Theorem itg2gt0 25698

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 13340	. . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ*
4		volf 25467	. . . . . . . . 9 ⊢ vol:dom vol⟶(0[,]+∞)
5	4	ffvelcdmi 7025	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) ∈ (0[,]+∞))
6	3, 5	sselid 3929	. . . . . . 7 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) ∈ ℝ*)
7	2, 6	syl 17	. . . . . 6 ⊢ (𝜑 → (vol‘𝐴) ∈ ℝ*)
8	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → (vol‘𝐴) ∈ ℝ*)
9		itg2gt0.4	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 ∈ MblFn)
10	9	elexd 3462	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∈ V)
11		cnvexg 7863	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
12	10, 11	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ◡𝐹 ∈ V)
13		imaexg 7852	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) ∈ V)
14	12, 13	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) ∈ V)
15	14	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) ∈ V)
16	15	fmpttd 7057	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))):ℕ⟶V)
17	16	ffnd 6660	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) Fn ℕ)
18		fniunfv 7190	. . . . . . . . . 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 13366	. . . . . . . . . . . . . . . 16 ⊢ (0[,)+∞) ⊆ ℝ
22		fss 6675	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐹:ℝ⟶ℝ)
23	20, 21, 22	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
24		mbfima 25568	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) ∈ dom vol)
27	26	fmpttd 7057	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))):ℕ⟶dom vol)
28	27	ffvelcdmda 7026	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ∈ dom vol)
29	28	ralrimiva 3126	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ∈ dom vol)
30		iunmbl 25491	. . . . . . . . . 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 2834	. . . . . . . 8 ⊢ (𝜑 → ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ∈ dom vol)
33		mblss 25469	. . . . . . . 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 25416	. . . . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) ∈ ℝ*)
38		0xr 11169	. . . . . 6 ⊢ 0 ∈ ℝ*
39	38	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → 0 ∈ ℝ*)
40		mblvol 25468	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) = (vol*‘𝐴))
41	2, 40	syl 17	. . . . . . 7 ⊢ (𝜑 → (vol‘𝐴) = (vol*‘𝐴))
42		mblss 25469	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
43	2, 42	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
44	43	sselda 3931	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
45	20	ffvelcdmda 7026	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ (0[,)+∞))
46		elrege0 13364	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
47	45, 46	sylib 218	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
48	47	simpld 494	. . . . . . . . . . . . . 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 12389	. . . . . . . . . . . . 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 6660	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 Fn ℝ)
54	53	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → 𝐹 Fn ℝ)
55		elpreima 7000	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ℝ)
58	57	biantrurd 532	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑥) ∈ ((1 / 𝑘)(,)+∞) ↔ (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ((1 / 𝑘)(,)+∞))))
59		nnrecre 12177	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
60	59	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
61	60	rexrd 11172	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ*)
62	61	adantlr 715	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ*)
63		elioopnf 13353	. . . . . . . . . . . . . . . 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 7852	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ V)
68	12, 67	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ V)
69	68	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ V)
70		oveq2 7363	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘))
71	70	oveq1d 7370	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑘 → ((1 / 𝑛)(,)+∞) = ((1 / 𝑘)(,)+∞))
72	71	imaeq2d 6016	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) = (◡𝐹 “ ((1 / 𝑘)(,)+∞)))
73		eqid 2733	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) = (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))
74	72, 73	fvmptg 6936	. . . . . . . . . . . . . . . 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 2819	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ↔ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))))
77	49	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑥) ∈ ℝ)
78	77	biantrurd 532	. . . . . . . . . . . . . 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 3156	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑘 ∈ ℕ (1 / 𝑘) < (𝐹‘𝑥) ↔ ∃𝑘 ∈ ℕ 𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
81	52, 80	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑘 ∈ ℕ 𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘))
82	81	ex 412	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → ∃𝑘 ∈ ℕ 𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
83		eluni2 4864	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ↔ ∃𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))𝑥 ∈ 𝑧)
