Theorem itg2gt0 24925

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 13162	. . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ*
4		volf 24693	. . . . . . . . 9 ⊢ vol:dom vol⟶(0[,]+∞)
5	4	ffvelrni 6960	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) ∈ (0[,]+∞))
6	3, 5	sselid 3919	. . . . . . 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 3452	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∈ V)
11		cnvexg 7771	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
12	10, 11	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ◡𝐹 ∈ V)
13		imaexg 7762	. . . . . . . . . . . . . 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 6989	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))):ℕ⟶V)
17	16	ffnd 6601	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) Fn ℕ)
18		fniunfv 7120	. . . . . . . . . 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 13188	. . . . . . . . . . . . . . . 16 ⊢ (0[,)+∞) ⊆ ℝ
22		fss 6617	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐹:ℝ⟶ℝ)
23	20, 21, 22	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
24		mbfima 24794	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ) → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) ∈ dom vol)
25	9, 23, 24	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) ∈ dom vol)
26	25	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) ∈ dom vol)
27	26	fmpttd 6989	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))):ℕ⟶dom vol)
28	27	ffvelrnda 6961	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ∈ dom vol)
29	28	ralrimiva 3103	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ∈ dom vol)
30		iunmbl 24717	. . . . . . . . . 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 2840	. . . . . . . 8 ⊢ (𝜑 → ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ∈ dom vol)
33		mblss 24695	. . . . . . . 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 24642	. . . . . . 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 11022	. . . . . 6 ⊢ 0 ∈ ℝ*
39	38	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → 0 ∈ ℝ*)
40		mblvol 24694	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) = (vol*‘𝐴))
41	2, 40	syl 17	. . . . . . 7 ⊢ (𝜑 → (vol‘𝐴) = (vol*‘𝐴))
42		mblss 24695	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
43	2, 42	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
44	43	sselda 3921	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
45	20	ffvelrnda 6961	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ (0[,)+∞))
46		elrege0 13186	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
47	45, 46	sylib 217	. . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℝ)
50		itg2gt0.5	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 < (𝐹‘𝑥))
51		nnrecl 12231	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) ∈ ℝ ∧ 0 < (𝐹‘𝑥)) → ∃𝑘 ∈ ℕ (1 / 𝑘) < (𝐹‘𝑥))
52	49, 50, 51	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑘 ∈ ℕ (1 / 𝑘) < (𝐹‘𝑥))
53	20	ffnd 6601	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 Fn ℝ)
54	53	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → 𝐹 Fn ℝ)
55		elpreima 6935	. . . . . . . . . . . . . . . 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 533	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥) ∈ ((1 / 𝑘)(,)+∞) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ((1 / 𝑘)(,)+∞))))
59		nnrecre 12015	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
60	59	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
61	60	rexrd 11025	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ*)
62	61	adantlr 712	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ*)
63		elioopnf 13175	. . . . . . . . . . . . . . . 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 307	. . . . . . . . . . . . . 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 7762	. . . . . . . . . . . . . . . . . 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 7283	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘))
71	70	oveq1d 7290	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → ((1 / 𝑛)(,)+∞) = ((1 / 𝑘)(,)+∞))
72	71	imaeq2d 5969	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) = (◡𝐹 “ ((1 / 𝑘)(,)+∞)))
73		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) = (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))
74	72, 73	fvmptg 6873	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℕ ∧ (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ V) → ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) = (◡𝐹 “ ((1 / 𝑘)(,)+∞)))
75	66, 69, 74	syl2anr 597	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) = (◡𝐹 “ ((1 / 𝑘)(,)+∞)))
76	75	eleq2d 2824	. . . . . . . . . . . . . 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 533	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → ((1 / 𝑘) < (𝐹‘𝑥) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ (1 / 𝑘) < (𝐹‘𝑥))))
79	65, 76, 78	3bitr4rd 312	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → ((1 / 𝑘) < (𝐹‘𝑥) ↔ 𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
