↔ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥)))
206	204, 205	ifboth 4501	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((1 / 𝑘) ≤ (𝐹‘𝑥) ∧ 0 ≤ (𝐹‘𝑥)) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥))
207	200, 203, 206	syl2anc 590	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥))
208	207	adantlr 721	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥))
209		iffalse 4470	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) = 0)
210	209	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) = 0)
211	202	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → 0 ≤ (𝐹‘𝑥))
212	210, 211	eqbrtrd 5101	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥))
213	208, 212	pm2.61dan 818	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥))
214	213	ralrimiva 3132	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑥 ∈ ℝ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥))
215	214	adantrr 723	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → ∀𝑥 ∈ ℝ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥))
216		reex 11127	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℝ ∈ V
217	216	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ℝ ∈ V)
218		ovex 7396	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 / 𝑘) ∈ V
219		c0ex 11136	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ V
220	218, 219	ifex 4512	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ∈ V
221	220	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ∈ V)
222		fvexd 6849	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ V)
223		eqidd 2741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)))
224	20	feqmptd 6902	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
225	217, 221, 222, 223, 224	ofrfval2 7648	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)) ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ ℝ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥)))
226	225	biimpar 478	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥)) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)) ∘r ≤ 𝐹)
227	215, 226	syldan 597	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)) ∘r ≤ 𝐹)
228		itg2le 25731	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)):ℝ⟶(0[,]+∞) ∧ 𝐹:ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)) ∘r ≤ 𝐹) → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))) ≤ (∫2‘𝐹))
229	146, 187, 227, 228	syl3anc 1379	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))) ≤ (∫2‘𝐹))
230	132, 148, 154, 186, 229	xrltletrd 13110	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → 0 < (∫2‘𝐹))
231	230	expr 457	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) → 0 < (∫2‘𝐹)))
232	231	con3d 152	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (¬ 0 < (∫2‘𝐹) → ¬ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞)))))
233	4	ffvelcdmi 7031	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ dom vol → (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ∈ (0[,]+∞))
234	3, 233	sselid 3920	. . . . . . . . . . . . . . . 16 ⊢ ((◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ dom vol → (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ∈ ℝ*)
235	157, 234	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ∈ ℝ*)
236		xrlenlt 11208	. . . . . . . . . . . . . . 15 ⊢ (((vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ≤ 0 ↔ ¬ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞)))))
237	235, 38, 236	sylancl 592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ≤ 0 ↔ ¬ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞)))))
238	232, 237	sylibrd 260	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (¬ 0 < (∫2‘𝐹) → (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ≤ 0))
239	238	imp 407	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ 0 < (∫2‘𝐹)) → (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ≤ 0)
240	239	an32s 658	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) ∧ 𝑘 ∈ ℕ) → (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ≤ 0)
241	131, 240	eqbrtrd 5101	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) ∧ 𝑘 ∈ ℕ) → (vol‘((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)) ≤ 0)
242	241	ralrimiva 3132	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → ∀𝑘 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)) ≤ 0)
243		ffn 6662	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))):ℕ⟶V → (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) Fn ℕ)
244		fveq2 6834	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) → (vol‘𝑧) = (vol‘((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
245	244	breq1d 5089	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) → ((vol‘𝑧) ≤ 0 ↔ (vol‘((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)) ≤ 0))
246	245	ralrn 7036	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) Fn ℕ → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))(vol‘𝑧) ≤ 0 ↔ ∀𝑘 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)) ≤ 0))
