Theorem ovoliunlem2 25734

Description: Lemma for ovoliun 25736. (Contributed by Mario Carneiro, 12-Jun-2014.)

Hypotheses

Ref	Expression
ovoliun.t	⊢ 𝑇 = seq1( + , 𝐺)
ovoliun.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
ovoliun.a	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
ovoliun.v	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
ovoliun.r	⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
ovoliun.b	⊢ (𝜑 → 𝐵 ∈ ℝ+)
ovoliun.s	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹‘𝑛)))
ovoliun.u	⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
ovoliun.h	⊢ 𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1st ‘(𝐽‘𝑘)))‘(2nd ‘(𝐽‘𝑘))))
ovoliun.j	⊢ (𝜑 → 𝐽:ℕ–1-1-onto→(ℕ × ℕ))
ovoliun.f	⊢ (𝜑 → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
ovoliun.x1	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛)))
ovoliun.x2	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))

Assertion

Ref	Expression
ovoliunlem2	⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))

Distinct variable groups:   𝐴,𝑘   𝑘,𝑛,𝐵   𝑘,𝐹,𝑛   𝑘,𝐽,𝑛   𝑛,𝐻   𝜑,𝑘,𝑛   𝑆,𝑘   𝑘,𝐺   𝑇,𝑘   𝑛,𝐺   𝑇,𝑛

Allowed substitution hints:   𝐴(𝑛)   𝑆(𝑛)   𝑈(𝑘, 𝑛)   𝐻(𝑘)

