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Theorem volivth 25642

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 767	. . . 4 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → 𝐴 ∈ dom vol)
2		mnfxr 11318	. . . . . 6 ⊢ -∞ ∈ ℝ^*
3	2	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → -∞ ∈ ℝ^*)
4		iccssxr 13470	. . . . . . 7 ⊢ (0[,](vol‘𝐴)) ⊆ ℝ^*
5		simpr 484	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → 𝐵 ∈ (0[,](vol‘𝐴)))
6	4, 5	sselid 3981	. . . . . 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 13470	. . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ^*
9		volf 25564	. . . . . . . . 9 ⊢ vol:dom vol⟶(0[,]+∞)
10	9	ffvelcdmi 7103	. . . . . . . 8 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) ∈ (0[,]+∞))
11	8, 10	sselid 3981	. . . . . . 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 11308	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
15		elicc1 13431	. . . . . . . . . 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 1144	. . . . . . 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 13167	. . . . . . . 8 ⊢ -∞ < 0
21		xrltletr 13199	. . . . . . . 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 13212	. . . . 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 25640	. . . 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 12273	. . . . . . 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 11690	. . . . 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 11264	. . . . . 6 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → 0 ∈ ℝ)
35		nngt0 12297	. . . . . . . 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 11833	. . . . . . 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 11422	. . . . 5 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → -𝑛 < 𝑛)
40		iccssre 13469	. . . . . 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 11212	. . . . . . 7 ⊢ ℝ ⊆ ℂ
43		ssid 4006	. . . . . . 7 ⊢ ℂ ⊆ ℂ
44		cncfss 24925	. . . . . . 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 2737	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦)))) = (𝑦 ∈ ℝ ↦ (vol‘(𝐴 ∩ (-𝑛[,]𝑦))))
48	47	volcn 25641	. . . . . . 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 3981	. . . . 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 3983	. . . . . 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 24919	. . . . . . . 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 7104	. . . . . 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 7439	. . . . . . . . . . . 12 ⊢ (𝑦 = -𝑛 → (-𝑛[,]𝑦) = (-𝑛[,]-𝑛))
57	56	ineq2d 4220	. . . . . . . . . . 11 ⊢ (𝑦 = -𝑛 → (𝐴 ∩ (-𝑛[,]𝑦)) = (𝐴 ∩ (-𝑛[,]-𝑛)))
58	57	fveq2d 6910	. . . . . . . . . 10 ⊢ (𝑦 = -𝑛 → (vol‘(𝐴 ∩ (-𝑛[,]𝑦))) = (vol‘(𝐴 ∩ (-𝑛[,]-𝑛))))
59		fvex 6919	. . . . . . . . . 10 ⊢ (vol‘(𝐴 ∩ (-𝑛[,]-𝑛))) ∈ V
60	58, 47, 59	fvmpt 7016	. . . . . . . . 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 4238	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ (-𝑛[,]-𝑛)) ⊆ (-𝑛[,]-𝑛)
63	32	rexrd 11311	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → -𝑛 ∈ ℝ^*)
64		iccid 13432	. . . . . . . . . . . . 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 4026	. . . . . . . . . . 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 4809	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → {-𝑛} ⊆ ℝ)
68	66, 67	sstrd 3994	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) ∧ (𝑛 ∈ ℕ ∧ 𝐵 < (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))) → (𝐴 ∩ (-𝑛[,]-𝑛)) ⊆ ℝ)
69		ovolsn 25530	. . . . . . . . . . . 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 25522	. . . . . . . . . . 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 25570	. . . . . . . . . 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 25565	. . . . . . . . 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 2781	. . . . . . 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 5165	. . . . . 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 25601	. . . . . . . . . . 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 25577	. . . . . . . . . 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 7103	. . . . . . . . . 10 ⊢ ((𝐴 ∩ (-𝑛[,]𝑛)) ∈ dom vol → (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ (0[,]+∞))
86	8, 85	sselid 3981	. . . . . . . . 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 773	. . . . . . . 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 13192	. . . . . . 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 7439	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑛 → (-𝑛[,]𝑦) = (-𝑛[,]𝑛))
91	90	ineq2d 4220	. . . . . . . . . 10 ⊢ (𝑦 = 𝑛 → (𝐴 ∩ (-𝑛[,]𝑦)) = (𝐴 ∩ (-𝑛[,]𝑛)))
92	91	fveq2d 6910	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → (vol‘(𝐴 ∩ (-𝑛[,]𝑦))) = (vol‘(𝐴 ∩ (-𝑛[,]𝑛))))
93		fvex 6919	. . . . . . . . 9 ⊢ (vol‘(𝐴 ∩ (-𝑛[,]𝑛))) ∈ V
94	92, 47, 93	fvmpt 7016	. . . . . . . 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 5171	. . . . . 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 25491	. . . 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 3983	. . . . . . . 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 7439	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (-𝑛[,]𝑦) = (-𝑛[,]𝑧))
101	100	ineq2d 4220	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝐴 ∩ (-𝑛[,]𝑦)) = (𝐴 ∩ (-𝑛[,]𝑧)))
102	101	fveq2d 6910	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (vol‘(𝐴 ∩ (-𝑛[,]𝑦))) = (vol‘(𝐴 ∩ (-𝑛[,]𝑧))))
103		fvex 6919	. . . . . . . . 9 ⊢ (vol‘(𝐴 ∩ (-𝑛[,]𝑧))) ∈ V
104	102, 47, 103	fvmpt 7016	. . . . . . . 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 2739	. . . . . 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 25601	. . . . . . . . . 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 25577	. . . . . . . . 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 4237	. . . . . . . . 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 773	. . . . . . . 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 4009	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴 ∩ (-𝑛[,]𝑧)) → (𝑥 ⊆ 𝐴 ↔ (𝐴 ∩ (-𝑛[,]𝑧)) ⊆ 𝐴))
118		fveqeq2 6915	. . . . . . . . . 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 3622	. . . . . . . 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 837	. . . . . . 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 3155	. . . 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 3161	. 2 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 < (vol‘𝐴)) → ∃𝑥 ∈ dom vol(𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵))
127		simpll 767	. . 3 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 = (vol‘𝐴)) → 𝐴 ∈ dom vol)
128		ssid 4006	. . . 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 2743	. . 3 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 = (vol‘𝐴)) → (vol‘𝐴) = 𝐵)
132		sseq1 4009	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
133		fveqeq2 6915	. . . . 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 3622	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ (𝐴 ⊆ 𝐴 ∧ (vol‘𝐴) = 𝐵)) → ∃𝑥 ∈ dom vol(𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵))
136	127, 129, 131, 135	syl12anc 837	. 2 ⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) ∧ 𝐵 = (vol‘𝐴)) → ∃𝑥 ∈ dom vol(𝑥 ⊆ 𝐴 ∧ (vol‘𝑥) = 𝐵))
137	17	simp3d 1145	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ (0[,](vol‘𝐴))) → 𝐵 ≤ (vol‘𝐴))
138		xrleloe 13186	. . . 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 961	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 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ∩ cin 3950 ⊆ wss 3951 {csn 4626 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 +∞cpnf 11292 -∞cmnf 11293 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 -cneg 11493 ℕcn 12266 [,]cicc 13390 –cn→ccncf 24902 vol*covol 25497 volcvol 25498

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-rest 17467 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-cmp 23395 df-cncf 24904 df-ovol 25499 df-vol 25500

This theorem is referenced by: itg2const2 25776

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