volivth - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > volivth		Structured version Visualization version GIF version

Theorem volivth 25515

Description: The Intermediate Value Theorem for the Lebesgue volume function. For any positive 𝐵 ≤ (vol‘𝐴), there is a measurable subset of 𝐴 whose volume is 𝐵. (Contributed by Mario Carneiro, 30-Aug-2014.)

**Assertion**
Ref	Expression
volivth	⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → ∃𝑥 ∈ dom vol(𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

Proof of Theorem volivth Dummy variables 𝑢 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 766	. . . 4 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → 𝐴 ∈ dom vol)
2		mnfxr 11238	. . . . . 6 ⊢ -∞ ∈ ℝ^*
3	2	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → -∞ ∈ ℝ^*)
4		iccssxr 13398	. . . . . . 7 ⊢ (0[,](vol‘𝐴)) ⊆ ℝ^*
5		simpr 484	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → 𝐵 ∈ (0[,](vol‘𝐴)))
6	4, 5	sselid 3947	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → 𝐵 ∈ ℝ^*)
7	6	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → 𝐵 ∈ ℝ^*)
8		iccssxr 13398	. . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ^*
9		volf 25437	. . . . . . . . 9 ⊢ vol:dom vol⟶(0[,]+∞)
10	9	ffvelcdmi 7058	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) ∈ (0[,]+∞))
11	8, 10	sselid 3947	. . . . . . 7 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) ∈ ℝ^*)
12	11	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → (vol‘𝐴) ∈ ℝ^*)
13	12	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → (vol‘𝐴) ∈ ℝ^*)
14		0xr 11228	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
15		elicc1 13357	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ (vol‘𝐴) ∈ ℝ^) → (𝐵 ∈ (0[,](vol‘𝐴)) ↔ (𝐵 ∈ ℝ^ ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ (vol‘𝐴))))
16	14, 12, 15	sylancr 587	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → (𝐵 ∈ (0[,](vol‘𝐴)) ↔ (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ (vol‘𝐴))))
17	5, 16	mpbid 232	. . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → (𝐵 ∈ ℝ^* ∧ 0 ≤ 𝐵 ∧ 𝐵 ≤ (vol‘𝐴)))
18	17	simp2d 1143	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → 0 ≤ 𝐵)
19	18	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → 0 ≤ 𝐵)
20		mnflt0 13092	. . . . . . . 8 ⊢ -∞ < 0
21		xrltletr 13124	. . . . . . . 8 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((-∞ < 0 ∧ 0 ≤ 𝐵) → -∞ < 𝐵))
22	20, 21	mpani 696	. . . . . . 7 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ 𝐵 → -∞ < 𝐵))
23	2, 14, 22	mp3an12 1453	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → (0 ≤ 𝐵 → -∞ < 𝐵))
24	7, 19, 23	sylc 65	. . . . 5 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → -∞ < 𝐵)
25		simpr 484	. . . . 5 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → 𝐵 < (vol‘𝐴))
26		xrre2 13137	. . . . 5 ⊢ (((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (vol‘𝐴) ∈ ℝ^*) ∧ (-∞ < 𝐵 ∧ 𝐵 < (vol‘𝐴))) → 𝐵 ∈ ℝ)
27	3, 7, 13, 24, 25, 26	syl32anc 1380	. . . 4 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → 𝐵 ∈ ℝ)
28		volsup2 25513	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ ∧ 𝐵 < (vol‘𝐴)) → ∃𝑛 ∈ ℕ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
29	1, 27, 25, 28	syl3anc 1373	. . 3 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → ∃𝑛 ∈ ℕ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
30		nnre 12200	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