80	79	rexbidva 3225	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑘 ∈ ℕ (1 / 𝑘) < (𝐹‘𝑥) ↔ ∃𝑘 ∈ ℕ 𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
81	52, 80	mpbid 231	. . . . . . . . . . 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 4843	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ↔ ∃𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))𝑥 ∈ 𝑧)
84		eleq2 2827	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
85	84	rexrn 6963	. . . . . . . . . . . 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 282	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ↔ ∃𝑘 ∈ ℕ 𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
88	82, 87	sylibrd 258	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))))
89	88	ssrdv 3927	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))))
90		ovolss 24649	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝜑 → (vol*‘𝐴) ≤ (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))))
92	41, 91	eqbrtrd 5096	. . . . . 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 24694	. . . . . . . . 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 11985	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
97	96	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
98		nnrecre 12015	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 1) ∈ ℕ → (1 / (𝑘 + 1)) ∈ ℝ)
99	97, 98	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℝ)
100	99	rexrd 11025	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℝ*)
101		nnre 11980	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
102	101	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
103	102	lep1d 11906	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ (𝑘 + 1))
104		nngt0 12004	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 0 < 𝑘)
105	104	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < 𝑘)
106	97	nnred 11988	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℝ)
107	97	nngt0d 12022	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < (𝑘 + 1))
108		lerec 11858	. . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 ≤ (𝑘 + 1) ↔ (1 / (𝑘 + 1)) ≤ (1 / 𝑘)))
110	103, 109	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ≤ (1 / 𝑘))
111		iooss1 13114	. . . . . . . . . . . . 13 ⊢ (((1 / (𝑘 + 1)) ∈ ℝ* ∧ (1 / (𝑘 + 1)) ≤ (1 / 𝑘)) → ((1 / 𝑘)(,)+∞) ⊆ ((1 / (𝑘 + 1))(,)+∞))
112	100, 110, 111	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1 / 𝑘)(,)+∞) ⊆ ((1 / (𝑘 + 1))(,)+∞))
113		imass2 6010	. . . . . . . . . . . 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 597	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) = (◡𝐹 “ ((1 / 𝑘)(,)+∞)))
116		imaexg 7762	. . . . . . . . . . . . 13 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ ((1 / (𝑘 + 1))(,)+∞)) ∈ V)
117	12, 116	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (◡𝐹 “ ((1 / (𝑘 + 1))(,)+∞)) ∈ V)
118		oveq2 7283	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑘 + 1) → (1 / 𝑛) = (1 / (𝑘 + 1)))
119	118	oveq1d 7290	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑘 + 1) → ((1 / 𝑛)(,)+∞) = ((1 / (𝑘 + 1))(,)+∞))
120	119	imaeq2d 5969	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑘 + 1) → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) = (◡𝐹 “ ((1 / (𝑘 + 1))(,)+∞)))
121	120, 73	fvmptg 6873	. . . . . . . . . . . 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 597	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘(𝑘 + 1)) = (◡𝐹 “ ((1 / (𝑘 + 1))(,)+∞)))
123	114, 115, 122	3sstr4d 3968	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ⊆ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘(𝑘 + 1)))
124	123	ralrimiva 3103	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ⊆ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘(𝑘 + 1)))
125		volsup 24720	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (𝜑 → (vol‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) = sup((vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))), ℝ*, < ))
127	95, 126	eqtr3d 2780	. . . . . . 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 597	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) = (◡𝐹 “ ((1 / 𝑘)(,)+∞)))
131	130	fveq2d 6778	. . . . . . . . . . 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 12016	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ ℕ → 0 < (1 / 𝑘))
134	133	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < (1 / 𝑘))
135		0re 10977	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ ℝ
136		ltle 11063	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((0 ∈ ℝ ∧ (1 / 𝑘) ∈ ℝ) → (0 < (1 / 𝑘) → 0 ≤ (1 / 𝑘)))
137	135, 60, 136	sylancr 587	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (0 < (1 / 𝑘) → 0 ≤ (1 / 𝑘)))
138	134, 137	mpd 15	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (1 / 𝑘))
139		elxrge0 13189	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 / 𝑘) ∈ (0[,]+∞) ↔ ((1 / 𝑘) ∈ ℝ* ∧ 0 ≤ (1 / 𝑘)))
140	61, 138, 139	sylanbrc 583	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ (0[,]+∞))
141		0e0iccpnf 13191	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ (0[,]+∞)
