Theorem volsup2 25114

Description: The volume of 𝐴 is the supremum of the sequence vol*‘(𝐴 ∩ (-𝑛[,]𝑛)) of volumes of bounded sets. (Contributed by Mario Carneiro, 30-Aug-2014.)

Assertion

Ref	Expression
volsup2	⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ∃𝑛 ∈ ℕ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))

Distinct variable groups:   𝐴,𝑛   𝐵,𝑛

Proof of Theorem volsup2

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

Step	Hyp	Ref	Expression
1		simp3 1139	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → 𝐵 < (vol‘𝐴))
2		rexr 11257	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*)
3	2	3ad2ant2 1135	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → 𝐵 ∈ ℝ*)
4		iccssxr 13404	. . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ*
5		volf 25038	. . . . . . . . 9 ⊢ vol:dom vol⟶(0[,]+∞)
6	5	ffvelcdmi 7083	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) ∈ (0[,]+∞))
7	4, 6	sselid 3980	. . . . . . 7 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) ∈ ℝ*)
8	7	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (vol‘𝐴) ∈ ℝ*)
9		xrltnle 11278	. . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ (vol‘𝐴) ∈ ℝ*) → (𝐵 < (vol‘𝐴) ↔ ¬ (vol‘𝐴) ≤ 𝐵))
10	3, 8, 9	syl2anc 585	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝐵 < (vol‘𝐴) ↔ ¬ (vol‘𝐴) ≤ 𝐵))
11	1, 10	mpbid 231	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ¬ (vol‘𝐴) ≤ 𝐵)
12		negeq 11449	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → -𝑚 = -𝑛)
13		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
14	12, 13	oveq12d 7424	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (-𝑚[,]𝑚) = (-𝑛[,]𝑛))
15	14	ineq2d 4212	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝐴 ∩ (-𝑚[,]𝑚)) = (𝐴 ∩ (-𝑛[,]𝑛)))
16		eqid 2733	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) = (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))
17		ovex 7439	. . . . . . . . . . . . 13 ⊢ (-𝑛[,]𝑛) ∈ V
18	17	inex2 5318	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ (-𝑛[,]𝑛)) ∈ V
19	15, 16, 18	fvmpt 6996	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) = (𝐴 ∩ (-𝑛[,]𝑛)))
20	19	iuneq2i 5018	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) = ∪ 𝑛 ∈ ℕ (𝐴 ∩ (-𝑛[,]𝑛))
21		iunin2 5074	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ ℕ (𝐴 ∩ (-𝑛[,]𝑛)) = (𝐴 ∩ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛))
22	20, 21	eqtri 2761	. . . . . . . . 9 ⊢ ∪ 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) = (𝐴 ∩ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛))
23		simpl1 1192	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ dom vol)
24		nnre 12216	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
25	24	adantl 483	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
26	25	renegcld 11638	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → -𝑛 ∈ ℝ)
27		iccmbl 25075	. . . . . . . . . . . . . 14 ⊢ ((-𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (-𝑛[,]𝑛) ∈ dom vol)
28	26, 25, 27	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (-𝑛[,]𝑛) ∈ dom vol)
29		inmbl 25051	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ dom vol ∧ (-𝑛[,]𝑛) ∈ dom vol) → (𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol)
30	23, 28, 29	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol)
31	15	cbvmptv 5261	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) = (𝑛 ∈ ℕ ↦ (𝐴 ∩ (-𝑛[,]𝑛)))
32	30, 31	fmptd 7111	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))):ℕ⟶dom vol)
33	32	ffnd 6716	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) Fn ℕ)
34		fniunfv 7243	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) Fn ℕ → ∪ 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) = ∪ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))))
35	33, 34	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ∪ 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) = ∪ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))))
36		mblss 25040	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
37	36	3ad2ant1 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → 𝐴 ⊆ ℝ)
38	37	sselda 3982	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
39		recn 11197	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
40	39	abscld 15380	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → (abs‘𝑥) ∈ ℝ)