31	30	ad2antrl 728	. . . . . 6 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 𝑛 ∈ ℝ)
32	31	renegcld 11612	. . . . 5 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → -𝑛 ∈ ℝ)
33	27	adantr 480	. . . . 5 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 𝐵 ∈ ℝ)
34		0red 11184	. . . . . 6 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 0 ∈ ℝ)
35		nngt0 12224	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 0 < 𝑛)
36	35	ad2antrl 728	. . . . . . 7 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 0 < 𝑛)
37	31	lt0neg2d 11755	. . . . . . 7 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (0 < 𝑛 ↔ -𝑛 < 0))
38	36, 37	mpbid 232	. . . . . 6 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → -𝑛 < 0)
39	32, 34, 31, 38, 36	lttrd 11342	. . . . 5 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → -𝑛 < 𝑛)
40		iccssre 13397	. . . . . 6 ⊢ ((-𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (-𝑛[,]𝑛) ⊆ ℝ)
41	32, 31, 40	syl2anc 584	. . . . 5 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (-𝑛[,]𝑛) ⊆ ℝ)
42		ax-resscn 11132	. . . . . . 7 ⊢ ℝ ⊆ ℂ
43		ssid 3972	. . . . . . 7 ⊢ ℂ ⊆ ℂ
44		cncfss 24799	. . . . . . 7 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℝ–cn→ℝ) ⊆ (ℝ–cn→ℂ))
45	42, 43, 44	mp2an 692	. . . . . 6 ⊢ (ℝ–cn→ℝ) ⊆ (ℝ–cn→ℂ)
46	1	adantr 480	. . . . . . 7 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 𝐴 ∈ dom vol)
47		eqid 2730	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦)))) = (𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))
48	47	volcn 25514	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ -𝑛 ∈ ℝ) → (𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦)))) ∈ (ℝ–cn→ℝ))
49	46, 32, 48	syl2anc 584	. . . . . 6 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦)))) ∈ (ℝ–cn→ℝ))
50	45, 49	sselid 3947	. . . . 5 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦)))) ∈ (ℝ–cn→ℂ))
51	41	sselda 3949	. . . . . 6 ⊢ (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑢 ∈ (-𝑛[,]𝑛)) → 𝑢 ∈ ℝ)
52		cncff 24793	. . . . . . . 8 ⊢ ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦)))) ∈ (ℝ–cn→ℝ) → (𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦)))):ℝ⟶ℝ)
53	49, 52	syl 17	. . . . . . 7 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦)))):ℝ⟶ℝ)
54	53	ffvelcdmda 7059	. . . . . 6 ⊢ (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑢 ∈ ℝ) → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑢) ∈ ℝ)
55	51, 54	syldan 591	. . . . 5 ⊢ (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑢 ∈ (-𝑛[,]𝑛)) → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑢) ∈ ℝ)
56		oveq2 7398	. . . . . . . . . . . 12 ⊢ (𝑦 = -𝑛 → (-𝑛[,]𝑦) = (-𝑛[,]-𝑛))
57	56	ineq2d 4186	. . . . . . . . . . 11 ⊢ (𝑦 = -𝑛 → (𝐴 ∩ (-𝑛[,]𝑦)) = (𝐴 ∩ (-𝑛[,]-𝑛)))
58	57	fveq2d 6865	. . . . . . . . . 10 ⊢ (𝑦 = -𝑛 → (vol‘(𝐴 ∩ (-𝑛[,]𝑦))) = (vol‘(𝐴 ∩ (-𝑛[,]-𝑛))))
59		fvex 6874	. . . . . . . . . 10 ⊢ (vol‘(𝐴 ∩ (-𝑛[,]-𝑛))) ∈ V
60	58, 47, 59	fvmpt 6971	. . . . . . . . 9 ⊢ (-𝑛 ∈ ℝ → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘-𝑛) = (vol‘(𝐴 ∩ (-𝑛[,]-𝑛))))
61	32, 60	syl 17	. . . . . . . 8 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘-𝑛) = (vol‘(𝐴 ∩ (-𝑛[,]-𝑛))))
62		inss2 4204	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ (-𝑛[,]-𝑛)) ⊆ (-𝑛[,]-𝑛)
63	32	rexrd 11231	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → -𝑛 ∈ ℝ^*)
64		iccid 13358	. . . . . . . . . . . . 13 ⊢ (-𝑛 ∈ ℝ^* → (-𝑛[,]-𝑛) = {-𝑛})
65	63, 64	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (-𝑛[,]-𝑛) = {-𝑛})
