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Theorem ovoliunlem2 25020

Description: Lemma for ovoliun 25022. (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‘(1^st ‘(𝐽‘𝑘)))‘(2^nd ‘(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 3147	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
3		iunss 5049	. . . 4 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
4	2, 3	sylibr 233	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
5		ovolcl 24995	. . 3 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^)
7		ovoliun.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
8	7	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
9		ovoliun.j	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽:ℕ–1-1-onto→(ℕ × ℕ))
10		f1of 6834	. . . . . . . . . . . 12 ⊢ (𝐽:ℕ–1-1-onto→(ℕ × ℕ) → 𝐽:ℕ⟶(ℕ × ℕ))
11	9, 10	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽:ℕ⟶(ℕ × ℕ))
12	11	ffvelcdmda 7087	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐽‘𝑘) ∈ (ℕ × ℕ))
13		xp1st 8007	. . . . . . . . . 10 ⊢ ((𝐽‘𝑘) ∈ (ℕ × ℕ) → (1^st ‘(𝐽‘𝑘)) ∈ ℕ)
14	12, 13	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1^st ‘(𝐽‘𝑘)) ∈ ℕ)
15	8, 14	ffvelcdmd 7088	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(1^st ‘(𝐽‘𝑘))) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
16		elovolmlem 24991	. . . . . . . 8 ⊢ ((𝐹‘(1^st ‘(𝐽‘𝑘))) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ (𝐹‘(1^st ‘(𝐽‘𝑘))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
17	15, 16	sylib 217	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(1^st ‘(𝐽‘𝑘))):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
18		xp2nd 8008	. . . . . . . 8 ⊢ ((𝐽‘𝑘) ∈ (ℕ × ℕ) → (2^nd ‘(𝐽‘𝑘)) ∈ ℕ)
19	12, 18	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (2^nd ‘(𝐽‘𝑘)) ∈ ℕ)
20	17, 19	ffvelcdmd 7088	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(1^st ‘(𝐽‘𝑘)))‘(2^nd ‘(𝐽‘𝑘))) ∈ ( ≤ ∩ (ℝ × ℝ)))
21		ovoliun.h	. . . . . 6 ⊢ 𝐻 = (𝑘 ∈ ℕ ↦ ((𝐹‘(1^st ‘(𝐽‘𝑘)))‘(2^nd ‘(𝐽‘𝑘))))
22	20, 21	fmptd 7114	. . . . 5 ⊢ (𝜑 → 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
23		eqid 2733	. . . . . 6 ⊢ ((abs ∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻)
24		ovoliun.u	. . . . . 6 ⊢ 𝑈 = seq1( + , ((abs ∘ − ) ∘ 𝐻))
25	23, 24	ovolsf 24989	. . . . 5 ⊢ (𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞))
26		frn 6725	. . . . 5 ⊢ (𝑈:ℕ⟶(0[,)+∞) → ran 𝑈 ⊆ (0[,)+∞))
27	22, 25, 26	3syl 18	. . . 4 ⊢ (𝜑 → ran 𝑈 ⊆ (0[,)+∞))
28		icossxr 13409	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ^*
29	27, 28	sstrdi 3995	. . 3 ⊢ (𝜑 → ran 𝑈 ⊆ ℝ^*)
30		supxrcl 13294	. . 3 ⊢ (ran 𝑈 ⊆ ℝ^* → sup(ran 𝑈, ℝ^, < ) ∈ ℝ^)
31	29, 30	syl 17	. 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 13016	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ)
35	32, 34	readdcld 11243	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^*, < ) + 𝐵) ∈ ℝ)
36	35	rexrd 11264	. 2 ⊢ (𝜑 → (sup(ran 𝑇, ℝ^, < ) + 𝐵) ∈ ℝ^)
37		eliun 5002	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝑛 ∈ ℕ 𝐴 ↔ ∃𝑛 ∈ ℕ 𝑧 ∈ 𝐴)
38		ovoliun.x1	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛)))