84		eleq2 2822	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
85	84	rexrn 7029	. . . . . . . . . . . 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 283	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ↔ ∃𝑘 ∈ ℕ 𝑥 ∈ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
88	82, 87	sylibrd 259	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))))
89	88	ssrdv 3937	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))))
90		ovolss 25423	. . . . . . . 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 5117	. . . . . 6 ⊢ (𝜑 → (vol‘𝐴) ≤ (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))))
93	92	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → (vol‘𝐴) ≤ (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))))
94		mblvol 25468	. . . . . . . . 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 12147	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
97	96	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
98		nnrecre 12177	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 1) ∈ ℕ → (1 / (𝑘 + 1)) ∈ ℝ)
99	97, 98	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℝ)
100	99	rexrd 11172	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℝ*)
101		nnre 12142	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
102	101	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
103	102	lep1d 12063	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ≤ (𝑘 + 1))
104		nngt0 12166	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 0 < 𝑘)
105	104	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < 𝑘)
106	97	nnred 12150	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℝ)
107	97	nngt0d 12184	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < (𝑘 + 1))
108		lerec 12015	. . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 ≤ (𝑘 + 1) ↔ (1 / (𝑘 + 1)) ≤ (1 / 𝑘)))
110	103, 109	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ≤ (1 / 𝑘))
111		iooss1 13290	. . . . . . . . . . . . 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 6058	. . . . . . . . . . . 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 7852	. . . . . . . . . . . . 13 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ ((1 / (𝑘 + 1))(,)+∞)) ∈ V)
117	12, 116	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (◡𝐹 “ ((1 / (𝑘 + 1))(,)+∞)) ∈ V)
118		oveq2 7363	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (𝑘 + 1) → (1 / 𝑛) = (1 / (𝑘 + 1)))
119	118	oveq1d 7370	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑘 + 1) → ((1 / 𝑛)(,)+∞) = ((1 / (𝑘 + 1))(,)+∞))
120	119	imaeq2d 6016	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑘 + 1) → (◡𝐹 “ ((1 / 𝑛)(,)+∞)) = (◡𝐹 “ ((1 / (𝑘 + 1))(,)+∞)))
121	120, 73	fvmptg 6936	. . . . . . . . . . . 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 3987	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ⊆ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘(𝑘 + 1)))
124	123	ralrimiva 3126	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) ⊆ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘(𝑘 + 1)))
125		volsup 25494	. . . . . . . . 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 2770	. . . . . . 7 ⊢ (𝜑 → (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) = sup((vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))), ℝ*, < ))
128	127	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) = sup((vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))), ℝ*, < ))
129	68	adantr 480	. . . . . . . . . . . . 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 6835	. . . . . . . . . . 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 12178	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ ℕ → 0 < (1 / 𝑘))
134	133	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < (1 / 𝑘))
135		0re 11124	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ ℝ
136		ltle 11211	. . . . . . . . . . . . . . . . . . . . . . . . 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 13367	. . . . . . . . . . . . . . . . . . . . . . 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 13369	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ (0[,]+∞)
142		ifcl 4522	. . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ∈ (0[,]+∞))
145	144	fmpttd 7057	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)):ℝ⟶(0[,]+∞))
146	145	adantrr 717	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0)):ℝ⟶(0[,]+∞))
147		itg2cl 25670	. . . . . . . . . . . . . . . . . 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 13346	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
150		fss 6675	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℝ⟶(0[,]+∞))
151	20, 149, 150	sylancl 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:ℝ⟶(0[,]+∞))
152		itg2cl 25670	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) ∈ ℝ*)
153	151, 152	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ*)
154	153	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (∫2‘𝐹) ∈ ℝ*)
155		0nrp 12937	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 0 ∈ ℝ+
156		simpr 484	. . . . . . . . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ dom vol)
158	157	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ dom vol)