142		ifcl 4504	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1 / 𝑘) ∈ (0[,]+∞) ∧ 0 ∈ (0[,]+∞)) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ∈ (0[,]+∞))
143	140, 141, 142	sylancl 586	. . . . . . . . . . . . . . . . . . . . 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 6989	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)):ℝ⟶(0[,]+∞))
146	145	adantrr 714	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)):ℝ⟶(0[,]+∞))
147		itg2cl 24897	. . . . . . . . . . . . . . . . . 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 13168	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
150		fss 6617	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℝ⟶(0[,]+∞))
151	20, 149, 150	sylancl 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,]+∞))
152		itg2cl 24897	. . . . . . . . . . . . . . . . . . 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 12765	. . . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ dom vol)
158	157	adantrr 714	. . . . . . . . . . . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . . . . . . . . . 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 12769	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ+)
162	161	adantrr 714	. . . . . . . . . . . . . . . . . . . . . . . . . 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 24906	. . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . . . . . . . . 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 13186	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((1 / 𝑘) ∈ (0[,)+∞) ↔ ((1 / 𝑘) ∈ ℝ ∧ 0 ≤ (1 / 𝑘)))
168	60, 138, 167	sylanbrc 583	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ (0[,)+∞))
169	168	adantrr 714	. . . . . . . . . . . . . . . . . . . . . . . 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 24905	. . . . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . . . . . 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 12769	. . . . . . . . . . . . . . . . . . . . . 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 12788	. . . . . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . . . . . 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 198	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → ¬ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))))
180		itg2ge0 24900	. . . . . . . . . . . . . . . . . . . . 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 12878	. . . . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . . 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 861	. . . . . . . . . . . . . . . . . 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 712	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ ℝ)
194	192, 193	syldan 591	. . . . . . . . . . . . . . . . . . . . . . . . 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	syl6bi 252	. . . . . . . . . . . . . . . . . . . . . . . . . 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 11123	. . . . . . . . . . . . . . . . . . . . . . . 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 712	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ (𝐹‘𝑥))
203	192, 202	syldan 591	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → 0 ≤ (𝐹‘𝑥))
204		breq1 5077	. . . . . . . . . . . . . . . . . . . . . . . . 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 5077	. . . . . . . . . . . . . . . . . . . . . . . . 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 4498	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((1 / 𝑘) ≤ (𝐹‘𝑥) ∧ 0 ≤ (𝐹‘𝑥)) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥))
207	200, 203, 206	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥))
208	207	adantlr 712	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥))
209		iffalse 4468	. . . . . . . . . . . . . . . . . . . . . . . 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 5096	. . . . . . . . . . . . . . . . . . . . . 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 810	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥))
214	213	ralrimiva 3103	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑥 ∈ ℝ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥))
215	214	adantrr 714	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → ∀𝑥 ∈ ℝ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥))
216		reex 10962	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℝ ∈ V
217	216	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ℝ ∈ V)
218		ovex 7308	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 / 𝑘) ∈ V
219		c0ex 10969	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ V
220	218, 219	ifex 4509	. . . . . . . . . . . . . . . . . . . . . 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 6789	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ V)
223		eqidd 2739	. . . . . . . . . . . . . . . . . . . . 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 6837	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
225	217, 221, 222, 223, 224	ofrfval2 7554	. . . . . . . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)) ∘r ≤ 𝐹)