39	38	3adant3 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → 𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛)))
40	1	3adant3 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → 𝐴 ⊆ ℝ)
41	7	ffvelcdmda 7087	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
42		elovolmlem 24991	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑛) ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ (𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
43	41, 42	sylib 217	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
44	43	3adant3 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ)))
45		ovolfioo 24984	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ (𝐹‘𝑛):ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛)) ↔ ∀𝑧 ∈ 𝐴 ∃𝑗 ∈ ℕ ((1^st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2^nd ‘((𝐹‘𝑛)‘𝑗)))))
46	40, 44, 45	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → (𝐴 ⊆ ∪ ran ((,) ∘ (𝐹‘𝑛)) ↔ ∀𝑧 ∈ 𝐴 ∃𝑗 ∈ ℕ ((1^st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2^nd ‘((𝐹‘𝑛)‘𝑗)))))
47	39, 46	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → ∀𝑧 ∈ 𝐴 ∃𝑗 ∈ ℕ ((1^st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2^nd ‘((𝐹‘𝑛)‘𝑗))))
48		simp3 1139	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
49		rsp 3245	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ∃𝑗 ∈ ℕ ((1^st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2^nd ‘((𝐹‘𝑛)‘𝑗))) → (𝑧 ∈ 𝐴 → ∃𝑗 ∈ ℕ ((1^st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2^nd ‘((𝐹‘𝑛)‘𝑗)))))
50	47, 48, 49	sylc 65	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → ∃𝑗 ∈ ℕ ((1^st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2^nd ‘((𝐹‘𝑛)‘𝑗))))
51		simpl1 1192	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → 𝜑)
52		f1ocnv 6846	. . . . . . . . . . . 12 ⊢ (𝐽:ℕ–1-1-onto→(ℕ × ℕ) → ^◡𝐽:(ℕ × ℕ)–1-1-onto→ℕ)
53		f1of 6834	. . . . . . . . . . . 12 ⊢ (^◡𝐽:(ℕ × ℕ)–1-1-onto→ℕ → ^◡𝐽:(ℕ × ℕ)⟶ℕ)
54	51, 9, 52, 53	4syl 19	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → ^◡𝐽:(ℕ × ℕ)⟶ℕ)
55		simpl2 1193	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → 𝑛 ∈ ℕ)
56		simpr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
57	54, 55, 56	fovcdmd 7579	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (𝑛^◡𝐽𝑗) ∈ ℕ)
58		2fveq3 6897	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑛^◡𝐽𝑗) → (1^st ‘(𝐽‘𝑘)) = (1^st ‘(𝐽‘(𝑛^◡𝐽𝑗))))
59	58	fveq2d 6896	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑛^◡𝐽𝑗) → (𝐹‘(1^st ‘(𝐽‘𝑘))) = (𝐹‘(1^st ‘(𝐽‘(𝑛^◡𝐽𝑗)))))
60		2fveq3 6897	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑛^◡𝐽𝑗) → (2^nd ‘(𝐽‘𝑘)) = (2^nd ‘(𝐽‘(𝑛^◡𝐽𝑗))))
61	59, 60	fveq12d 6899	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑛^◡𝐽𝑗) → ((𝐹‘(1^st ‘(𝐽‘𝑘)))‘(2^nd ‘(𝐽‘𝑘))) = ((𝐹‘(1^st ‘(𝐽‘(𝑛^◡𝐽𝑗))))‘(2^nd ‘(𝐽‘(𝑛^◡𝐽𝑗)))))
62		fvex 6905	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘(1^st ‘(𝐽‘(𝑛^◡𝐽𝑗))))‘(2^nd ‘(𝐽‘(𝑛^◡𝐽𝑗)))) ∈ V
63	61, 21, 62	fvmpt 6999	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛^◡𝐽𝑗) ∈ ℕ → (𝐻‘(𝑛^◡𝐽𝑗)) = ((𝐹‘(1^st ‘(𝐽‘(𝑛^◡𝐽𝑗))))‘(2^nd ‘(𝐽‘(𝑛^◡𝐽𝑗)))))