Proof of Theorem ovoliunlem2

Dummy variables 𝑗 𝑚 𝑥 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovoliun.a	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
2	1	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
3		iunss 5003	. . . 4 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
4	2, 3	sylibr 237	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
5		ovolcl 25709	. . 3 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
6	4, 5	syl 18	. 2 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
7		ovoliun.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
8	7	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
9		ovoliun.j	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽:ℕ–1-1-onto→(ℕ × ℕ))
10		f1of 6818	. . . . . . . . . . . 12 ⊢ (𝐽:ℕ–1-1-onto→(ℕ × ℕ) → 𝐽:ℕ⟶(ℕ × ℕ))
11	9, 10	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽:ℕ⟶(ℕ × ℕ))
12	11	ffvelcdmda 7078	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐽‘𝑘) ∈ (ℕ × ℕ))
13		xp1st 8019	. . . . . . . . . 10 ⊢ ((𝐽‘𝑘) ∈ (ℕ × ℕ) → (1st ‘(𝐽‘𝑘)) ∈ ℕ)
14	12, 13	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1st ‘(𝐽‘𝑘)) ∈ ℕ)
15	8, 14	ffvelcdmd 7079	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(1st ‘(𝐽‘𝑘))) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
16		elovolmlem 25705	. . . . . . . 8 ⊢ ((𝐹‘(1st ‘(𝐽‘𝑘))) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ (𝐹‘(1st ‘(𝐽‘𝑘))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
17	15, 16	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(1st ‘(𝐽‘𝑘))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
18		xp2nd 8020	. . . . . . . 8 ⊢ ((𝐽‘𝑘) ∈ (ℕ × ℕ) → (2nd ‘(𝐽‘𝑘)) ∈ ℕ)
19	12, 18	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (2nd ‘(𝐽‘𝑘)) ∈ ℕ)
20	17, 19	ffvelcdmd 7079	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(1st ‘(𝐽‘𝑘)))‘(2nd ‘(𝐽‘𝑘))) ∈ ( ≤ ∩ (ℝ × ℝ)))
21		ovoliun.h	. . . . . 6 ⊢ 𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1st ‘(𝐽‘𝑘)))‘(2nd ‘(𝐽‘𝑘))))
22	20, 21	fmptd 7108	. . . . 5 ⊢ (𝜑 → 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
23		eqid 2760	. . . . . 6 ⊢ ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
24		ovoliun.u	. . . . . 6 ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
25	23, 24	ovolsf 25703	. . . . 5 ⊢ (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
26		frn 6711	. . . . 5 ⊢ (𝑈:ℕ⟶(0[,)+∞) → ran 𝑈 ⊆ (0[,)+∞))
27	22, 25, 26	3syl 19	. . . 4 ⊢ (𝜑 → ran 𝑈 ⊆ (0[,)+∞))
28		icossxr 13488	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ*
29	27, 28	sstrdi 3943	. . 3 ⊢ (𝜑 → ran 𝑈 ⊆ ℝ*)
30		supxrcl 13370	. . 3 ⊢ (ran 𝑈 ⊆ ℝ* → sup(ran 𝑈, ℝ*, < ) ∈ ℝ*)
31	29, 30	syl 18	. 2 ⊢ (𝜑 → sup(ran 𝑈, ℝ*, < ) ∈ ℝ*)
32		ovoliun.r	. . . 4 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
33		ovoliun.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ+)
34	33	rpred 13089	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
35	32, 34	readdcld 11265	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ*, < ) + 𝐵) ∈ ℝ)
36	35	rexrd 11286	. 2 ⊢ (𝜑 → (sup(ran 𝑇, ℝ*, < ) + 𝐵) ∈ ℝ*)
37		eliun 4955	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝑛 ∈ ℕ 𝐴 ↔ ∃𝑛 ∈ ℕ 𝑧 ∈ 𝐴)
38		ovoliun.x1	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛)))
39	38	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → 𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛)))
40	1	3adant3 1150	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → 𝐴 ⊆ ℝ)
41	7	ffvelcdmda 7078	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
42		elovolmlem 25705	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) ↔ (𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
43	41, 42	sylib 221	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
44	43	3adant3 1150	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