247	16, 243, 246	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))(vol‘𝑧) ≤ 0 ↔ ∀𝑘 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)) ≤ 0))
248	247	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → (∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))(vol‘𝑧) ≤ 0 ↔ ∀𝑘 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)) ≤ 0))
249	242, 248	mpbird 258	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))(vol‘𝑧) ≤ 0)
250		ffn 6662	. . . . . . . . . 10 ⊢ (vol:dom vol⟶(0[,]+∞) → vol Fn dom vol)
251	4, 250	ax-mp 5	. . . . . . . . 9 ⊢ vol Fn dom vol
252	27	frnd 6670	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ⊆ dom vol)
253	252	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ⊆ dom vol)
254		breq1 5082	. . . . . . . . . 10 ⊢ (𝑥 = (vol‘𝑧) → (𝑥 ≤ 0 ↔ (vol‘𝑧) ≤ 0))
255	254	ralima 7188	. . . . . . . . 9 ⊢ ((vol Fn dom vol ∧ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ⊆ dom vol) → (∀𝑥 ∈ (vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))))𝑥 ≤ 0 ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))(vol‘𝑧) ≤ 0))
256	251, 253, 255	sylancr 593	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → (∀𝑥 ∈ (vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))))𝑥 ≤ 0 ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))(vol‘𝑧) ≤ 0))
257	249, 256	mpbird 258	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → ∀𝑥 ∈ (vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))))𝑥 ≤ 0)
258		imassrn 6030	. . . . . . . . 9 ⊢ (vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) ⊆ ran vol
259		frn 6669	. . . . . . . . . . 11 ⊢ (vol:dom vol⟶(0[,]+∞) → ran vol ⊆ (0[,]+∞))
260	4, 259	ax-mp 5	. . . . . . . . . 10 ⊢ ran vol ⊆ (0[,]+∞)
261	260, 3	sstri 3931	. . . . . . . . 9 ⊢ ran vol ⊆ ℝ*
262	258, 261	sstri 3931	. . . . . . . 8 ⊢ (vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) ⊆ ℝ*
263		supxrleub 13276	. . . . . . . 8 ⊢ (((vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) ⊆ ℝ* ∧ 0 ∈ ℝ*) → (sup((vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))), ℝ*, < ) ≤ 0 ↔ ∀𝑥 ∈ (vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))))𝑥 ≤ 0))
264	262, 38, 263	mp2an 698	. . . . . . 7 ⊢ (sup((vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))), ℝ*, < ) ≤ 0 ↔ ∀𝑥 ∈ (vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))))𝑥 ≤ 0)
265	257, 264	sylibr 235	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → sup((vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))), ℝ*, < ) ≤ 0)
266	128, 265	eqbrtrd 5101	. . . . 5 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) ≤ 0)
267	8, 37, 39, 93, 266	xrletrd 13111	. . . 4 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → (vol‘𝐴) ≤ 0)
268	267	ex 413	. . 3 ⊢ (𝜑 → (¬ 0 < (∫2‘𝐹) → (vol‘𝐴) ≤ 0))
269		xrlenlt 11208	. . . 4 ⊢ (((vol‘𝐴) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((vol‘𝐴) ≤ 0 ↔ ¬ 0 < (vol‘𝐴)))
270	7, 38, 269	sylancl 592	. . 3 ⊢ (𝜑 → ((vol‘𝐴) ≤ 0 ↔ ¬ 0 < (vol‘𝐴)))
271	268, 270	sylibd 240	. 2 ⊢ (𝜑 → (¬ 0 < (∫2‘𝐹) → ¬ 0 < (vol‘𝐴)))
272	1, 271	mt4d 117	1 ⊢ (𝜑 → 0 < (∫2‘𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  Vcvv 3432   ⊆ wss 3890  ifcif 4461  ∪ cuni 4845  ∪ ciun 4928   class class class wbr 5079   ↦ cmpt 5160  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   “ cima 5628   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363   ∘_r cofr 7626  supcsup 9350  ℝcr 11035  0cc0 11036  1c1 11037   + caddc 11039   · cmul 11041  +∞cpnf 11174  ℝ^*cxr 11176   < clt 11177   ≤ cle 11178   / cdiv 11805  ℕcn 12172  ℝ⁺crp 12940  (,)cioo 13296  [,)cico 13298  [,]cicc 13299  vol*covol 25454  volcvol 25455  MblFncmbf 25606  ∫₂citg2 25608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-inf2 9560  ax-cc 10355  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-pre-sup 11114  ax-addf 11115

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-disj 5047  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-of 7627  df-ofr 7628  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-er 8640  df-map 8772  df-pm 8773  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fi 9321  df-sup 9352  df-inf 9353  df-oi 9422  df-dju 9823  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-div 11806  df-nn 12173  df-2 12242  df-3 12243  df-n0 12436  df-z 12523  df-uz 12787  df-q 12897  df-rp 12941  df-xneg 13061  df-xadd 13062  df-xmul 13063  df-ioo 13300  df-ico 13302  df-icc 13303  df-fz 13460  df-fzo 13607  df-fl 13749  df-seq 13962  df-exp 14022  df-hash 14291  df-cj 15059  df-re 15060  df-im 15061  df-sqrt 15195  df-abs 15196  df-clim 15448  df-rlim 15449  df-sum 15647  df-rest 17383  df-topgen 17404  df-psmet 21346  df-xmet 21347  df-met 21348  df-bl 21349  df-mopn 21350  df-top 22884  df-topon 22901  df-bases 22936  df-cmp 23377  df-cncf 24870  df-ovol 25456  df-vol 25457  df-mbf 25611  df-itg1 25612  df-itg2 25613  df-0p 25662

This theorem is referenced by:  itggt0  25836