Theorem volsup2 25533

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 1138	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → 𝐵 < (vol‘𝐴))
2		rexr 11158	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*)
3	2	3ad2ant2 1134	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → 𝐵 ∈ ℝ*)
4		iccssxr 13330	. . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ*
5		volf 25457	. . . . . . . . 9 ⊢ vol:dom vol⟶(0[,]+∞)
6	5	ffvelcdmi 7016	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) ∈ (0[,]+∞))
7	4, 6	sselid 3927	. . . . . . 7 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) ∈ ℝ*)
8	7	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (vol‘𝐴) ∈ ℝ*)
9		xrltnle 11179	. . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ (vol‘𝐴) ∈ ℝ*) → (𝐵 < (vol‘𝐴) ↔ ¬ (vol‘𝐴) ≤ 𝐵))
10	3, 8, 9	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝐵 < (vol‘𝐴) ↔ ¬ (vol‘𝐴) ≤ 𝐵))
11	1, 10	mpbid 232	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ¬ (vol‘𝐴) ≤ 𝐵)
12		negeq 11352	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → -𝑚 = -𝑛)
13		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
14	12, 13	oveq12d 7364	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (-𝑚[,]𝑚) = (-𝑛[,]𝑛))
15	14	ineq2d 4167	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝐴 ∩ (-𝑚[,]𝑚)) = (𝐴 ∩ (-𝑛[,]𝑛)))
16		eqid 2731	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) = (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))
17		ovex 7379	. . . . . . . . . . . . 13 ⊢ (-𝑛[,]𝑛) ∈ V
18	17	inex2 5254	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ (-𝑛[,]𝑛)) ∈ V
19	15, 16, 18	fvmpt 6929	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) = (𝐴 ∩ (-𝑛[,]𝑛)))
20	19	iuneq2i 4961	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) = ∪ 𝑛 ∈ ℕ (𝐴 ∩ (-𝑛[,]𝑛))
21		iunin2 5017	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ ℕ (𝐴 ∩ (-𝑛[,]𝑛)) = (𝐴 ∩ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛))
22	20, 21	eqtri 2754	. . . . . . . . 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 12132	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
25	24	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
26	25	renegcld 11544	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → -𝑛 ∈ ℝ)
27		iccmbl 25494	. . . . . . . . . . . . . 14 ⊢ ((-𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (-𝑛[,]𝑛) ∈ dom vol)
28	26, 25, 27	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (-𝑛[,]𝑛) ∈ dom vol)
29		inmbl 25470	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ dom vol ∧ (-𝑛[,]𝑛) ∈ dom vol) → (𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol)
30	23, 28, 29	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol)
31	15	cbvmptv 5193	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) = (𝑛 ∈ ℕ ↦ (𝐴 ∩ (-𝑛[,]𝑛)))
32	30, 31	fmptd 7047	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))):ℕ⟶dom vol)
33	32	ffnd 6652	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) Fn ℕ)
34		fniunfv 7181	. . . . . . . . . 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 25459	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
37	36	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → 𝐴 ⊆ ℝ)
38	37	sselda 3929	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
39		recn 11096	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
40	39	abscld 15346	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → (abs‘𝑥) ∈ ℝ)
41		arch 12378	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘𝑥) ∈ ℝ → ∃𝑛 ∈ ℕ (abs‘𝑥) < 𝑛)
42	40, 41	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → ∃𝑛 ∈ ℕ (abs‘𝑥) < 𝑛)
43		ltle 11201	. . . . . . . . . . . . . . . . . 18 ⊢ (((abs‘𝑥) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((abs‘𝑥) < 𝑛 → (abs‘𝑥) ≤ 𝑛))
44	40, 24, 43	syl2an 596	. . . . . . . . . . . . . . . . 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 1122	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ → ((-𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛) → (𝑥 ∈ ℝ ∧ -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
47	46	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((-𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛) → (𝑥 ∈ ℝ ∧ -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
48		absle 15223	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((abs‘𝑥) ≤ 𝑛 ↔ (-𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
49	24, 48	sylan2 593	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((abs‘𝑥) ≤ 𝑛 ↔ (-𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
50	24	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
51	50	renegcld 11544	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → -𝑛 ∈ ℝ)
52		elicc2 13311	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝑥 ∈ (-𝑛[,]𝑛) ↔ (𝑥 ∈ ℝ ∧ -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
53	51, 50, 52	syl2anc 584	. . . . . . . . . . . . . . . . . 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 3145	. . . . . . . . . . . . . . 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 412	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝑥 ∈ 𝐴 → ∃𝑛 ∈ ℕ 𝑥 ∈ (-𝑛[,]𝑛)))
60		eliun 4943	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛) ↔ ∃𝑛 ∈ ℕ 𝑥 ∈ (-𝑛[,]𝑛))
61	59, 60	imbitrrdi 252	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛)))
62	61	ssrdv 3935	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → 𝐴 ⊆ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛))
63		dfss2 3915	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛) ↔ (𝐴 ∩ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛)) = 𝐴)
64	62, 63	sylib 218	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝐴 ∩ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛)) = 𝐴)
65	22, 35, 64	3eqtr3a 2790	. . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ∪ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) = 𝐴)
66	65	fveq2d 6826	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (vol‘∪ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))) = (vol‘𝐴))
