Theorem nulmbl2 25053

Description: A set of outer measure zero is measurable. The term "outer measure zero" here is slightly different from "nullset/negligible set"; a nullset has vol*(𝐴) = 0 while "outer measure zero" means that for any 𝑥 there is a 𝑦 containing 𝐴 with volume less than 𝑥. Assuming AC, these notions are equivalent (because the intersection of all such 𝑦 is a nullset) but in ZF this is a strictly weaker notion. Proposition 563Gb of [Fremlin5] p. 193. (Contributed by Mario Carneiro, 19-Mar-2015.)

Assertion

Ref	Expression
nulmbl2	⊢ (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ∈ dom vol)

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem nulmbl2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1rp 12978	. . . . 5 ⊢ 1 ∈ ℝ+
2	1	ne0ii 4338	. . . 4 ⊢ ℝ+ ≠ ∅
3		r19.2z 4495	. . . 4 ⊢ ((ℝ+ ≠ ∅ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → ∃𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))
4	2, 3	mpan 689	. . 3 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ∃𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))
5		simprl 770	. . . . . 6 ⊢ ((𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴 ⊆ 𝑦)
6		mblss 25048	. . . . . . 7 ⊢ (𝑦 ∈ dom vol → 𝑦 ⊆ ℝ)
7	6	adantr 482	. . . . . 6 ⊢ ((𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝑦 ⊆ ℝ)
8	5, 7	sstrd 3993	. . . . 5 ⊢ ((𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴 ⊆ ℝ)
9	8	rexlimiva 3148	. . . 4 ⊢ (∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ⊆ ℝ)
10	9	rexlimivw 3152	. . 3 ⊢ (∃𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ⊆ ℝ)
11	4, 10	syl 17	. 2 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ⊆ ℝ)
12		inss1 4229	. . . . . . . . . . . 12 ⊢ (𝑧 ∩ 𝐴) ⊆ 𝑧
13		elpwi 4610	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝒫 ℝ → 𝑧 ⊆ ℝ)
14	13	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → 𝑧 ⊆ ℝ)
15		simpr 486	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) ∈ ℝ)
16		ovolsscl 25003	. . . . . . . . . . . 12 ⊢ (((𝑧 ∩ 𝐴) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∩ 𝐴)) ∈ ℝ)
17	12, 14, 15, 16	mp3an2i 1467	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∩ 𝐴)) ∈ ℝ)
18		difssd 4133	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (𝑧 ∖ 𝐴) ⊆ 𝑧)
19		ovolsscl 25003	. . . . . . . . . . . 12 ⊢ (((𝑧 ∖ 𝐴) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∖ 𝐴)) ∈ ℝ)
20	18, 14, 15, 19	syl3anc 1372	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∖ 𝐴)) ∈ ℝ)
21	17, 20	readdcld 11243	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ∈ ℝ)
22	21	ad2antrr 725	. . . . . . . . 9 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ∈ ℝ)
23	15	ad2antrr 725	. . . . . . . . . 10 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑧) ∈ ℝ)
24		difssd 4133	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦 ∖ 𝐴) ⊆ 𝑦)
25	7	adantl 483	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑦 ⊆ ℝ)
26		rpre 12982	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ)
27	26	ad2antlr 726	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑥 ∈ ℝ)
28		simprrr 781	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ≤ 𝑥)
29		ovollecl 25000	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊆ ℝ ∧ 𝑥 ∈ ℝ ∧ (vol*‘𝑦) ≤ 𝑥) → (vol*‘𝑦) ∈ ℝ)
