Theorem volsup2 25981

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 1156	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → 𝐵 < (vol‘𝐴))
2		rexr 11394	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*)
3	2	3ad2ant2 1152	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → 𝐵 ∈ ℝ*)
4		iccssxr 13601	. . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ*
5		volf 25905	. . . . . . . . 9 ⊢ vol:dom vol⟶(0[,]+∞)
6	5	ffvelcdmi 7115	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) ∈ (0[,]+∞))
7	4, 6	sselid 3911	. . . . . . 7 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) ∈ ℝ*)
8	7	3ad2ant1 1151	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (vol‘𝐴) ∈ ℝ*)
9		xrltnle 11415	. . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ (vol‘𝐴) ∈ ℝ*) → (𝐵 < (vol‘𝐴) ↔ ¬ (vol‘𝐴) ≤ 𝐵))
10	3, 8, 9	syl2anc 596	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝐵 < (vol‘𝐴) ↔ ¬ (vol‘𝐴) ≤ 𝐵))
11	1, 10	mpbid 235	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ¬ (vol‘𝐴) ≤ 𝐵)
12		negeq 11588	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → -𝑚 = -𝑛)
13		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
14	12, 13	oveq12d 7469	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ( -𝑚[,]𝑚) = ( -𝑛[,]𝑛))
15	14	ineq2d 4120	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (𝐴 ∩ ( -𝑚[,]𝑚)) = (𝐴 ∩ ( -𝑛[,]𝑛)))
16		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚))) = (𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))
17		ovex 7484	. . . . . . . . . . . . 13 ⊢ ( -𝑛[,]𝑛) ∈ V
18	17	inex2 5285	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ ( -𝑛[,]𝑛)) ∈ V
19	15, 16, 18	fvmpt 7024	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))‘𝑛) = (𝐴 ∩ ( -𝑛[,]𝑛)))
20	19	iuneq2i 4978	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))‘𝑛) = ∪ 𝑛 ∈ ℕ (𝐴 ∩ ( -𝑛[,]𝑛))
21		iunin2 5034	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ ℕ (𝐴 ∩ ( -𝑛[,]𝑛)) = (𝐴 ∩ ∪ 𝑛 ∈ ℕ ( -𝑛[,]𝑛))
22	20, 21	eqtri 2786	. . . . . . . . 9 ⊢ ∪ 𝑛 ∈ ℕ ((𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))‘𝑛) = (𝐴 ∩ ∪ 𝑛 ∈ ℕ ( -𝑛[,]𝑛))
23		simpl1 1210	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ dom vol)
24		nnre 12382	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
25	24	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
26	25	renegcld 11782	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → -𝑛 ∈ ℝ)
27		iccmbl 25942	. . . . . . . . . . . . . 14 ⊢ (( -𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) → ( -𝑛[,]𝑛) ∈ dom vol)
28	26, 25, 27	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → ( -𝑛[,]𝑛) ∈ dom vol)
29		inmbl 25918	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ dom vol ∧ ( -𝑛[,]𝑛) ∈ dom vol) → (𝐴 ∩ ( -𝑛[,]𝑛)) ∈ dom vol)
30	23, 28, 29	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (𝐴 ∩ ( -𝑛[,]𝑛)) ∈ dom vol)
31	15	cbvmptv 5215	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚))) = (𝑛 ∈ ℕ ↦ (𝐴 ∩ ( -𝑛[,]𝑛)))
32	30, 31	fmptd 7146	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚))):ℕ⟶dom vol)
33	32	ffnd 6740	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚))) Fn ℕ)
34		fniunfv 7282	. . . . . . . . . 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 18	. . . . . . . . 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 25907	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
37	36	3ad2ant1 1151	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → 𝐴 ⊆ ℝ)
38	37	sselda 3913	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ)
39		recn 11329	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