228		itg2le 24904	. . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . 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 12895	. . . . . . . . . . . . . . . 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	ffvelrni 6960	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ dom vol → (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ∈ (0[,]+∞))
234	3, 233	sselid 3919	. . . . . . . . . . . . . . . 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 11040	. . . . . . . . . . . . . . 15 ⊢ (((vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ≤ 0 ↔ ¬ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞)))))
237	235, 38, 236	sylancl 586	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ≤ 0 ↔ ¬ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞)))))
238	232, 237	sylibrd 258	. . . . . . . . . . . . 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 649	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) ∧ 𝑘 ∈ ℕ) → (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ≤ 0)
241	131, 240	eqbrtrd 5096	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) ∧ 𝑘 ∈ ℕ) → (vol‘((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)) ≤ 0)
242	241	ralrimiva 3103	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → ∀𝑘 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)) ≤ 0)
243		ffn 6600	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))):ℕ⟶V → (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) Fn ℕ)
244		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) → (vol‘𝑧) = (vol‘((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
245	244	breq1d 5084	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) → ((vol‘𝑧) ≤ 0 ↔ (vol‘((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)) ≤ 0))
246	245	ralrn 6964	. . . . . . . . . . 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 256	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))(vol‘𝑧) ≤ 0)
250		ffn 6600	. . . . . . . . . 10 ⊢ (vol:dom vol⟶(0[,]+∞) → vol Fn dom vol)
251	4, 250	ax-mp 5	. . . . . . . . 9 ⊢ vol Fn dom vol
252	27	frnd 6608	. . . . . . . . . 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 5077	. . . . . . . . . 10 ⊢ (𝑥 = (vol‘𝑧) → (𝑥 ≤ 0 ↔ (vol‘𝑧) ≤ 0))
255	254	ralima 7114	. . . . . . . . 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 587	. . . . . . . 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 256	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → ∀𝑥 ∈ (vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))))𝑥 ≤ 0)
258		imassrn 5980	. . . . . . . . 9 ⊢ (vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) ⊆ ran vol
259		frn 6607	. . . . . . . . . . 11 ⊢ (vol:dom vol⟶(0[,]+∞) → ran vol ⊆ (0[,]+∞))
260	4, 259	ax-mp 5	. . . . . . . . . 10 ⊢ ran vol ⊆ (0[,]+∞)
261	260, 3	sstri 3930	. . . . . . . . 9 ⊢ ran vol ⊆ ℝ*
262	258, 261	sstri 3930	. . . . . . . 8 ⊢ (vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) ⊆ ℝ*
263		supxrleub 13060	. . . . . . . 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 689	. . . . . . 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 233	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → sup((vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))), ℝ*, < ) ≤ 0)
266	128, 265	eqbrtrd 5096	. . . . 5 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) ≤ 0)
267	8, 37, 39, 93, 266	xrletrd 12896	. . . 4 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → (vol‘𝐴) ≤ 0)
268	267	ex 413	. . 3 ⊢ (𝜑 → (¬ 0 < (∫2‘𝐹) → (vol‘𝐴) ≤ 0))
269		xrlenlt 11040	. . . 4 ⊢ (((vol‘𝐴) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((vol‘𝐴) ≤ 0 ↔ ¬ 0 < (vol‘𝐴)))
270	7, 38, 269	sylancl 586	. . 3 ⊢ (𝜑 → ((vol‘𝐴) ≤ 0 ↔ ¬ 0 < (vol‘𝐴)))
271	268, 270	sylibd 238	. 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 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  ifcif 4459  ∪ cuni 4839  ∪ ciun 4924   class class class wbr 5074   ↦ cmpt 5157  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   “ cima 5592   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∘_r cofr 7532  supcsup 9199  ℝcr 10870  0cc0 10871  1c1 10872   + caddc 10874   · cmul 10876  +∞cpnf 11006  ℝ^*cxr 11008   < clt 11009   ≤ cle 11010   / cdiv 11632  ℕcn 11973  ℝ⁺crp 12730  (,)cioo 13079  [,)cico 13081  [,]cicc 13082  vol*covol 24626  volcvol 24627  MblFncmbf 24778  ∫₂citg2 24780

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cc 10191  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949  ax-addf 10950

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-disj 5040  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-ofr 7534  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-fl 13512  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-rlim 15198  df-sum 15398  df-rest 17133  df-topgen 17154  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-top 22043  df-topon 22060  df-bases 22096  df-cmp 22538  df-cncf 24041  df-ovol 24628  df-vol 24629  df-mbf 24783  df-itg1 24784  df-itg2 24785  df-0p 24834

This theorem is referenced by:  itggt0  25008