30	25, 27, 28, 29	syl3anc 1372	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ∈ ℝ)
31		ovolsscl 25003	. . . . . . . . . . 11 ⊢ (((𝑦 ∖ 𝐴) ⊆ 𝑦 ∧ 𝑦 ⊆ ℝ ∧ (vol*‘𝑦) ∈ ℝ) → (vol*‘(𝑦 ∖ 𝐴)) ∈ ℝ)
32	24, 25, 30, 31	syl3anc 1372	. . . . . . . . . 10 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦 ∖ 𝐴)) ∈ ℝ)
33	23, 32	readdcld 11243	. . . . . . . . 9 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + (vol*‘(𝑦 ∖ 𝐴))) ∈ ℝ)
34	23, 27	readdcld 11243	. . . . . . . . 9 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + 𝑥) ∈ ℝ)
35	17	ad2antrr 725	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∩ 𝐴)) ∈ ℝ)
36	20	ad2antrr 725	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∖ 𝐴)) ∈ ℝ)
37		inss1 4229	. . . . . . . . . . . 12 ⊢ (𝑧 ∩ 𝑦) ⊆ 𝑧
38	14	ad2antrr 725	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑧 ⊆ ℝ)
39		ovolsscl 25003	. . . . . . . . . . . 12 ⊢ (((𝑧 ∩ 𝑦) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∩ 𝑦)) ∈ ℝ)
40	37, 38, 23, 39	mp3an2i 1467	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∩ 𝑦)) ∈ ℝ)
41		difssd 4133	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ 𝑦) ⊆ 𝑧)
42		ovolsscl 25003	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∖ 𝑦) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∖ 𝑦)) ∈ ℝ)
43	41, 38, 23, 42	syl3anc 1372	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∖ 𝑦)) ∈ ℝ)
44	43, 32	readdcld 11243	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧 ∖ 𝑦)) + (vol*‘(𝑦 ∖ 𝐴))) ∈ ℝ)
45		simprrl 780	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝐴 ⊆ 𝑦)
46		sslin 4235	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑦 → (𝑧 ∩ 𝐴) ⊆ (𝑧 ∩ 𝑦))
47	45, 46	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∩ 𝐴) ⊆ (𝑧 ∩ 𝑦))
48	37, 38	sstrid 3994	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∩ 𝑦) ⊆ ℝ)
49		ovolss 25002	. . . . . . . . . . . 12 ⊢ (((𝑧 ∩ 𝐴) ⊆ (𝑧 ∩ 𝑦) ∧ (𝑧 ∩ 𝑦) ⊆ ℝ) → (vol*‘(𝑧 ∩ 𝐴)) ≤ (vol*‘(𝑧 ∩ 𝑦)))
50	47, 48, 49	syl2anc 585	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∩ 𝐴)) ≤ (vol*‘(𝑧 ∩ 𝑦)))
51	38	ssdifssd 4143	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ 𝑦) ⊆ ℝ)
52	25	ssdifssd 4143	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦 ∖ 𝐴) ⊆ ℝ)
53	51, 52	unssd 4187	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴)) ⊆ ℝ)
54		ovolun 25016	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ∖ 𝑦) ⊆ ℝ ∧ (vol*‘(𝑧 ∖ 𝑦)) ∈ ℝ) ∧ ((𝑦 ∖ 𝐴) ⊆ ℝ ∧ (vol*‘(𝑦 ∖ 𝐴)) ∈ ℝ)) → (vol*‘((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴))) ≤ ((vol*‘(𝑧 ∖ 𝑦)) + (vol*‘(𝑦 ∖ 𝐴))))
55	51, 43, 52, 32, 54	syl22anc 838	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴))) ≤ ((vol*‘(𝑧 ∖ 𝑦)) + (vol*‘(𝑦 ∖ 𝐴))))
56		ovollecl 25000	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴)) ⊆ ℝ ∧ ((vol*‘(𝑧 ∖ 𝑦)) + (vol*‘(𝑦 ∖ 𝐴))) ∈ ℝ ∧ (vol*‘((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴))) ≤ ((vol*‘(𝑧 ∖ 𝑦)) + (vol*‘(𝑦 ∖ 𝐴)))) → (vol*‘((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴))) ∈ ℝ)
57	53, 44, 55, 56	syl3anc 1372	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴))) ∈ ℝ)
58		ssun1 4173	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ⊆ (𝑧 ∪ 𝑦)