40	39	abscld 15650	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → (abs‘𝑥) ∈ ℝ)
41		arch 12643	. . . . . . . . . . . . . . . 16 ⊢ ((abs‘𝑥) ∈ ℝ → ∃𝑛 ∈ ℕ (abs‘𝑥) < 𝑛)
42	40, 41	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → ∃𝑛 ∈ ℕ (abs‘𝑥) < 𝑛)
43		ltle 11437	. . . . . . . . . . . . . . . . . 18 ⊢ (((abs‘𝑥) ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((abs‘𝑥) < 𝑛 → (abs‘𝑥) ≤ 𝑛))
44	40, 24, 43	syl2an 608	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((abs‘𝑥) < 𝑛 → (abs‘𝑥) ≤ 𝑛))
45		id 23	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛) → (𝑥 ∈ ℝ ∧ -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛))
46	45	3expib 1140	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ → (( -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛) → (𝑥 ∈ ℝ ∧ -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
47	46	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (( -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛) → (𝑥 ∈ ℝ ∧ -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
48		absle 15526	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((abs‘𝑥) ≤ 𝑛 ↔ ( -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
49	24, 48	sylan2 605	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((abs‘𝑥) ≤ 𝑛 ↔ ( -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
50	24	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
51	50	renegcld 11782	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → -𝑛 ∈ ℝ)
52		elicc2 13582	. . . . . . . . . . . . . . . . . . 19 ⊢ (( -𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝑥 ∈ ( -𝑛[,]𝑛) ↔ (𝑥 ∈ ℝ ∧ -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
53	51, 50, 52	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑥 ∈ ( -𝑛[,]𝑛) ↔ (𝑥 ∈ ℝ ∧ -𝑛 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛)))
54	47, 49, 53	3imtr4d 297	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((abs‘𝑥) ≤ 𝑛 → 𝑥 ∈ ( -𝑛[,]𝑛)))
55	44, 54	syld 48	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((abs‘𝑥) < 𝑛 → 𝑥 ∈ ( -𝑛[,]𝑛)))
56	55	reximdva 3179	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → (∃𝑛 ∈ ℕ (abs‘𝑥) < 𝑛 → ∃𝑛 ∈ ℕ 𝑥 ∈ ( -𝑛[,]𝑛)))
57	42, 56	mpd 16	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → ∃𝑛 ∈ ℕ 𝑥 ∈ ( -𝑛[,]𝑛))
58	38, 57	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑥 ∈ 𝐴) → ∃𝑛 ∈ ℕ 𝑥 ∈ ( -𝑛[,]𝑛))
59	58	ex 418	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝑥 ∈ 𝐴 → ∃𝑛 ∈ ℕ 𝑥 ∈ ( -𝑛[,]𝑛)))
60		eliun 4960	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ ℕ ( -𝑛[,]𝑛) ↔ ∃𝑛 ∈ ℕ 𝑥 ∈ ( -𝑛[,]𝑛))
61	59, 60	imbitrrdi 255	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝑛 ∈ ℕ ( -𝑛[,]𝑛)))
62	61	ssrdv 3919	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → 𝐴 ⊆ ∪ 𝑛 ∈ ℕ ( -𝑛[,]𝑛))
63		dfss2 4097	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ∪ 𝑛 ∈ ℕ ( -𝑛[,]𝑛) ↔ (𝐴 ∩ ∪ 𝑛 ∈ ℕ ( -𝑛[,]𝑛)) = 𝐴)
64	62, 63	sylib 221	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (𝐴 ∩ ∪ 𝑛 ∈ ℕ ( -𝑛[,]𝑛)) = 𝐴)
65	22, 35, 64	3eqtr3a 2822	. . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ∪ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚))) = 𝐴)
66	65	fveq2d 6920	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (vol‘∪ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))) = (vol‘𝐴))
67		peano2re 11522	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
68	25, 67	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℝ)
69	68	renegcld 11782	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → -(𝑛 + 1) ∈ ℝ)
70	25	lep1d 12288	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → 𝑛 ≤ (𝑛 + 1))