159	158	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . . . . . . . . . . . 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 12941	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ+)
162	161	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (1 / 𝑘) ∈ ℝ+)
163	162	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 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 25679	. . . . . . . . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . . . . . . . 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 13364	. . . . . . . . . . . . . . . . . . . . . . . . . 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 717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → (1 / 𝑘) ∈ (0[,)+∞))
170	169	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 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 25678	. . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . . . . . . . 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 12941	. . . . . . . . . . . . . . . . . . . . . 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 12960	. . . . . . . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . 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 199	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → ¬ 0 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0))))
180		itg2ge0 25673	. . . . . . . . . . . . . . . . . . . . 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 13053	. . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . 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 864	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → 𝐹:ℝ⟶(0[,]+∞))
188	60	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → (1 / 𝑘) ∈ ℝ)
189	53	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → (𝑥 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ((1 / 𝑘)(,)+∞)))
192	191	simpld 494	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → 𝑥 ∈ ℝ)
193	48	adantlr 715	. . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → (1 / 𝑘) ∈ ℝ*)
196	191	simprd 495	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → (𝐹‘𝑥) ∈ ((1 / 𝑘)(,)+∞))
197		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐹‘𝑥) ∈ ℝ ∧ (1 / 𝑘) < (𝐹‘𝑥)) → (1 / 𝑘) < (𝐹‘𝑥))
198	63, 197	biimtrdi 253	. . . . . . . . . . . . . . . . . . . . . . . . . 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 11271	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → (1 / 𝑘) ≤ (𝐹‘𝑥))
201	47	simprd 495	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 0 ≤ (𝐹‘𝑥))
202	201	adantlr 715	. . . . . . . . . . . . . . . . . . . . . . . . 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 5098	. . . . . . . . . . . . . . . . . . . . . . . . 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 5098	. . . . . . . . . . . . . . . . . . . . . . . . 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 4516	. . . . . . . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥))
209		iffalse 4485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) = 0)
210	209	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) = 0)
211	202	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ 𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞))) → 0 ≤ (𝐹‘𝑥))
212	210, 211	eqbrtrd 5117	. . . . . . . . . . . . . . . . . . . . . 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 812	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥))
214	213	ralrimiva 3126	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑥 ∈ ℝ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥))
215	214	adantrr 717	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → ∀𝑥 ∈ ℝ if(𝑥 ∈ (◡𝐹 “ ((1 / 𝑘)(,)+∞)), (1 / 𝑘), 0) ≤ (𝐹‘𝑥))
216		reex 11107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℝ ∈ V
217	216	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ℝ ∈ V)
218		ovex 7388	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 / 𝑘) ∈ V
219		c0ex 11116	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ V
220	218, 219	ifex 4527	. . . . . . . . . . . . . . . . . . . . . 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 6846	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹‘𝑥) ∈ V)
223		eqidd 2734	. . . . . . . . . . . . . . . . . . . . 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 6899	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹‘𝑥)))
225	217, 221, 222, 223, 224	ofrfval2 7640	. . . . . . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . . . . . 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 25677	. . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . 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 13070	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 0 < (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))))) → 0 < (∫2‘𝐹))
231	230	expr 456	. . . . . . . . . . . . . . 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 7025	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝐹 “ ((1 / 𝑘)(,)+∞)) ∈ dom vol → (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ∈ (0[,]+∞))
234	3, 233	sselid 3929	. . . . . . . . . . . . . . . 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 11187	. . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (¬ 0 < (∫2‘𝐹) → (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ≤ 0))