59		undif1 4476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∖ 𝑦) ∪ 𝑦) = (𝑧 ∪ 𝑦)
60	58, 59	sseqtrri 4020	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ⊆ ((𝑧 ∖ 𝑦) ∪ 𝑦)
61		ssdif 4140	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ ((𝑧 ∖ 𝑦) ∪ 𝑦) → (𝑧 ∖ 𝐴) ⊆ (((𝑧 ∖ 𝑦) ∪ 𝑦) ∖ 𝐴))
62	60, 61	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∖ 𝐴) ⊆ (((𝑧 ∖ 𝑦) ∪ 𝑦) ∖ 𝐴)
63		difundir 4281	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∖ 𝑦) ∪ 𝑦) ∖ 𝐴) = (((𝑧 ∖ 𝑦) ∖ 𝐴) ∪ (𝑦 ∖ 𝐴))
64	62, 63	sseqtri 4019	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∖ 𝐴) ⊆ (((𝑧 ∖ 𝑦) ∖ 𝐴) ∪ (𝑦 ∖ 𝐴))
65		difun1 4290	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∖ (𝑦 ∪ 𝐴)) = ((𝑧 ∖ 𝑦) ∖ 𝐴)
66		ssequn2 4184	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ⊆ 𝑦 ↔ (𝑦 ∪ 𝐴) = 𝑦)
67	45, 66	sylib 217	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦 ∪ 𝐴) = 𝑦)
68	67	difeq2d 4123	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ (𝑦 ∪ 𝐴)) = (𝑧 ∖ 𝑦))
69	65, 68	eqtr3id 2787	. . . . . . . . . . . . . . 15 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((𝑧 ∖ 𝑦) ∖ 𝐴) = (𝑧 ∖ 𝑦))
70	69	uneq1d 4163	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (((𝑧 ∖ 𝑦) ∖ 𝐴) ∪ (𝑦 ∖ 𝐴)) = ((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴)))
71	64, 70	sseqtrid 4035	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ 𝐴) ⊆ ((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴)))
72		ovolss 25002	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∖ 𝐴) ⊆ ((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴)) ∧ ((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴)) ⊆ ℝ) → (vol*‘(𝑧 ∖ 𝐴)) ≤ (vol*‘((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴))))
73	71, 53, 72	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∖ 𝐴)) ≤ (vol*‘((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴))))
74	36, 57, 44, 73, 55	letrd 11371	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∖ 𝐴)) ≤ ((vol*‘(𝑧 ∖ 𝑦)) + (vol*‘(𝑦 ∖ 𝐴))))
75	35, 36, 40, 44, 50, 74	le2addd 11833	. . . . . . . . . 10 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ≤ ((vol*‘(𝑧 ∩ 𝑦)) + ((vol*‘(𝑧 ∖ 𝑦)) + (vol*‘(𝑦 ∖ 𝐴)))))
76		simprl 770	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑦 ∈ dom vol)
77		mblsplit 25049	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ dom vol ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) = ((vol*‘(𝑧 ∩ 𝑦)) + (vol*‘(𝑧 ∖ 𝑦))))
78	76, 38, 23, 77	syl3anc 1372	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑧) = ((vol*‘(𝑧 ∩ 𝑦)) + (vol*‘(𝑧 ∖ 𝑦))))
79	78	oveq1d 7424	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + (vol*‘(𝑦 ∖ 𝐴))) = (((vol*‘(𝑧 ∩ 𝑦)) + (vol*‘(𝑧 ∖ 𝑦))) + (vol*‘(𝑦 ∖ 𝐴))))
80	40	recnd 11242	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∩ 𝑦)) ∈ ℂ)
81	43	recnd 11242	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∖ 𝑦)) ∈ ℂ)
82	32	recnd 11242	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦 ∖ 𝐴)) ∈ ℂ)
83	80, 81, 82	addassd 11236	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (((vol*‘(𝑧 ∩ 𝑦)) + (vol*‘(𝑧 ∖ 𝑦))) + (vol*‘(𝑦 ∖ 𝐴))) = ((vol*‘(𝑧 ∩ 𝑦)) + ((vol*‘(𝑧 ∖ 𝑦)) + (vol*‘(𝑦 ∖ 𝐴)))))
84	79, 83	eqtrd 2773	. . . . . . . . . 10 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + (vol*‘(𝑦 ∖ 𝐴))) = ((vol*‘(𝑧 ∩ 𝑦)) + ((vol*‘(𝑧 ∖ 𝑦)) + (vol*‘(𝑦 ∖ 𝐴)))))