66	62, 65	sseqtrid 3992	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (𝐴 ∩ (-𝑛[,]-𝑛)) ⊆ {-𝑛})
67	32	snssd 4776	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → {-𝑛} ⊆ ℝ)
68	66, 67	sstrd 3960	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (𝐴 ∩ (-𝑛[,]-𝑛)) ⊆ ℝ)
69		ovolsn 25403	. . . . . . . . . . . 12 ⊢ (-𝑛 ∈ ℝ → (vol*‘{-𝑛}) = 0)
70	32, 69	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (vol*‘{-𝑛}) = 0)
71		ovolssnul 25395	. . . . . . . . . . 11 ⊢ (((𝐴 ∩ (-𝑛[,]-𝑛)) ⊆ {-𝑛} ∧ {-𝑛} ⊆ ℝ ∧ (vol‘{-𝑛}) = 0) → (vol‘(𝐴 ∩ (-𝑛[,]-𝑛))) = 0)
72	66, 67, 70, 71	syl3anc 1373	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (vol*‘(𝐴 ∩ (-𝑛[,]-𝑛))) = 0)
73		nulmbl 25443	. . . . . . . . . 10 ⊢ (((𝐴 ∩ (-𝑛[,]-𝑛)) ⊆ ℝ ∧ (vol*‘(𝐴 ∩ (-𝑛[,]-𝑛))) = 0) → (𝐴 ∩ (-𝑛[,]-𝑛)) ∈ dom vol)
74	68, 72, 73	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (𝐴 ∩ (-𝑛[,]-𝑛)) ∈ dom vol)
75		mblvol 25438	. . . . . . . . 9 ⊢ ((𝐴 ∩ (-𝑛[,]-𝑛)) ∈ dom vol → (vol‘(𝐴 ∩ (-𝑛[,]-𝑛))) = (vol*‘(𝐴 ∩ (-𝑛[,]-𝑛))))
76	74, 75	syl 17	. . . . . . . 8 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (vol‘(𝐴 ∩ (-𝑛[,]-𝑛))) = (vol*‘(𝐴 ∩ (-𝑛[,]-𝑛))))
77	61, 76, 72	3eqtrd 2769	. . . . . . 7 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘-𝑛) = 0)
78	19	adantr 480	. . . . . . 7 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 0 ≤ 𝐵)
79	77, 78	eqbrtrd 5132	. . . . . 6 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘-𝑛) ≤ 𝐵)
80	7	adantr 480	. . . . . . . 8 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 𝐵 ∈ ℝ^*)
81		iccmbl 25474	. . . . . . . . . . 11 ⊢ ((-𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (-𝑛[,]𝑛) ∈ dom vol)
82	32, 31, 81	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (-𝑛[,]𝑛) ∈ dom vol)
83		inmbl 25450	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom vol ∧ (-𝑛[,]𝑛) ∈ dom vol) → (𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol)
84	46, 82, 83	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol)
85	9	ffvelcdmi 7058	. . . . . . . . . 10 ⊢ ((𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol → (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ (0[,]+∞))
86	8, 85	sselid 3947	. . . . . . . . 9 ⊢ ((𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol → (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ ℝ^*)
87	84, 86	syl 17	. . . . . . . 8 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ ℝ^*)
88		simprr 772	. . . . . . . 8 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
89	80, 87, 88	xrltled 13117	. . . . . . 7 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 𝐵 ≤ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
90		oveq2 7398	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑛 → (-𝑛[,]𝑦) = (-𝑛[,]𝑛))
91	90	ineq2d 4186	. . . . . . . . . 10 ⊢ (𝑦 = 𝑛 → (𝐴 ∩ (-𝑛[,]𝑦)) = (𝐴 ∩ (-𝑛[,]𝑛)))
92	91	fveq2d 6865	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → (vol‘(𝐴 ∩ (-𝑛[,]𝑦))) = (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
93		fvex 6874	. . . . . . . . 9 ⊢ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ V
94	92, 47, 93	fvmpt 6971	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑛) = (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
95	31, 94	syl 17	. . . . . . 7 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑛) = (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
96	89, 95	breqtrrd 5138	. . . . . 6 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 𝐵 ≤ ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑛))