64	57, 63	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (𝐻‘(𝑛^◡𝐽𝑗)) = ((𝐹‘(1^st ‘(𝐽‘(𝑛^◡𝐽𝑗))))‘(2^nd ‘(𝐽‘(𝑛^◡𝐽𝑗)))))
65		df-ov 7412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛^◡𝐽𝑗) = (^◡𝐽‘⟨𝑛, 𝑗⟩)
66	65	fveq2i 6895	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐽‘(𝑛^◡𝐽𝑗)) = (𝐽‘(^◡𝐽‘⟨𝑛, 𝑗⟩))
67	51, 9	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → 𝐽:ℕ–1-1-onto→(ℕ × ℕ))
68	55, 56	opelxpd 5716	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ))
69		f1ocnvfv2 7275	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐽:ℕ–1-1-onto→(ℕ × ℕ) ∧ ⟨𝑛, 𝑗⟩ ∈ (ℕ × ℕ)) → (𝐽‘(^◡𝐽‘⟨𝑛, 𝑗⟩)) = ⟨𝑛, 𝑗⟩)
70	67, 68, 69	syl2anc 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (𝐽‘(^◡𝐽‘⟨𝑛, 𝑗⟩)) = ⟨𝑛, 𝑗⟩)
71	66, 70	eqtrid 2785	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (𝐽‘(𝑛^◡𝐽𝑗)) = ⟨𝑛, 𝑗⟩)
72	71	fveq2d 6896	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (1^st ‘(𝐽‘(𝑛^◡𝐽𝑗))) = (1^st ‘⟨𝑛, 𝑗⟩))
73		vex 3479	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑛 ∈ V
74		vex 3479	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑗 ∈ V
75	73, 74	op1st 7983	. . . . . . . . . . . . . . . . . 18 ⊢ (1^st ‘⟨𝑛, 𝑗⟩) = 𝑛
76	72, 75	eqtrdi 2789	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (1^st ‘(𝐽‘(𝑛^◡𝐽𝑗))) = 𝑛)
77	76	fveq2d 6896	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (𝐹‘(1^st ‘(𝐽‘(𝑛^◡𝐽𝑗)))) = (𝐹‘𝑛))
78	71	fveq2d 6896	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (2^nd ‘(𝐽‘(𝑛^◡𝐽𝑗))) = (2^nd ‘⟨𝑛, 𝑗⟩))
79	73, 74	op2nd 7984	. . . . . . . . . . . . . . . . 17 ⊢ (2^nd ‘⟨𝑛, 𝑗⟩) = 𝑗
80	78, 79	eqtrdi 2789	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (2^nd ‘(𝐽‘(𝑛^◡𝐽𝑗))) = 𝑗)
81	77, 80	fveq12d 6899	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → ((𝐹‘(1^st ‘(𝐽‘(𝑛^◡𝐽𝑗))))‘(2^nd ‘(𝐽‘(𝑛^◡𝐽𝑗)))) = ((𝐹‘𝑛)‘𝑗))
82	64, 81	eqtrd 2773	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (𝐻‘(𝑛^◡𝐽𝑗)) = ((𝐹‘𝑛)‘𝑗))
83	82	fveq2d 6896	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (1^st ‘(𝐻‘(𝑛^◡𝐽𝑗))) = (1^st ‘((𝐹‘𝑛)‘𝑗)))
84	83	breq1d 5159	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → ((1^st ‘(𝐻‘(𝑛^◡𝐽𝑗))) < 𝑧 ↔ (1^st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧))
85	82	fveq2d 6896	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (2^nd ‘(𝐻‘(𝑛^◡𝐽𝑗))) = (2^nd ‘((𝐹‘𝑛)‘𝑗)))
86	85	breq2d 5161	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (𝑧 < (2^nd ‘(𝐻‘(𝑛^◡𝐽𝑗))) ↔ 𝑧 < (2^nd ‘((𝐹‘𝑛)‘𝑗))))
87	84, 86	anbi12d 632	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (((1^st ‘(𝐻‘(𝑛^◡𝐽𝑗))) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘(𝑛^◡𝐽𝑗)))) ↔ ((1^st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2^nd ‘((𝐹‘𝑛)‘𝑗)))))
88	87	biimprd 247	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (((1^st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2^nd ‘((𝐹‘𝑛)‘𝑗))) → ((1^st ‘(𝐻‘(𝑛^◡𝐽𝑗))) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘(𝑛^◡𝐽𝑗))))))
89		2fveq3 6897	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛^◡𝐽𝑗) → (1^st ‘(𝐻‘𝑚)) = (1^st ‘(𝐻‘(𝑛^◡𝐽𝑗))))
90	89	breq1d 5159	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛^◡𝐽𝑗) → ((1^st ‘(𝐻‘𝑚)) < 𝑧 ↔ (1^st ‘(𝐻‘(𝑛^◡𝐽𝑗))) < 𝑧))