239	238	imp 406	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ ¬ 0 < (∫2‘𝐹)) → (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ≤ 0)
240	239	an32s 652	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) ∧ 𝑘 ∈ ℕ) → (vol‘(◡𝐹 “ ((1 / 𝑘)(,)+∞))) ≤ 0)
241	131, 240	eqbrtrd 5117	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) ∧ 𝑘 ∈ ℕ) → (vol‘((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)) ≤ 0)
242	241	ralrimiva 3126	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → ∀𝑘 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)) ≤ 0)
243		ffn 6659	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))):ℕ⟶V → (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) Fn ℕ)
244		fveq2 6831	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) → (vol‘𝑧) = (vol‘((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)))
245	244	breq1d 5105	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘) → ((vol‘𝑧) ≤ 0 ↔ (vol‘((𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))‘𝑘)) ≤ 0))
246	245	ralrn 7030	. . . . . . . . . . 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 480	. . . . . . . . 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 257	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))(vol‘𝑧) ≤ 0)
250		ffn 6659	. . . . . . . . . 10 ⊢ (vol:dom vol⟶(0[,]+∞) → vol Fn dom vol)
251	4, 250	ax-mp 5	. . . . . . . . 9 ⊢ vol Fn dom vol
252	27	frnd 6667	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ⊆ dom vol)
253	252	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))) ⊆ dom vol)
254		breq1 5098	. . . . . . . . . 10 ⊢ (𝑥 = (vol‘𝑧) → (𝑥 ≤ 0 ↔ (vol‘𝑧) ≤ 0))
255	254	ralima 7180	. . . . . . . . 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 257	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → ∀𝑥 ∈ (vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞))))𝑥 ≤ 0)
258		imassrn 6027	. . . . . . . . 9 ⊢ (vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) ⊆ ran vol
259		frn 6666	. . . . . . . . . . 11 ⊢ (vol:dom vol⟶(0[,]+∞) → ran vol ⊆ (0[,]+∞))
260	4, 259	ax-mp 5	. . . . . . . . . 10 ⊢ ran vol ⊆ (0[,]+∞)
261	260, 3	sstri 3941	. . . . . . . . 9 ⊢ ran vol ⊆ ℝ*
262	258, 261	sstri 3941	. . . . . . . 8 ⊢ (vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) ⊆ ℝ*
263		supxrleub 13235	. . . . . . . 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 692	. . . . . . 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 234	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → sup((vol “ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))), ℝ*, < ) ≤ 0)
266	128, 265	eqbrtrd 5117	. . . . 5 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → (vol*‘∪ ran (𝑛 ∈ ℕ ↦ (◡𝐹 “ ((1 / 𝑛)(,)+∞)))) ≤ 0)
267	8, 37, 39, 93, 266	xrletrd 13071	. . . 4 ⊢ ((𝜑 ∧ ¬ 0 < (∫2‘𝐹)) → (vol‘𝐴) ≤ 0)
268	267	ex 412	. . 3 ⊢ (𝜑 → (¬ 0 < (∫2‘𝐹) → (vol‘𝐴) ≤ 0))
269		xrlenlt 11187	. . . 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 239	. 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 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  Vcvv 3438   ⊆ wss 3899  ifcif 4476  ∪ cuni 4860  ∪ ciun 4943   class class class wbr 5095   ↦ cmpt 5176  ^◡ccnv 5620  dom cdm 5621  ran crn 5622   “ cima 5624   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355   ∘_r cofr 7618  supcsup 9334  ℝcr 11015  0cc0 11016  1c1 11017   + caddc 11019   · cmul 11021  +∞cpnf 11153  ℝ^*cxr 11155   < clt 11156   ≤ cle 11157   / cdiv 11784  ℕcn 12135  ℝ⁺crp 12900  (,)cioo 13255  [,)cico 13257  [,]cicc 13258  vol*covol 25400  volcvol 25401  MblFncmbf 25552  ∫₂citg2 25554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-inf2 9541  ax-cc 10336  ax-cnex 11072  ax-resscn 11073  ax-1cn 11074  ax-icn 11075  ax-addcl 11076  ax-addrcl 11077  ax-mulcl 11078  ax-mulrcl 11079  ax-mulcom 11080  ax-addass 11081  ax-mulass 11082  ax-distr 11083  ax-i2m1 11084  ax-1ne0 11085  ax-1rid 11086  ax-rnegex 11087  ax-rrecex 11088  ax-cnre 11089  ax-pre-lttri 11090  ax-pre-lttrn 11091  ax-pre-ltadd 11092  ax-pre-mulgt0 11093  ax-pre-sup 11094  ax-addf 11095

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-disj 5063  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-of 7619  df-ofr 7620  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-er 8631  df-map 8761  df-pm 8762  df-en 8879  df-dom 8880  df-sdom 8881  df-fin 8882  df-fi 9305  df-sup 9336  df-inf 9337  df-oi 9406  df-dju 9804  df-card 9842  df-pnf 11158  df-mnf 11159  df-xr 11160  df-ltxr 11161  df-le 11162  df-sub 11356  df-neg 11357  df-div 11785  df-nn 12136  df-2 12198  df-3 12199  df-n0 12392  df-z 12479  df-uz 12743  df-q 12857  df-rp 12901  df-xneg 13021  df-xadd 13022  df-xmul 13023  df-ioo 13259  df-ico 13261  df-icc 13262  df-fz 13418  df-fzo 13565  df-fl 13706  df-seq 13919  df-exp 13979  df-hash 14248  df-cj 15016  df-re 15017  df-im 15018  df-sqrt 15152  df-abs 15153  df-clim 15405  df-rlim 15406  df-sum 15604  df-rest 17336  df-topgen 17357  df-psmet 21293  df-xmet 21294  df-met 21295  df-bl 21296  df-mopn 21297  df-top 22819  df-topon 22836  df-bases 22871  df-cmp 23312  df-cncf 24808  df-ovol 25402  df-vol 25403  df-mbf 25557  df-itg1 25558  df-itg2 25559  df-0p 25608

This theorem is referenced by:  itggt0  25782