67		peano2re 11286	. . . . . . . . . . . . . 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 11544	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → -(𝑛 + 1) ∈ ℝ)
70	25	lep1d 12053	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑛 ≤ (𝑛 + 1))
71	25, 68	lenegd 11696	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑛 ≤ (𝑛 + 1) ↔ -(𝑛 + 1) ≤ -𝑛))
72	70, 71	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → -(𝑛 + 1) ≤ -𝑛)
73		iccss 13314	. . . . . . . . . . . 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 4190	. . . . . . . . . . 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 481	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) = (𝐴 ∩ (-𝑛[,]𝑛)))
78		peano2nn 12137	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
79	78	adantl 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
80		negeq 11352	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 + 1) → -𝑚 = -(𝑛 + 1))
81		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 + 1) → 𝑚 = (𝑛 + 1))
82	80, 81	oveq12d 7364	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → (-𝑚[,]𝑚) = (-(𝑛 + 1)[,](𝑛 + 1)))
83	82	ineq2d 4167	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝐴 ∩ (-𝑚[,]𝑚)) = (𝐴 ∩ (-(𝑛 + 1)[,](𝑛 + 1))))
84		ovex 7379	. . . . . . . . . . . . 13 ⊢ (-(𝑛 + 1)[,](𝑛 + 1)) ∈ V
85	84	inex2 5254	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ (-(𝑛 + 1)[,](𝑛 + 1))) ∈ V
86	83, 16, 85	fvmpt 6929	. . . . . . . . . . 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 3985	. . . . . . . . 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 3124	. . . . . . . 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 25484	. . . . . . . 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 584	. . . . . . 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 2768	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (vol‘𝐴) = sup((vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))), ℝ*, < ))
93	92	breq1d 5099	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ((vol‘𝐴) ≤ 𝐵 ↔ sup((vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))), ℝ*, < ) ≤ 𝐵))
94		imassrn 6019	. . . . . . 7 ⊢ (vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))) ⊆ ran vol
95		frn 6658	. . . . . . . . 9 ⊢ (vol:dom vol⟶(0[,]+∞) → ran vol ⊆ (0[,]+∞))
96	5, 95	ax-mp 5	. . . . . . . 8 ⊢ ran vol ⊆ (0[,]+∞)
97	96, 4	sstri 3939	. . . . . . 7 ⊢ ran vol ⊆ ℝ*
98	94, 97	sstri 3939	. . . . . 6 ⊢ (vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))) ⊆ ℝ*
99		supxrleub 13225	. . . . . 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 587	. . . . 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 6651	. . . . . . . 8 ⊢ (vol:dom vol⟶(0[,]+∞) → vol Fn dom vol)
102	5, 101	ax-mp 5	. . . . . . 7 ⊢ vol Fn dom vol
103	32	frnd 6659	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) ⊆ dom vol)
104		breq1 5092	. . . . . . . 8 ⊢ (𝑛 = (vol‘𝑧) → (𝑛 ≤ 𝐵 ↔ (vol‘𝑧) ≤ 𝐵))
105	104	ralima 7171	. . . . . . 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 587	. . . . . 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 6822	. . . . . . . . . 10 ⊢ (𝑧 = ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) → (vol‘𝑧) = (vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛)))
108	107	breq1d 5099	. . . . . . . . 9 ⊢ (𝑧 = ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) → ((vol‘𝑧) ≤ 𝐵 ↔ (vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛)) ≤ 𝐵))
109	108	ralrn 7021	. . . . . . . 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 6826	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛)) = (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
112	111	breq1d 5099	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛)) ≤ 𝐵 ↔ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵))
113	112	ralbiia 3076	. . . . . . 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 3084	. . 3 ⊢ (∃𝑛 ∈ ℕ ¬ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵 ↔ ¬ ∀𝑛 ∈ ℕ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵)
119	117, 118	sylibr 234	. 2 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ∃𝑛 ∈ ℕ ¬ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵)
120	3	adantr 480	. . . 4 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ*)
121	5	ffvelcdmi 7016	. . . . . 6 ⊢ ((𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol → (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ (0[,]+∞))
122	4, 121	sselid 3927	. . . . 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 11179	. . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ ℝ*) → (𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ↔ ¬ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵))
125	120, 123, 124	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ↔ ¬ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵))
126	125	rexbidva 3154	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ∩ cin 3896   ⊆ wss 3897  ∪ cuni 4856  ∪ ciun 4939   class class class wbr 5089   ↦ cmpt 5170  dom cdm 5614  ran crn 5615   “ cima 5617   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  supcsup 9324  ℝcr 11005  0cc0 11006  1c1 11007   + caddc 11009  +∞cpnf 11143  ℝ^*cxr 11145   < clt 11146   ≤ cle 11147  -cneg 11345  ℕcn 12125  [,]cicc 13248  abscabs 15141  volcvol 25391

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531  ax-cc 10326  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-disj 5057  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-er 8622  df-map 8752  df-pm 8753  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-inf 9327  df-oi 9396  df-dju 9794  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-3 12189  df-n0 12382  df-z 12469  df-uz 12733  df-q 12847  df-rp 12891  df-xadd 13012  df-ioo 13249  df-ico 13251  df-icc 13252  df-fz 13408  df-fzo 13555  df-fl 13696  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-rlim 15396  df-sum 15594  df-xmet 21284  df-met 21285  df-ovol 25392  df-vol 25393

This theorem is referenced by:  volivth  25535