45		ovolfioo 25698	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ (𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛)) ↔ ∀𝑧 ∈ 𝐴 ∃𝑗 ∈ ℕ ((1st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2nd ‘((𝐹‘𝑛)‘𝑗)))))
46	40, 44, 45	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → (𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛)) ↔ ∀𝑧 ∈ 𝐴 ∃𝑗 ∈ ℕ ((1st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2nd ‘((𝐹‘𝑛)‘𝑗)))))
47	39, 46	mpbid 235	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → ∀𝑧 ∈ 𝐴 ∃𝑗 ∈ ℕ ((1st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2nd ‘((𝐹‘𝑛)‘𝑗))))
48		simp3 1156	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
49		rsp 3250	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ∃𝑗 ∈ ℕ ((1st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2nd ‘((𝐹‘𝑛)‘𝑗))) → (𝑧 ∈ 𝐴 → ∃𝑗 ∈ ℕ ((1st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2nd ‘((𝐹‘𝑛)‘𝑗)))))
50	47, 48, 49	sylc 66	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → ∃𝑗 ∈ ℕ ((1st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2nd ‘((𝐹‘𝑛)‘𝑗))))
51		simpl1 1210	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → 𝜑)
52		f1ocnv 6831	. . . . . . . . . . . 12 ⊢ (𝐽:ℕ–1-1-onto→(ℕ × ℕ) → ◡𝐽:(ℕ × ℕ)–1-1-onto→ℕ)
53		f1of 6818	. . . . . . . . . . . 12 ⊢ (◡𝐽:(ℕ × ℕ)–1-1-onto→ℕ → ◡𝐽:(ℕ × ℕ)⟶ℕ)
54	51, 9, 52, 53	4syl 20	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → ◡𝐽:(ℕ × ℕ)⟶ℕ)
55		simpl2 1211	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → 𝑛 ∈ ℕ)
56		simpr 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
57	54, 55, 56	fovcdmd 7587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (𝑛◡𝐽𝑗) ∈ ℕ)
58		2fveq3 6884	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑛◡𝐽𝑗) → (1st ‘(𝐽‘𝑘)) = (1st ‘(𝐽‘(𝑛◡𝐽𝑗))))
59	58	fveq2d 6883	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑛◡𝐽𝑗) → (𝐹‘(1st ‘(𝐽‘𝑘))) = (𝐹‘(1st ‘(𝐽‘(𝑛◡𝐽𝑗)))))
60		2fveq3 6884	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑛◡𝐽𝑗) → (2nd ‘(𝐽‘𝑘)) = (2nd ‘(𝐽‘(𝑛◡𝐽𝑗))))
61	59, 60	fveq12d 6886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑛◡𝐽𝑗) → ((𝐹‘(1st ‘(𝐽‘𝑘)))‘(2nd ‘(𝐽‘𝑘))) = ((𝐹‘(1st ‘(𝐽‘(𝑛◡𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛◡𝐽𝑗)))))
62		fvex 6892	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘(1st ‘(𝐽‘(𝑛◡𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛◡𝐽𝑗)))) ∈ V
63	61, 21, 62	fvmpt 6987	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛◡𝐽𝑗) ∈ ℕ → (𝐻‘(𝑛◡𝐽𝑗)) = ((𝐹‘(1st ‘(𝐽‘(𝑛◡𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛◡𝐽𝑗)))))
64	57, 63	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (𝐻‘(𝑛◡𝐽𝑗)) = ((𝐹‘(1st ‘(𝐽‘(𝑛◡𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛◡𝐽𝑗)))))
65		df-ov 7417	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛◡𝐽𝑗) = (◡𝐽‘⟨𝑛, 𝑗⟩)
66	65	fveq2i 6882	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐽‘(𝑛◡𝐽𝑗)) = (𝐽‘(◡𝐽‘⟨𝑛, 𝑗⟩))
67	51, 9	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → 𝐽:ℕ–1-1-onto→(ℕ × ℕ))
68	55, 56	opelxpd 5694	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
69		f1ocnvfv2 7279	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐽:ℕ–1-1-onto→(ℕ × ℕ) ∧ ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ)) → (𝐽‘(◡𝐽‘⟨𝑛, 𝑗⟩)) = ⟨𝑛, 𝑗⟩)
70	67, 68, 69	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (𝐽‘(◡𝐽‘⟨𝑛, 𝑗⟩)) = ⟨𝑛, 𝑗⟩)
71	66, 70	eqtrid 2807	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (𝐽‘(𝑛◡𝐽𝑗)) = ⟨𝑛, 𝑗⟩)
72	71	fveq2d 6883	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (1st ‘(𝐽‘(𝑛◡𝐽𝑗))) = (1st ‘⟨𝑛, 𝑗⟩))
73		vex 3454	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑛 ∈ V
74		vex 3454	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑗 ∈ V
75	73, 74	op1st 7995	. . . . . . . . . . . . . . . . . 18 ⊢ (1st ‘⟨𝑛, 𝑗⟩) = 𝑛