41		arch 12466	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘𝑥) ∈ ℝ → ∃𝑛 ∈ ℕ (abs‘𝑥) < 𝑛)
42	40, 41	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → ∃𝑛 ∈ ℕ (abs‘𝑥) < 𝑛)
43		ltle 11299	. . . . . . . . . . . . . . . . . 18 ⊢ (((abs‘𝑥) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((abs‘𝑥) < 𝑛 → (abs‘𝑥) ≤ 𝑛))
44	40, 24, 43	syl2an 597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((abs‘𝑥) < 𝑛 → (abs‘𝑥) ≤ 𝑛))
45		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛) → (𝑥 ∈ ℝ ∧ -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛))
46	45	3expib 1123	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ → ((-𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛) → (𝑥 ∈ ℝ ∧ -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
47	46	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((-𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛) → (𝑥 ∈ ℝ ∧ -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
48		absle 15259	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((abs‘𝑥) ≤ 𝑛 ↔ (-𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
49	24, 48	sylan2 594	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((abs‘𝑥) ≤ 𝑛 ↔ (-𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
50	24	adantl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
51	50	renegcld 11638	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → -𝑛 ∈ ℝ)
52		elicc2 13386	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝑥 ∈ (-𝑛[,]𝑛) ↔ (𝑥 ∈ ℝ ∧ -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
53	51, 50, 52	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑥 ∈ (-𝑛[,]𝑛) ↔ (𝑥 ∈ ℝ ∧ -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
54	47, 49, 53	3imtr4d 294	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((abs‘𝑥) ≤ 𝑛 → 𝑥 ∈ (-𝑛[,]𝑛)))
55	44, 54	syld 47	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((abs‘𝑥) < 𝑛 → 𝑥 ∈ (-𝑛[,]𝑛)))
56	55	reximdva 3169	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → (∃𝑛 ∈ ℕ (abs‘𝑥) < 𝑛 → ∃𝑛 ∈ ℕ 𝑥 ∈ (-𝑛[,]𝑛)))
57	42, 56	mpd 15	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → ∃𝑛 ∈ ℕ 𝑥 ∈ (-𝑛[,]𝑛))
58	38, 57	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑥 ∈ 𝐴) → ∃𝑛 ∈ ℕ 𝑥 ∈ (-𝑛[,]𝑛))
59	58	ex 414	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝑥 ∈ 𝐴 → ∃𝑛 ∈ ℕ 𝑥 ∈ (-𝑛[,]𝑛)))
60		eliun 5001	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛) ↔ ∃𝑛 ∈ ℕ 𝑥 ∈ (-𝑛[,]𝑛))
61	59, 60	syl6ibr 252	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛)))
62	61	ssrdv 3988	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → 𝐴 ⊆ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛))
63		df-ss 3965	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛) ↔ (𝐴 ∩ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛)) = 𝐴)
64	62, 63	sylib 217	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝐴 ∩ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛)) = 𝐴)
65	22, 35, 64	3eqtr3a 2797	. . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ∪ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) = 𝐴)
66	65	fveq2d 6893	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (vol‘∪ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))) = (vol‘𝐴))
67		peano2re 11384	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
68	25, 67	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℝ)
69	68	renegcld 11638	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → -(𝑛 + 1) ∈ ℝ)
70	25	lep1d 12142	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑛 ≤ (𝑛 + 1))
71	25, 68	lenegd 11790	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑛 ≤ (𝑛 + 1) ↔ -(𝑛 + 1) ≤ -𝑛))
72	70, 71	mpbid 231	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → -(𝑛 + 1) ≤ -𝑛)
73		iccss 13389	. . . . . . . . . . . 12 ⊢ (((-(𝑛 + 1) ∈ ℝ ∧ (𝑛 + 1) ∈ ℝ) ∧ (-(𝑛 + 1) ≤ -𝑛 ∧ 𝑛 ≤ (𝑛 + 1))) → (-𝑛[,]𝑛) ⊆ (-(𝑛 + 1)[,](𝑛 + 1)))
74	69, 68, 72, 70, 73	syl22anc 838	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (-𝑛[,]𝑛) ⊆ (-(𝑛 + 1)[,](𝑛 + 1)))
75		sslin 4234	. . . . . . . . . . 11 ⊢ ((-𝑛[,]𝑛) ⊆ (-(𝑛 + 1)[,](𝑛 + 1)) → (𝐴 ∩ (-𝑛[,]𝑛)) ⊆ (𝐴 ∩ (-(𝑛 + 1)[,](𝑛 + 1))))
76	74, 75	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (𝐴 ∩ (-𝑛[,]𝑛)) ⊆ (𝐴 ∩ (-(𝑛 + 1)[,](𝑛 + 1))))
77	19	adantl 483	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) = (𝐴 ∩ (-𝑛[,]𝑛)))