91		2fveq3 6897	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛^◡𝐽𝑗) → (2^nd ‘(𝐻‘𝑚)) = (2^nd ‘(𝐻‘(𝑛^◡𝐽𝑗))))
92	91	breq2d 5161	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛^◡𝐽𝑗) → (𝑧 < (2^nd ‘(𝐻‘𝑚)) ↔ 𝑧 < (2^nd ‘(𝐻‘(𝑛^◡𝐽𝑗)))))
93	90, 92	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛^◡𝐽𝑗) → (((1^st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑚))) ↔ ((1^st ‘(𝐻‘(𝑛^◡𝐽𝑗))) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘(𝑛^◡𝐽𝑗))))))
94	93	rspcev 3613	. . . . . . . . . 10 ⊢ (((𝑛^◡𝐽𝑗) ∈ ℕ ∧ ((1^st ‘(𝐻‘(𝑛^◡𝐽𝑗))) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘(𝑛^◡𝐽𝑗))))) → ∃𝑚 ∈ ℕ ((1^st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑚))))
95	57, 88, 94	syl6an 683	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) ∧ 𝑗 ∈ ℕ) → (((1^st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2^nd ‘((𝐹‘𝑛)‘𝑗))) → ∃𝑚 ∈ ℕ ((1^st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑚)))))
96	95	rexlimdva 3156	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → (∃𝑗 ∈ ℕ ((1^st ‘((𝐹‘𝑛)‘𝑗)) < 𝑧 ∧ 𝑧 < (2^nd ‘((𝐹‘𝑛)‘𝑗))) → ∃𝑚 ∈ ℕ ((1^st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑚)))))
97	50, 96	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑧 ∈ 𝐴) → ∃𝑚 ∈ ℕ ((1^st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑚))))
98	97	rexlimdv3a 3160	. . . . . 6 ⊢ (𝜑 → (∃𝑛 ∈ ℕ 𝑧 ∈ 𝐴 → ∃𝑚 ∈ ℕ ((1^st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑚)))))
99	37, 98	biimtrid 241	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝑛 ∈ ℕ 𝐴 → ∃𝑚 ∈ ℕ ((1^st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑚)))))
100	99	ralrimiv 3146	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ ∪ 𝑛 ∈ ℕ 𝐴∃𝑚 ∈ ℕ ((1^st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑚))))
101		ovolfioo 24984	. . . . 5 ⊢ ((∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ ∪ 𝑛 ∈ ℕ 𝐴∃𝑚 ∈ ℕ ((1^st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑚)))))
102	4, 22, 101	syl2anc 585	. . . 4 ⊢ (𝜑 → (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻) ↔ ∀𝑧 ∈ ∪ 𝑛 ∈ ℕ 𝐴∃𝑚 ∈ ℕ ((1^st ‘(𝐻‘𝑚)) < 𝑧 ∧ 𝑧 < (2^nd ‘(𝐻‘𝑚)))))
103	100, 102	mpbird 257	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻))
104	24	ovollb 24996	. . 3 ⊢ ((𝐻:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ∪ 𝑛 ∈ ℕ 𝐴 ⊆ ∪ ran ((,) ∘ 𝐻)) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑈, ℝ^, < ))
105	22, 103, 104	syl2anc 585	. 2 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑈, ℝ^, < ))
106		fzfi 13937	. . . . . . 7 ⊢ (1...𝑗) ∈ Fin
107		elfznn 13530	. . . . . . . . . 10 ⊢ (𝑤 ∈ (1...𝑗) → 𝑤 ∈ ℕ)
108		ffvelcdm 7084	. . . . . . . . . . 11 ⊢ ((𝐽:ℕ⟶(ℕ × ℕ) ∧ 𝑤 ∈ ℕ) → (𝐽‘𝑤) ∈ (ℕ × ℕ))
109		xp1st 8007	. . . . . . . . . . 11 ⊢ ((𝐽‘𝑤) ∈ (ℕ × ℕ) → (1^st ‘(𝐽‘𝑤)) ∈ ℕ)
110		nnre 12219	. . . . . . . . . . 11 ⊢ ((1^st ‘(𝐽‘𝑤)) ∈ ℕ → (1^st ‘(𝐽‘𝑤)) ∈ ℝ)
111	108, 109, 110	3syl 18	. . . . . . . . . 10 ⊢ ((𝐽:ℕ⟶(ℕ × ℕ) ∧ 𝑤 ∈ ℕ) → (1^st ‘(𝐽‘𝑤)) ∈ ℝ)
112	11, 107, 111	syl2an 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (1...𝑗)) → (1^st ‘(𝐽‘𝑤)) ∈ ℝ)