76	72, 75	eqtrdi 2811	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (1st ‘(𝐽‘(𝑛◡𝐽𝑗))) = 𝑛)
77	76	fveq2d 6883	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (𝐹‘(1st ‘(𝐽‘(𝑛◡𝐽𝑗)))) = (𝐹‘𝑛))
78	71	fveq2d 6883	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (2nd ‘(𝐽‘(𝑛◡𝐽𝑗))) = (2nd ‘⟨𝑛, 𝑗⟩))
79	73, 74	op2nd 7996	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘⟨𝑛, 𝑗⟩) = 𝑗
80	78, 79	eqtrdi 2811	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (2nd ‘(𝐽‘(𝑛◡𝐽𝑗))) = 𝑗)
81	77, 80	fveq12d 6886	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → ((𝐹‘(1st ‘(𝐽‘(𝑛◡𝐽𝑗))))‘(2nd ‘(𝐽‘(𝑛◡𝐽𝑗)))) = ((𝐹‘𝑛)‘𝑗))
82	64, 81	eqtrd 2795	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (𝐻‘(𝑛◡𝐽𝑗)) = ((𝐹‘𝑛)‘𝑗))
83	82	fveq2d 6883	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (1st ‘(𝐻‘(𝑛◡𝐽𝑗))) = (1st ‘((𝐹‘𝑛)‘𝑗)))
84	83	breq1d 5113	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → ((1st ‘(𝐻‘(𝑛◡𝐽𝑗))) < 𝑧 ↔ (1st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧))
85	82	fveq2d 6883	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (2nd ‘(𝐻‘(𝑛◡𝐽𝑗))) = (2nd ‘((𝐹‘𝑛)‘𝑗)))
86	85	breq2d 5115	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (𝑧 < (2nd ‘(𝐻‘(𝑛◡𝐽𝑗))) ↔ 𝑧 < (2nd ‘((𝐹‘𝑛)‘𝑗))))
87	84, 86	anbi12d 644	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (((1st ‘(𝐻‘(𝑛◡𝐽𝑗))) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘(𝑛◡𝐽𝑗)))) ↔ ((1st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2nd ‘((𝐹‘𝑛)‘𝑗)))))
88	87	biimprd 251	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (((1st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2nd ‘((𝐹‘𝑛)‘𝑗))) → ((1st ‘(𝐻‘(𝑛◡𝐽𝑗))) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘(𝑛◡𝐽𝑗))))))
89		2fveq3 6884	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛◡𝐽𝑗) → (1st ‘(𝐻‘𝑚)) = (1st ‘(𝐻‘(𝑛◡𝐽𝑗))))
90	89	breq1d 5113	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛◡𝐽𝑗) → ((1st ‘(𝐻‘𝑚)) < 𝑧 ↔ (1st ‘(𝐻‘(𝑛◡𝐽𝑗))) < 𝑧))
91		2fveq3 6884	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛◡𝐽𝑗) → (2nd ‘(𝐻‘𝑚)) = (2nd ‘(𝐻‘(𝑛◡𝐽𝑗))))
92	91	breq2d 5115	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛◡𝐽𝑗) → (𝑧 < (2nd ‘(𝐻‘𝑚)) ↔ 𝑧 < (2nd ‘(𝐻‘(𝑛◡𝐽𝑗)))))
93	90, 92	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛◡𝐽𝑗) → (((1st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑚))) ↔ ((1st ‘(𝐻‘(𝑛◡𝐽𝑗))) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘(𝑛◡𝐽𝑗))))))
94	93	rspcev 3576	. . . . . . . . . 10 ⊢ (((𝑛◡𝐽𝑗) ∈ ℕ ∧ ((1st ‘(𝐻‘(𝑛◡𝐽𝑗))) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘(𝑛◡𝐽𝑗))))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑚))))
95	57, 88, 94	syl6an 697	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (((1st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2nd ‘((𝐹‘𝑛)‘𝑗))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑚)))))
96	95	rexlimdva 3163	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → (∃𝑗 ∈ ℕ ((1st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2nd ‘((𝐹‘𝑛)‘𝑗))) → ∃𝑚 ∈ ℕ ((1st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑚)))))
97	50, 96	mpd 16	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → ∃𝑚 ∈ ℕ ((1st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑚))))
98	97	rexlimdv3a 3167	. . . . . 6 ⊢ (𝜑 → (∃𝑛 ∈ ℕ 𝑧 ∈ 𝐴 → ∃𝑚 ∈ ℕ ((1st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑚)))))
99	37, 98	biimtrid 245	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝑛 ∈ ℕ 𝐴 → ∃𝑚 ∈ ℕ ((1st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑚)))))
100	99	ralrimiv 3153	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ ∪ 𝑛 ∈ ℕ 𝐴∃𝑚 ∈ ℕ ((1st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑚))))
101		ovolfioo 25698	. . . . 5 ⊢ ((∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ ∪ 𝑛 ∈ ℕ 𝐴∃𝑚 ∈ ℕ ((1st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑚)))))