71	25, 68	lenegd 11935	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑛 ≤ (𝑛 + 1) ↔ -(𝑛 + 1) ≤ -𝑛))
72	70, 71	mpbid 235	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → -(𝑛 + 1) ≤ -𝑛)
73		iccss 13585	. . . . . . . . . . . 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 852	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → ( -𝑛[,]𝑛) ⊆ ( -(𝑛 + 1)[,](𝑛 + 1)))
75		sslin 4142	. . . . . . . . . . 11 ⊢ (( -𝑛[,]𝑛) ⊆ ( -(𝑛 + 1)[,](𝑛 + 1)) → (𝐴 ∩ ( -𝑛[,]𝑛)) ⊆ (𝐴 ∩ ( -(𝑛 + 1)[,](𝑛 + 1))))
76	74, 75	syl 18	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (𝐴 ∩ ( -𝑛[,]𝑛)) ⊆ (𝐴 ∩ ( -(𝑛 + 1)[,](𝑛 + 1))))
77	19	adantl 487	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))‘𝑛) = (𝐴 ∩ ( -𝑛[,]𝑛)))
78		peano2nn 12387	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
79	78	adantl 487	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
80		negeq 11588	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 + 1) → -𝑚 = -(𝑛 + 1))
81		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 + 1) → 𝑚 = (𝑛 + 1))
82	80, 81	oveq12d 7469	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → ( -𝑚[,]𝑚) = ( -(𝑛 + 1)[,](𝑛 + 1)))
83	82	ineq2d 4120	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝐴 ∩ ( -𝑚[,]𝑚)) = (𝐴 ∩ ( -(𝑛 + 1)[,](𝑛 + 1))))
84		ovex 7484	. . . . . . . . . . . . 13 ⊢ ( -(𝑛 + 1)[,](𝑛 + 1)) ∈ V
85	84	inex2 5285	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ ( -(𝑛 + 1)[,](𝑛 + 1))) ∈ V
86	83, 16, 85	fvmpt 7024	. . . . . . . . . . 11 ⊢ ((𝑛 + 1) ∈ ℕ → ((𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))‘(𝑛 + 1)) = (𝐴 ∩ ( -(𝑛 + 1)[,](𝑛 + 1))))
87	79, 86	syl 18	. . . . . . . . . 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 3968	. . . . . . . . 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 3158	. . . . . . . 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 25932	. . . . . . . 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 596	. . . . . . 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 2800	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → (vol‘𝐴) = sup((vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))), ℝ*, < ))
93	92	breq1d 5118	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ((vol‘𝐴) ≤ 𝐵 ↔ sup((vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))), ℝ*, < ) ≤ 𝐵))
94		imassrn 6216	. . . . . . 7 ⊢ (vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))) ⊆ ran vol
95		frn 6747	. . . . . . . . 9 ⊢ (vol:dom vol⟶(0[,]+∞) → ran vol ⊆ (0[,]+∞))
96	5, 95	ax-mp 5	. . . . . . . 8 ⊢ ran vol ⊆ (0[,]+∞)
97	96, 4	sstri 3922	. . . . . . 7 ⊢ ran vol ⊆ ℝ*
98	94, 97	sstri 3922	. . . . . 6 ⊢ (vol “ ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))) ⊆ ℝ*
99		supxrleub 13496	. . . . . 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 599	. . . . 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 6739	. . . . . . . 8 ⊢ (vol:dom vol⟶(0[,]+∞) → vol Fn dom vol)
102	5, 101	ax-mp 5	. . . . . . 7 ⊢ vol Fn dom vol
103	32	frnd 6748	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ran (𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚))) ⊆ dom vol)
104		breq1 5111	. . . . . . . 8 ⊢ (𝑛 = (vol‘𝑧) → (𝑛 ≤ 𝐵 ↔ (vol‘𝑧) ≤ 𝐵))
105	104	ralima 7274	. . . . . . 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 599	. . . . . 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 6916	. . . . . . . . . 10 ⊢ (𝑧 = ((𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))‘𝑛) → (vol‘𝑧) = (vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))‘𝑛)))
