Theorem volsup2 24969

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 11201	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*)
3	2	3ad2ant2 1134	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → 𝐵 ∈ ℝ*)
4		iccssxr 13347	. . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ*
5		volf 24893	. . . . . . . . 9 ⊢ vol:dom vol⟶(0[,]+∞)
6	5	ffvelcdmi 7034	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) ∈ (0[,]+∞))
7	4, 6	sselid 3942	. . . . . . 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 11222	. . . . . 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 231	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ¬ (vol‘𝐴) ≤ 𝐵)
12		negeq 11393	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → -𝑚 = -𝑛)
13		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
14	12, 13	oveq12d 7375	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (-𝑚[,]𝑚) = (-𝑛[,]𝑛))
15	14	ineq2d 4172	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝐴 ∩ (-𝑚[,]𝑚)) = (𝐴 ∩ (-𝑛[,]𝑛)))
16		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) = (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))
17		ovex 7390	. . . . . . . . . . . . 13 ⊢ (-𝑛[,]𝑛) ∈ V
18	17	inex2 5275	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ (-𝑛[,]𝑛)) ∈ V
19	15, 16, 18	fvmpt 6948	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) = (𝐴 ∩ (-𝑛[,]𝑛)))
20	19	iuneq2i 4975	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) = ∪ 𝑛 ∈ ℕ (𝐴 ∩ (-𝑛[,]𝑛))
21		iunin2 5031	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ ℕ (𝐴 ∩ (-𝑛[,]𝑛)) = (𝐴 ∩ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛))
22	20, 21	eqtri 2764	. . . . . . . . 9 ⊢ ∪ 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) = (𝐴 ∩ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛))
23		simpl1 1191	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ dom vol)
24		nnre 12160	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
25	24	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
26	25	renegcld 11582	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → -𝑛 ∈ ℝ)
27		iccmbl 24930	. . . . . . . . . . . . . 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 24906	. . . . . . . . . . . . 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 5218	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) = (𝑛 ∈ ℕ ↦ (𝐴 ∩ (-𝑛[,]𝑛)))
32	30, 31	fmptd 7062	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))):ℕ⟶dom vol)
33	32	ffnd 6669	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) Fn ℕ)
34		fniunfv 7194	. . . . . . . . . 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 24895	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
37	36	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → 𝐴 ⊆ ℝ)
38	37	sselda 3944	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
39		recn 11141	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
40	39	abscld 15321	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → (abs‘𝑥) ∈ ℝ)
41		arch 12410	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘𝑥) ∈ ℝ → ∃𝑛 ∈ ℕ (abs‘𝑥) < 𝑛)
42	40, 41	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → ∃𝑛 ∈ ℕ (abs‘𝑥) < 𝑛)
43		ltle 11243	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((-𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛) → (𝑥 ∈ ℝ ∧ -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
48		absle 15200	. . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
51	50	renegcld 11582	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → -𝑛 ∈ ℝ)
52		elicc2 13329	. . . . . . . . . . . . . . . . . . 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 293	. . . . . . . . . . . . . . . . 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 3165	. . . . . . . . . . . . . . 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 413	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝑥 ∈ 𝐴 → ∃𝑛 ∈ ℕ 𝑥 ∈ (-𝑛[,]𝑛)))
60		eliun 4958	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛) ↔ ∃𝑛 ∈ ℕ 𝑥 ∈ (-𝑛[,]𝑛))
61	59, 60	syl6ibr 251	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛)))
62	61	ssrdv 3950	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → 𝐴 ⊆ ∪ 𝑛 ∈ ℕ (-𝑛[,]𝑛))
63		df-ss 3927	. . . . . . . . . 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 2800	. . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ∪ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) = 𝐴)
66	65	fveq2d 6846	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (vol‘∪ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))) = (vol‘𝐴))
67		peano2re 11328	. . . . . . . . . . . . . 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 11582	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → -(𝑛 + 1) ∈ ℝ)
70	25	lep1d 12086	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑛 ≤ (𝑛 + 1))
71	25, 68	lenegd 11734	. . . . . . . . . . . . 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 13332	. . . . . . . . . . . 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 837	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (-𝑛[,]𝑛) ⊆ (-(𝑛 + 1)[,](𝑛 + 1)))