102	4, 22, 101	syl2anc 596	. . . 4 ⊢ (𝜑 → (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ ∪ 𝑛 ∈ ℕ 𝐴∃𝑚 ∈ ℕ ((1st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐻‘𝑚)))))
103	100, 102	mpbird 260	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻))
104	24	ovollb 25710	. . 3 ⊢ ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ∪ 𝑛 ∈ ℕ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻)) → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑈, ℝ*, < ))
105	22, 103, 104	syl2anc 596	. 2 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑈, ℝ*, < ))
106		fzfi 14039	. . . . . . 7 ⊢ (1...𝑗) ∈ Fin
107		elfznn 13611	. . . . . . . . . 10 ⊢ (𝑤 ∈ (1...𝑗) → 𝑤 ∈ ℕ)
108		ffvelcdm 7075	. . . . . . . . . . 11 ⊢ ((𝐽:ℕ⟶(ℕ × ℕ) ∧ 𝑤 ∈ ℕ) → (𝐽‘𝑤) ∈ (ℕ × ℕ))
109		xp1st 8019	. . . . . . . . . . 11 ⊢ ((𝐽‘𝑤) ∈ (ℕ × ℕ) → (1st ‘(𝐽‘𝑤)) ∈ ℕ)
110		nnre 12267	. . . . . . . . . . 11 ⊢ ((1st ‘(𝐽‘𝑤)) ∈ ℕ → (1st ‘(𝐽‘𝑤)) ∈ ℝ)
111	108, 109, 110	3syl 19	. . . . . . . . . 10 ⊢ ((𝐽:ℕ⟶(ℕ × ℕ) ∧ 𝑤 ∈ ℕ) → (1st ‘(𝐽‘𝑤)) ∈ ℝ)
112	11, 107, 111	syl2an 608	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (1...𝑗)) → (1st ‘(𝐽‘𝑤)) ∈ ℝ)
113	112	ralrimiva 3154	. . . . . . . 8 ⊢ (𝜑 → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ∈ ℝ)
114	113	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ∈ ℝ)
115		fimaxre3 12188	. . . . . . 7 ⊢ (((1...𝑗) ∈ Fin ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ 𝑥)
116	106, 114, 115	sylancr 599	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ 𝑥)
117		fllep1 13865	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
118	117	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → 𝑥 ≤ ((⌊‘𝑥) + 1))
119	112	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → (1st ‘(𝐽‘𝑤)) ∈ ℝ)
120		simplr 781	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → 𝑥 ∈ ℝ)
121		flcl 13859	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℤ)
122	121	peano2zd 12731	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℤ)
123	122	zred 12728	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℝ)
124	123	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → ((⌊‘𝑥) + 1) ∈ ℝ)
125		letr 11331	. . . . . . . . . . . 12 ⊢ (((1st ‘(𝐽‘𝑤)) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ ((⌊‘𝑥) + 1) ∈ ℝ) → (((1st ‘(𝐽‘𝑤)) ≤ 𝑥 ∧ 𝑥 ≤ ((⌊‘𝑥) + 1)) → (1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1)))
126	119, 120, 124, 125	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → (((1st ‘(𝐽‘𝑤)) ≤ 𝑥 ∧ 𝑥 ≤ ((⌊‘𝑥) + 1)) → (1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1)))
127	118, 126	mpan2d 707	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → ((1st ‘(𝐽‘𝑤)) ≤ 𝑥 → (1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1)))
128	127	ralimdva 3174	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ 𝑥 → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1)))
129	128	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ 𝑥 → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1)))
130		ovoliun.t	. . . . . . . . . 10 ⊢ 𝑇 = seq1( + , 𝐺)
131		ovoliun.g	. . . . . . . . . 10 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
132		simpll 779	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝜑)
133	132, 1	sylan 592	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
134		ovoliun.v	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
135	132, 134	sylan 592	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
136	132, 32	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
137	132, 33	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝐵 ∈ ℝ+)
138		ovoliun.s	. . . . . . . . . 10 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹‘𝑛)))
139	132, 9	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝐽:ℕ–1-1-onto→(ℕ × ℕ))
140	132, 7	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑m ℕ))