78		peano2nn 12221	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
79	78	adantl 483	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
80		negeq 11449	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 + 1) → -𝑚 = -(𝑛 + 1))
81		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 + 1) → 𝑚 = (𝑛 + 1))
82	80, 81	oveq12d 7424	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → (-𝑚[,]𝑚) = (-(𝑛 + 1)[,](𝑛 + 1)))
83	82	ineq2d 4212	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝐴 ∩ (-𝑚[,]𝑚)) = (𝐴 ∩ (-(𝑛 + 1)[,](𝑛 + 1))))
84		ovex 7439	. . . . . . . . . . . . 13 ⊢ (-(𝑛 + 1)[,](𝑛 + 1)) ∈ V
85	84	inex2 5318	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ (-(𝑛 + 1)[,](𝑛 + 1))) ∈ V
86	83, 16, 85	fvmpt 6996	. . . . . . . . . . 11 ⊢ ((𝑛 + 1) ∈ ℕ → ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘(𝑛 + 1)) = (𝐴 ∩ (-(𝑛 + 1)[,](𝑛 + 1))))
87	79, 86	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘(𝑛 + 1)) = (𝐴 ∩ (-(𝑛 + 1)[,](𝑛 + 1))))
88	76, 77, 87	3sstr4d 4029	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘(𝑛 + 1)))
89	88	ralrimiva 3147	. . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ∀𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘(𝑛 + 1)))
90		volsup 25065	. . . . . . . 8 ⊢ (((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))):ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) ⊆ ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘(𝑛 + 1))) → (vol‘∪ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))) = sup((vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))), ℝ*, < ))
91	32, 89, 90	syl2anc 585	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (vol‘∪ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))) = sup((vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))), ℝ*, < ))
92	66, 91	eqtr3d 2775	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (vol‘𝐴) = sup((vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))), ℝ*, < ))
93	92	breq1d 5158	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ((vol‘𝐴) ≤ 𝐵 ↔ sup((vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))), ℝ*, < ) ≤ 𝐵))
94		imassrn 6069	. . . . . . 7 ⊢ (vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))) ⊆ ran vol
95		frn 6722	. . . . . . . . 9 ⊢ (vol:dom vol⟶(0[,]+∞) → ran vol ⊆ (0[,]+∞))
96	5, 95	ax-mp 5	. . . . . . . 8 ⊢ ran vol ⊆ (0[,]+∞)
97	96, 4	sstri 3991	. . . . . . 7 ⊢ ran vol ⊆ ℝ*
98	94, 97	sstri 3991	. . . . . 6 ⊢ (vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))) ⊆ ℝ*
99		supxrleub 13302	. . . . . 6 ⊢ (((vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))) ⊆ ℝ* ∧ 𝐵 ∈ ℝ*) → (sup((vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))), ℝ*, < ) ≤ 𝐵 ↔ ∀𝑛 ∈ (vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))))𝑛 ≤ 𝐵))
100	98, 3, 99	sylancr 588	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (sup((vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))), ℝ*, < ) ≤ 𝐵 ↔ ∀𝑛 ∈ (vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))))𝑛 ≤ 𝐵))
101		ffn 6715	. . . . . . . 8 ⊢ (vol:dom vol⟶(0[,]+∞) → vol Fn dom vol)
102	5, 101	ax-mp 5	. . . . . . 7 ⊢ vol Fn dom vol
103	32	frnd 6723	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) ⊆ dom vol)
104		breq1 5151	. . . . . . . 8 ⊢ (𝑛 = (vol‘𝑧) → (𝑛 ≤ 𝐵 ↔ (vol‘𝑧) ≤ 𝐵))
105	104	ralima 7237	. . . . . . 7 ⊢ ((vol Fn dom vol ∧ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) ⊆ dom vol) → (∀𝑛 ∈ (vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))))𝑛 ≤ 𝐵 ↔ ∀𝑧 ∈ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))(vol‘𝑧) ≤ 𝐵))
106	102, 103, 105	sylancr 588	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (∀𝑛 ∈ (vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))))𝑛 ≤ 𝐵 ↔ ∀𝑧 ∈ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))(vol‘𝑧) ≤ 𝐵))
107		fveq2 6889	. . . . . . . . . 10 ⊢ (𝑧 = ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) → (vol‘𝑧) = (vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛)))
108	107	breq1d 5158	. . . . . . . . 9 ⊢ (𝑧 = ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) → ((vol‘𝑧) ≤ 𝐵 ↔ (vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛)) ≤ 𝐵))
109	108	ralrn 7087	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) Fn ℕ → (∀𝑧 ∈ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))(vol‘𝑧) ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ (vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛)) ≤ 𝐵))