97	79, 96	jca 511	. . . . 5 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘-𝑛) ≤ 𝐵 ∧ 𝐵 ≤ ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑛)))
98	32, 31, 33, 39, 41, 50, 55, 97	ivthle 25364	. . . 4 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → ∃𝑧 ∈ (-𝑛[,]𝑛)((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑧) = 𝐵)
99	41	sselda 3949	. . . . . . . 8 ⊢ (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑧 ∈ (-𝑛[,]𝑛)) → 𝑧 ∈ ℝ)
100		oveq2 7398	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (-𝑛[,]𝑦) = (-𝑛[,]𝑧))
101	100	ineq2d 4186	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝐴 ∩ (-𝑛[,]𝑦)) = (𝐴 ∩ (-𝑛[,]𝑧)))
102	101	fveq2d 6865	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (vol‘(𝐴 ∩ (-𝑛[,]𝑦))) = (vol‘(𝐴 ∩ (-𝑛[,]𝑧))))
103		fvex 6874	. . . . . . . . 9 ⊢ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) ∈ V
104	102, 47, 103	fvmpt 6971	. . . . . . . 8 ⊢ (𝑧 ∈ ℝ → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑧) = (vol‘(𝐴 ∩ (-𝑛[,]𝑧))))
105	99, 104	syl 17	. . . . . . 7 ⊢ (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑧 ∈ (-𝑛[,]𝑛)) → ((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑧) = (vol‘(𝐴 ∩ (-𝑛[,]𝑧))))
106	105	eqeq1d 2732	. . . . . 6 ⊢ (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑧 ∈ (-𝑛[,]𝑛)) → (((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑧) = 𝐵 ↔ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵))
107	46	adantr 480	. . . . . . . . 9 ⊢ (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → 𝐴 ∈ dom vol)
108	32	adantr 480	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → -𝑛 ∈ ℝ)
109	99	adantrr 717	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → 𝑧 ∈ ℝ)
110		iccmbl 25474	. . . . . . . . . 10 ⊢ ((-𝑛 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-𝑛[,]𝑧) ∈ dom vol)
111	108, 109, 110	syl2anc 584	. . . . . . . . 9 ⊢ (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → (-𝑛[,]𝑧) ∈ dom vol)
112		inmbl 25450	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom vol ∧ (-𝑛[,]𝑧) ∈ dom vol) → (𝐴 ∩ (-𝑛[,]𝑧)) ∈ dom vol)
113	107, 111, 112	syl2anc 584	. . . . . . . 8 ⊢ (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → (𝐴 ∩ (-𝑛[,]𝑧)) ∈ dom vol)
114		inss1 4203	. . . . . . . . 9 ⊢ (𝐴 ∩ (-𝑛[,]𝑧)) ⊆ 𝐴
115	114	a1i 11	. . . . . . . 8 ⊢ (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → (𝐴 ∩ (-𝑛[,]𝑧)) ⊆ 𝐴)
116		simprr 772	. . . . . . . 8 ⊢ (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)
117		sseq1 3975	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴 ∩ (-𝑛[,]𝑧)) → (𝑥 ⊆ 𝐴 ↔ (𝐴 ∩ (-𝑛[,]𝑧)) ⊆ 𝐴))
118		fveqeq2 6870	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴 ∩ (-𝑛[,]𝑧)) → ((vol‘𝑥) = 𝐵 ↔ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵))
119	117, 118	anbi12d 632	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 ∩ (-𝑛[,]𝑧)) → ((𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵) ↔ ((𝐴 ∩ (-𝑛[,]𝑧)) ⊆ 𝐴 ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)))
120	119	rspcev 3591	. . . . . . . 8 ⊢ (((𝐴 ∩ (-𝑛[,]𝑧)) ∈ dom vol ∧ ((𝐴 ∩ (-𝑛[,]𝑧)) ⊆ 𝐴 ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → ∃𝑥 ∈ dom vol(𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵))
121	113, 115, 116, 120	syl12anc 836	. . . . . . 7 ⊢ (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ (𝑧 ∈ (-𝑛[,]𝑛) ∧ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵)) → ∃𝑥 ∈ dom vol(𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵))