85	75, 84	breqtrrd 5177	. . . . . . . . 9 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ≤ ((vol*‘𝑧) + (vol*‘(𝑦 ∖ 𝐴))))
86		difss 4132	. . . . . . . . . . . 12 ⊢ (𝑦 ∖ 𝐴) ⊆ 𝑦
87		ovolss 25002	. . . . . . . . . . . 12 ⊢ (((𝑦 ∖ 𝐴) ⊆ 𝑦 ∧ 𝑦 ⊆ ℝ) → (vol*‘(𝑦 ∖ 𝐴)) ≤ (vol*‘𝑦))
88	86, 25, 87	sylancr 588	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦 ∖ 𝐴)) ≤ (vol*‘𝑦))
89	32, 30, 27, 88, 28	letrd 11371	. . . . . . . . . 10 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦 ∖ 𝐴)) ≤ 𝑥)
90	32, 27, 23, 89	leadd2dd 11829	. . . . . . . . 9 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + (vol*‘(𝑦 ∖ 𝐴))) ≤ ((vol*‘𝑧) + 𝑥))
91	22, 33, 34, 85, 90	letrd 11371	. . . . . . . 8 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ≤ ((vol*‘𝑧) + 𝑥))
92	91	rexlimdvaa 3157	. . . . . . 7 ⊢ (((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ≤ ((vol*‘𝑧) + 𝑥)))
93	92	ralimdva 3168	. . . . . 6 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ∀𝑥 ∈ ℝ+ ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ≤ ((vol*‘𝑧) + 𝑥)))
94	93	impcom 409	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ∀𝑥 ∈ ℝ+ ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ≤ ((vol*‘𝑧) + 𝑥))
95	21	adantl 483	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ∈ ℝ)
96	95	rexrd 11264	. . . . . 6 ⊢ ((∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ∈ ℝ*)
97		simprr 772	. . . . . 6 ⊢ ((∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → (vol*‘𝑧) ∈ ℝ)
98		xralrple 13184	. . . . . 6 ⊢ ((((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ∈ ℝ* ∧ (vol*‘𝑧) ∈ ℝ) → (((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ≤ (vol*‘𝑧) ↔ ∀𝑥 ∈ ℝ+ ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ≤ ((vol*‘𝑧) + 𝑥)))
99	96, 97, 98	syl2anc 585	. . . . 5 ⊢ ((∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → (((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ≤ (vol*‘𝑧) ↔ ∀𝑥 ∈ ℝ+ ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ≤ ((vol*‘𝑧) + 𝑥)))
100	94, 99	mpbird 257	. . . 4 ⊢ ((∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ≤ (vol*‘𝑧))
101	100	expr 458	. . 3 ⊢ ((∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ 𝑧 ∈ 𝒫 ℝ) → ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ≤ (vol*‘𝑧)))
102	101	ralrimiva 3147	. 2 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ∀𝑧 ∈ 𝒫 ℝ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ≤ (vol*‘𝑧)))
103		ismbl2 25044	. 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ≤ (vol*‘𝑧))))
104	11, 102, 103	sylanbrc 584	1 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ∈ dom vol)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  ∃wrex 3071   ∖ cdif 3946   ∪ cun 3947   ∩ cin 3948   ⊆ wss 3949  ∅c0 4323  𝒫 cpw 4603   class class class wbr 5149  dom cdm 5677  ‘cfv 6544  (class class class)co 7409  ℝcr 11109  1c1 11111   + caddc 11113  ℝ^*cxr 11247   ≤ cle 11249  ℝ⁺crp 12974  vol*covol 24979  volcvol 24980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-ioo 13328  df-ico 13330  df-icc 13331  df-fz 13485  df-fl 13757  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-ovol 24981  df-vol 24982

This theorem is referenced by: (None)