75		sslin 4194	. . . . . . . . . . 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 482	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) = (𝐴 ∩ (-𝑛[,]𝑛)))
78		peano2nn 12165	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
79	78	adantl 482	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
80		negeq 11393	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 + 1) → -𝑚 = -(𝑛 + 1))
81		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 + 1) → 𝑚 = (𝑛 + 1))
82	80, 81	oveq12d 7375	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → (-𝑚[,]𝑚) = (-(𝑛 + 1)[,](𝑛 + 1)))
83	82	ineq2d 4172	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝐴 ∩ (-𝑚[,]𝑚)) = (𝐴 ∩ (-(𝑛 + 1)[,](𝑛 + 1))))
84		ovex 7390	. . . . . . . . . . . . 13 ⊢ (-(𝑛 + 1)[,](𝑛 + 1)) ∈ V
85	84	inex2 5275	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ (-(𝑛 + 1)[,](𝑛 + 1))) ∈ V
86	83, 16, 85	fvmpt 6948	. . . . . . . . . . 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 3991	. . . . . . . . 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 3143	. . . . . . . 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 24920	. . . . . . . 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 2778	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (vol‘𝐴) = sup((vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))), ℝ*, < ))
93	92	breq1d 5115	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ((vol‘𝐴) ≤ 𝐵 ↔ sup((vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))), ℝ*, < ) ≤ 𝐵))
94		imassrn 6024	. . . . . . 7 ⊢ (vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))) ⊆ ran vol
95		frn 6675	. . . . . . . . 9 ⊢ (vol:dom vol⟶(0[,]+∞) → ran vol ⊆ (0[,]+∞))
96	5, 95	ax-mp 5	. . . . . . . 8 ⊢ ran vol ⊆ (0[,]+∞)
97	96, 4	sstri 3953	. . . . . . 7 ⊢ ran vol ⊆ ℝ*
98	94, 97	sstri 3953	. . . . . 6 ⊢ (vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))) ⊆ ℝ*
99		supxrleub 13245	. . . . . 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 6668	. . . . . . . 8 ⊢ (vol:dom vol⟶(0[,]+∞) → vol Fn dom vol)
102	5, 101	ax-mp 5	. . . . . . 7 ⊢ vol Fn dom vol
103	32	frnd 6676	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚))) ⊆ dom vol)
104		breq1 5108	. . . . . . . 8 ⊢ (𝑛 = (vol‘𝑧) → (𝑛 ≤ 𝐵 ↔ (vol‘𝑧) ≤ 𝐵))
105	104	ralima 7188	. . . . . . 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 6842	. . . . . . . . . 10 ⊢ (𝑧 = ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) → (vol‘𝑧) = (vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛)))
108	107	breq1d 5115	. . . . . . . . 9 ⊢ (𝑧 = ((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛) → ((vol‘𝑧) ≤ 𝐵 ↔ (vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛)) ≤ 𝐵))
109	108	ralrn 7038	. . . . . . . 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 6846	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛)) = (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
112	111	breq1d 5115	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛)) ≤ 𝐵 ↔ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵))
113	112	ralbiia 3094	. . . . . . 7 ⊢ (∀𝑛 ∈ ℕ (vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ (-𝑚[,]𝑚)))‘𝑛)) ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵)
114	110, 113	bitrdi 286	. . . . . 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 278	. . . . 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 304	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ((vol‘𝐴) ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵))
117	11, 116	mtbid 323	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ¬ ∀𝑛 ∈ ℕ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ≤ 𝐵)
118		rexnal 3103	. . 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 481	. . . 4 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ*)
121	5	ffvelcdmi 7034	. . . . . 6 ⊢ ((𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol → (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ (0[,]+∞))
122	4, 121	sselid 3942	. . . . 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 11222	. . . 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 3173	. 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 256	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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073   ∩ cin 3909   ⊆ wss 3910  ∪ cuni 4865  ∪ ciun 4954   class class class wbr 5105   ↦ cmpt 5188  dom cdm 5633  ran crn 5634   “ cima 5636   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  supcsup 9376  ℝcr 11050  0cc0 11051  1c1 11052   + caddc 11054  +∞cpnf 11186  ℝ^*cxr 11188   < clt 11189   ≤ cle 11190  -cneg 11386  ℕcn 12153  [,]cicc 13267  abscabs 15119  volcvol 24827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cc 10371  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xadd 13034  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-xmet 20789  df-met 20790  df-ovol 24828  df-vol 24829

This theorem is referenced by:  volivth  24971