110	33, 109	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (∀𝑧 ∈ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))(vol‘𝑧) ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ (vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛)) ≤ 𝐵))
111	19	fveq2d 6893	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛)) = (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
112	111	breq1d 5158	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛)) ≤ 𝐵 ↔ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵))
113	112	ralbiia 3092	. . . . . . 7 ⊢ (∀𝑛 ∈ ℕ (vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛)) ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵)
114	110, 113	bitrdi 287	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (∀𝑧 ∈ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))(vol‘𝑧) ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵))
115	106, 114	bitrd 279	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (∀𝑛 ∈ (vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))))𝑛 ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵))
116	93, 100, 115	3bitrd 305	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ((vol‘𝐴) ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵))
117	11, 116	mtbid 324	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ¬ ∀𝑛 ∈ ℕ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵)
118		rexnal 3101	. . 3 ⊢ (∃𝑛 ∈ ℕ ¬ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵 ↔ ¬ ∀𝑛 ∈ ℕ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵)
119	117, 118	sylibr 233	. 2 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ∃𝑛 ∈ ℕ ¬ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵)
120	3	adantr 482	. . . 4 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ*)
121	5	ffvelcdmi 7083	. . . . . 6 ⊢ ((𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol → (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ (0[,]+∞))
122	4, 121	sselid 3980	. . . . 5 ⊢ ((𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol → (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ ℝ*)
123	30, 122	syl 17	. . . 4 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ ℝ*)
124		xrltnle 11278	. . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ ℝ*) → (𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ↔ ¬ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵))
125	120, 123, 124	syl2anc 585	. . 3 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ↔ ¬ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵))
126	125	rexbidva 3177	. 2 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (∃𝑛 ∈ ℕ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ↔ ∃𝑛 ∈ ℕ ¬ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵))
127	119, 126	mpbird 257	1 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ∃𝑛 ∈ ℕ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   ∩ cin 3947   ⊆ wss 3948  ∪ cuni 4908  ∪ ciun 4997   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  ran crn 5677   “ cima 5679   Fn wfn 6536  ⟶wf 6537  ‘cfv 6541  (class class class)co 7406  supcsup 9432  ℝcr 11106  0cc0 11107  1c1 11108   + caddc 11110  +∞cpnf 11242  ℝ^*cxr 11244   < clt 11245   ≤ cle 11246  -cneg 11442  ℕcn 12209  [,]cicc 13324  abscabs 15178  volcvol 24972

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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-inf2 9633  ax-cc 10427  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185

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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-of 7667  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-2o 8464  df-er 8700  df-map 8819  df-pm 8820  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-inf 9435  df-oi 9502  df-dju 9893  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-z 12556  df-uz 12820  df-q 12930  df-rp 12972  df-xadd 13090  df-ioo 13325  df-ico 13327  df-icc 13328  df-fz 13482  df-fzo 13625  df-fl 13754  df-seq 13964  df-exp 14025  df-hash 14288  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-clim 15429  df-rlim 15430  df-sum 15630  df-xmet 20930  df-met 20931  df-ovol 24973  df-vol 24974

This theorem is referenced by:  volivth  25116