113	112	ralrimiva 3147	. . . . . . . 8 ⊢ (𝜑 → ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ∈ ℝ)
114	113	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ∈ ℝ)
115		fimaxre3 12160	. . . . . . 7 ⊢ (((1...𝑗) ∈ Fin ∧ ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ 𝑥)
116	106, 114, 115	sylancr 588	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ 𝑥)
117		fllep1 13766	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
118	117	ad2antlr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → 𝑥 ≤ ((⌊‘𝑥) + 1))
119	112	adantlr 714	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → (1^st ‘(𝐽‘𝑤)) ∈ ℝ)
120		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → 𝑥 ∈ ℝ)
121		flcl 13760	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℤ)
122	121	peano2zd 12669	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℤ)
123	122	zred 12666	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → ((⌊‘𝑥) + 1) ∈ ℝ)
124	123	ad2antlr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → ((⌊‘𝑥) + 1) ∈ ℝ)
125		letr 11308	. . . . . . . . . . . 12 ⊢ (((1^st ‘(𝐽‘𝑤)) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ ((⌊‘𝑥) + 1) ∈ ℝ) → (((1^st ‘(𝐽‘𝑤)) ≤ 𝑥 ∧ 𝑥 ≤ ((⌊‘𝑥) + 1)) → (1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1)))
126	119, 120, 124, 125	syl3anc 1372	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → (((1^st ‘(𝐽‘𝑤)) ≤ 𝑥 ∧ 𝑥 ≤ ((⌊‘𝑥) + 1)) → (1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1)))
127	118, 126	mpan2d 693	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑤 ∈ (1...𝑗)) → ((1^st ‘(𝐽‘𝑤)) ≤ 𝑥 → (1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1)))
128	127	ralimdva 3168	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ 𝑥 → ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1)))
129	128	adantlr 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ 𝑥 → ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1)))
130		ovoliun.t	. . . . . . . . . 10 ⊢ 𝑇 = seq1( + , 𝐺)
131		ovoliun.g	. . . . . . . . . 10 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
132		simpll 766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝜑)
133	132, 1	sylan 581	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
134		ovoliun.v	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
135	132, 134	sylan 581	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
136	132, 32	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → sup(ran 𝑇, ℝ^*, < ) ∈ ℝ)
137	132, 33	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝐵 ∈ ℝ⁺)
138		ovoliun.s	. . . . . . . . . 10 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ (𝐹‘𝑛)))
139	132, 9	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝐽:ℕ–1-1-onto→(ℕ × ℕ))
140	132, 7	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝐹:ℕ⟶(( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
141	132, 38	sylan 581	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1^st ‘(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 581	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) ∧ 𝑛 ∈ ℕ) → sup(ran 𝑆, ℝ^, < ) ≤ ((vol‘𝐴) + (𝐵 / (2↑𝑛))))