122	121	expr 456	. . . . . 6 ⊢ (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑧 ∈ (-𝑛[,]𝑛)) → ((vol‘(𝐴 ∩ (-𝑛[,]𝑧))) = 𝐵 → ∃𝑥 ∈ dom vol(𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵)))
123	106, 122	sylbid 240	. . . . 5 ⊢ (((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) ∧ 𝑧 ∈ (-𝑛[,]𝑛)) → (((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑧) = 𝐵 → ∃𝑥 ∈ dom vol(𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵)))
124	123	rexlimdva 3135	. . . 4 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (∃𝑧 ∈ (-𝑛[,]𝑛)((𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))‘𝑧) = 𝐵 → ∃𝑥 ∈ dom vol(𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵)))
125	98, 124	mpd 15	. . 3 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → ∃𝑥 ∈ dom vol(𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵))
126	29, 125	rexlimddv 3141	. 2 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → ∃𝑥 ∈ dom vol(𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵))
127		simpll 766	. . 3 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 = (vol‘𝐴)) → 𝐴 ∈ dom vol)
128		ssid 3972	. . . 4 ⊢ 𝐴 ⊆ 𝐴
129	128	a1i 11	. . 3 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 = (vol‘𝐴)) → 𝐴 ⊆ 𝐴)
130		simpr 484	. . . 4 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 = (vol‘𝐴)) → 𝐵 = (vol‘𝐴))
131	130	eqcomd 2736	. . 3 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 = (vol‘𝐴)) → (vol‘𝐴) = 𝐵)
132		sseq1 3975	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
133		fveqeq2 6870	. . . . 5 ⊢ (𝑥 = 𝐴 → ((vol‘𝑥) = 𝐵 ↔ (vol‘𝐴) = 𝐵))
134	132, 133	anbi12d 632	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵) ↔ (𝐴 ⊆ 𝐴 ∧ (vol‘𝐴) = 𝐵)))
135	134	rspcev 3591	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ (𝐴 ⊆ 𝐴 ∧ (vol‘𝐴) = 𝐵)) → ∃𝑥 ∈ dom vol(𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵))
136	127, 129, 131, 135	syl12anc 836	. 2 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 = (vol‘𝐴)) → ∃𝑥 ∈ dom vol(𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵))
137	17	simp3d 1144	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → 𝐵 ≤ (vol‘𝐴))
138		xrleloe 13111	. . . 4 ⊢ ((𝐵 ∈ ℝ^* ∧ (vol‘𝐴) ∈ ℝ^*) → (𝐵 ≤ (vol‘𝐴) ↔ (𝐵 < (vol‘𝐴) ∨ 𝐵 = (vol‘𝐴))))
139	6, 12, 138	syl2anc 584	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → (𝐵 ≤ (vol‘𝐴) ↔ (𝐵 < (vol‘𝐴) ∨ 𝐵 = (vol‘𝐴))))
140	137, 139	mpbid 232	. 2 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → (𝐵 < (vol‘𝐴) ∨ 𝐵 = (vol‘𝐴)))
141	126, 136, 140	mpjaodan 960	1 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → ∃𝑥 ∈ dom vol(𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∃wrex 3054 ∩ cin 3916 ⊆ wss 3917 {csn 4592 class class class wbr 5110 ↦ cmpt 5191 dom cdm 5641 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 ℝcr 11074 0cc0 11075 +∞cpnf 11212 -∞cmnf 11213 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 -cneg 11413 ℕcn 12193 [,]cicc 13316 –cn→ccncf 24776 vol*covol 25370 volcvol 25371

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cc 10395 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-disj 5078 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-rlim 15462 df-sum 15660 df-rest 17392 df-topgen 17413 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-top 22788 df-topon 22805 df-bases 22840 df-cmp 23281 df-cncf 24778 df-ovol 25372 df-vol 25373

This theorem is referenced by: itg2const2 25649

W3C validator