141	132, 38	sylan 592	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛)))
142		ovoliun.x2	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
143	132, 142	sylan 592	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) ∧ 𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ*, < ) ≤ ((vol*‘𝐴) + (𝐵 / (2↑𝑛))))
144		simplr 781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝑗 ∈ ℕ)
145	122	ad2antrl 741	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → ((⌊‘𝑥) + 1) ∈ ℤ)
146		simprr 785	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))
147	130, 131, 133, 135, 136, 137, 138, 24, 21, 139, 140, 141, 143, 144, 145, 146	ovoliunlem1 25733	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → (𝑈‘𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
148	147	expr 462	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1) → (𝑈‘𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
149	129, 148	syld 48	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ 𝑥 → (𝑈‘𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
150	149	rexlimdva 3163	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (∃𝑥 ∈ ℝ ∀𝑤 ∈ (1...𝑗)(1st ‘(𝐽‘𝑤)) ≤ 𝑥 → (𝑈‘𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
151	116, 150	mpd 16	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑈‘𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
152	151	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
153		ffn 6703	. . . . 5 ⊢ (𝑈:ℕ⟶(0[,)+∞) → 𝑈 Fn ℕ)
154		breq1 5106	. . . . . 6 ⊢ (𝑧 = (𝑈‘𝑗) → (𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ↔ (𝑈‘𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
155	154	ralrn 7082	. . . . 5 ⊢ (𝑈 Fn ℕ → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ↔ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
156	22, 25, 153, 155	4syl 20	. . . 4 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ↔ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
157	152, 156	mpbird 260	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
158		supxrleub 13381	. . . 4 ⊢ ((ran 𝑈 ⊆ ℝ* ∧ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ∈ ℝ*) → (sup(ran 𝑈, ℝ*, < ) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
159	29, 36, 158	syl2anc 596	. . 3 ⊢ (𝜑 → (sup(ran 𝑈, ℝ*, < ) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵)))
160	157, 159	mpbird 260	. 2 ⊢ (𝜑 → sup(ran 𝑈, ℝ*, < ) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))
161	6, 31, 36, 105, 160	xrletrd 13216	1 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ∩ cin 3898   ⊆ wss 3899  ⟨cop 4590  ∪ cuni 4867  ∪ ciun 4951   class class class wbr 5103   ↦ cmpt 5186   × cxp 5653  ^◡ccnv 5654  ran crn 5656   ∘ ccom 5659   Fn wfn 6528  ⟶wf 6529  –1-1-onto→wf1o 6532  ‘cfv 6533  (class class class)co 7414  1^st c1st 7985  2^nd c2nd 7986   ↑_m cmap 8829  Fincfn 8955  supcsup 9413  ℝcr 11126  0cc0 11127  1c1 11128   + caddc 11130  +∞cpnf 11267  ℝ^*cxr 11269   < clt 11270   ≤ cle 11271   − cmin 11468   / cdiv 11898  ℕcn 12260  2c2 12322  ℤcz 12618  ℝ⁺crp 13045  (,)cioo 13401  [,)cico 13403  ...cfz 13564  ⌊cfl 13854  seqcseq 14068  ↑cexp 14128  abscabs 15324  vol*covol 25693

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-inf2 9623  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8458  df-er 8699  df-map 8831  df-pm 8832  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9415  df-inf 9416  df-oi 9485  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-3 12331  df-n0 12532  df-z 12619  df-uz 12891  df-rp 13046  df-ioo 13405  df-ico 13407  df-fz 13565  df-fzo 13713  df-fl 13856  df-seq 14069  df-exp 14129  df-hash 14398  df-cj 15189  df-re 15190  df-im 15191  df-sqrt 15325  df-abs 15326  df-clim 15578  df-rlim 15579  df-sum 15777  df-ovol 25695

This theorem is used by:  ovoliunlem3  25735