144		simplr 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → 𝑗 ∈ ℕ)
145	122	ad2antrl 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → ((⌊‘𝑥) + 1) ∈ ℤ)
146		simprr 772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))
147	130, 131, 133, 135, 136, 137, 138, 24, 21, 139, 140, 141, 143, 144, 145, 146	ovoliunlem1 25019	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑥 ∈ ℝ ∧ ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1))) → (𝑈‘𝑗) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝐵))
148	147	expr 458	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ ((⌊‘𝑥) + 1) → (𝑈‘𝑗) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝐵)))
149	129, 148	syld 47	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ 𝑥 → (𝑈‘𝑗) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝐵)))
150	149	rexlimdva 3156	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (∃𝑥 ∈ ℝ ∀𝑤 ∈ (1...𝑗)(1^st ‘(𝐽‘𝑤)) ≤ 𝑥 → (𝑈‘𝑗) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝐵)))
151	116, 150	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑈‘𝑗) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝐵))
152	151	ralrimiva 3147	. . . 4 ⊢ (𝜑 → ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝐵))
153		ffn 6718	. . . . 5 ⊢ (𝑈:ℕ⟶(0[,)+∞) → 𝑈 Fn ℕ)
154		breq1 5152	. . . . . 6 ⊢ (𝑧 = (𝑈‘𝑗) → (𝑧 ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵) ↔ (𝑈‘𝑗) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵)))
155	154	ralrn 7090	. . . . 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 19	. . . 4 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵) ↔ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵)))
157	152, 156	mpbird 257	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝐵))
158		supxrleub 13305	. . . 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 585	. . 3 ⊢ (𝜑 → (sup(ran 𝑈, ℝ^, < ) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵) ↔ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ (sup(ran 𝑇, ℝ^*, < ) + 𝐵)))
160	157, 159	mpbird 257	. 2 ⊢ (𝜑 → sup(ran 𝑈, ℝ^, < ) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵))
161	6, 31, 36, 105, 160	xrletrd 13141	1 ⊢ (𝜑 → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ^, < ) + 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ∩ cin 3948 ⊆ wss 3949 ⟨cop 4635 ∪ cuni 4909 ∪ ciun 4998 class class class wbr 5149 ↦ cmpt 5232 × cxp 5675 ^◡ccnv 5676 ran crn 5678 ∘ ccom 5681 Fn wfn 6539 ⟶wf 6540 –1-1-onto→wf1o 6543 ‘cfv 6544 (class class class)co 7409 1^st c1st 7973 2^nd c2nd 7974 ↑_m cmap 8820 Fincfn 8939 supcsup 9435 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 +∞cpnf 11245 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 − cmin 11444 / cdiv 11871 ℕcn 12212 2c2 12267 ℤcz 12558 ℝ⁺crp 12974 (,)cioo 13324 [,)cico 13326 ...cfz 13484 ⌊cfl 13755 seqcseq 13966 ↑cexp 14027 abscabs 15181 vol*covol 24979

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-ioo 13328 df-ico 13330 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-ovol 24981

This theorem is referenced by: ovoliunlem3 25021

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