108	107	breq1d 5118	. . . . . . . . 9 ⊢ (𝑧 = ((𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))‘𝑛) → ((vol‘𝑧) ≤ 𝐵 ↔ (vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))‘𝑛)) ≤ 𝐵))
109	108	ralrn 7120	. . . . . . . 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 18	. . . . . . 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 6920	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))‘𝑛)) = (vol‘(𝐴 ∩ ( -𝑛[,]𝑛))))
112	111	breq1d 5118	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → ((vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))‘𝑛)) ≤ 𝐵 ↔ (vol‘(𝐴 ∩ ( -𝑛[,]𝑛))) ≤ 𝐵))
113	112	ralbiia 3110	. . . . . . 7 ⊢ (∀𝑛 ∈ ℕ (vol‘((𝑚 ∈ ℕ ↦ (𝐴 ∩ ( -𝑚[,]𝑚)))‘𝑛)) ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ (vol‘(𝐴 ∩ ( -𝑛[,]𝑛))) ≤ 𝐵)
114	110, 113	bitrdi 290	. . . . . 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 282	. . . . 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 308	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ((vol‘𝐴) ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ (vol‘(𝐴 ∩ ( -𝑛[,]𝑛))) ≤ 𝐵))
117	11, 116	mtbid 327	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ¬ ∀𝑛 ∈ ℕ (vol‘(𝐴 ∩ ( -𝑛[,]𝑛))) ≤ 𝐵)
118		rexnal 3118	. . 3 ⊢ (∃𝑛 ∈ ℕ ¬ (vol‘(𝐴 ∩ ( -𝑛[,]𝑛))) ≤ 𝐵 ↔ ¬ ∀𝑛 ∈ ℕ (vol‘(𝐴 ∩ ( -𝑛[,]𝑛))) ≤ 𝐵)
119	117, 118	sylibr 237	. 2 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ∃𝑛 ∈ ℕ ¬ (vol‘(𝐴 ∩ ( -𝑛[,]𝑛))) ≤ 𝐵)
120	3	adantr 486	. . . 4 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ*)
121	5	ffvelcdmi 7115	. . . . . 6 ⊢ ((𝐴 ∩ ( -𝑛[,]𝑛)) ∈ dom vol → (vol‘(𝐴 ∩ ( -𝑛[,]𝑛))) ∈ (0[,]+∞))
122	4, 121	sselid 3911	. . . . 5 ⊢ ((𝐴 ∩ ( -𝑛[,]𝑛)) ∈ dom vol → (vol‘(𝐴 ∩ ( -𝑛[,]𝑛))) ∈ ℝ*)
123	30, 122	syl 18	. . . 4 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐴 ∩ ( -𝑛[,]𝑛))) ∈ ℝ*)
124		xrltnle 11415	. . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ (vol‘(𝐴 ∩ ( -𝑛[,]𝑛))) ∈ ℝ*) → (𝐵 < (vol‘(𝐴 ∩ ( -𝑛[,]𝑛))) ↔ ¬ (vol‘(𝐴 ∩ ( -𝑛[,]𝑛))) ≤ 𝐵))
125	120, 123, 124	syl2anc 596	. . 3 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) ∧ 𝑛 ∈ ℕ) → (𝐵 < (vol‘(𝐴 ∩ ( -𝑛[,]𝑛))) ↔ ¬ (vol‘(𝐴 ∩ ( -𝑛[,]𝑛))) ≤ 𝐵))
126	125	rexbidva 3188	. 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 260	1 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ∃𝑛 ∈ ℕ 𝐵 < (vol‘(𝐴 ∩ ( -𝑛[,]𝑛))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092  ∪ cuni 4872  ∪ ciun 4956   class class class wbr 5108   ↦ cmpt 5192  dom cdm 5658  ran crn 5659   “ cima 5661   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  supcsup 9469  ℝcr 11238  0cc0 11239  1c1 11240   + caddc 11242  +∞cpnf 11379  ℝ^*cxr 11381   < clt 11382   ≤ cle 11383   -cneg 11581  ℕcn 12375  [,]cicc 13519  abscabs 15444  volcvol 25839

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679  ax-cc 10552  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-disj 5076  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-of 7725  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-er 8755  df-map 8887  df-pm 8888  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-sup 9471  df-inf 9472  df-oi 9541  df-dju 10019  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-div 12014  df-nn 12376  df-2 12445  df-3 12446  df-n0 12647  df-z 12734  df-uz 13006  df-q 13116  df-rp 13161  df-xadd 13282  df-ioo 13520  df-ico 13522  df-icc 13523  df-fz 13680  df-fzo 13829  df-fl 13972  df-seq 14185  df-exp 14245  df-hash 14516  df-cj 15309  df-re 15310  df-im 15311  df-sqrt 15445  df-abs 15446  df-clim 15699  df-rlim 15700  df-sum 15898  df-xmet 21722  df-met 21723  df-ovol 25840  df-vol 25841

This theorem is used by:  volivth  25983