Theorem nulmbl2 24140

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 12381	. . . . 5 ⊢ 1 ∈ ℝ+
2	1	ne0ii 4253	. . . 4 ⊢ ℝ+ ≠ ∅
3		r19.2z 4398	. . . 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 24135	. . . . . . 7 ⊢ (𝑦 ∈ dom vol → 𝑦 ⊆ ℝ)
7	6	adantr 484	. . . . . 6 ⊢ ((𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝑦 ⊆ ℝ)
8	5, 7	sstrd 3925	. . . . 5 ⊢ ((𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴 ⊆ ℝ)
9	8	rexlimiva 3240	. . . 4 ⊢ (∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ⊆ ℝ)
10	9	rexlimivw 3241	. . 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 4155	. . . . . . . . . . . 12 ⊢ (𝑧 ∩ 𝐴) ⊆ 𝑧
13		elpwi 4506	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝒫 ℝ → 𝑧 ⊆ ℝ)
14	13	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → 𝑧 ⊆ ℝ)
15		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) ∈ ℝ)
16		ovolsscl 24090	. . . . . . . . . . . 12 ⊢ (((𝑧 ∩ 𝐴) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∩ 𝐴)) ∈ ℝ)
17	12, 14, 15, 16	mp3an2i 1463	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∩ 𝐴)) ∈ ℝ)
18		difssd 4060	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (𝑧 ∖ 𝐴) ⊆ 𝑧)
19		ovolsscl 24090	. . . . . . . . . . . 12 ⊢ (((𝑧 ∖ 𝐴) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∖ 𝐴)) ∈ ℝ)
20	18, 14, 15, 19	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∖ 𝐴)) ∈ ℝ)
21	17, 20	readdcld 10659	. . . . . . . . . 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 4060	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦 ∖ 𝐴) ⊆ 𝑦)
25	7	adantl 485	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑦 ⊆ ℝ)
26		rpre 12385	. . . . . . . . . . . . 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 24087	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊆ ℝ ∧ 𝑥 ∈ ℝ ∧ (vol*‘𝑦) ≤ 𝑥) → (vol*‘𝑦) ∈ ℝ)
30	25, 27, 28, 29	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ∈ ℝ)
31		ovolsscl 24090	. . . . . . . . . . 11 ⊢ (((𝑦 ∖ 𝐴) ⊆ 𝑦 ∧ 𝑦 ⊆ ℝ ∧ (vol*‘𝑦) ∈ ℝ) → (vol*‘(𝑦 ∖ 𝐴)) ∈ ℝ)
32	24, 25, 30, 31	syl3anc 1368	. . . . . . . . . 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 10659	. . . . . . . . 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 10659	. . . . . . . . 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 4155	. . . . . . . . . . . 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 24090	. . . . . . . . . . . 12 ⊢ (((𝑧 ∩ 𝑦) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∩ 𝑦)) ∈ ℝ)
40	37, 38, 23, 39	mp3an2i 1463	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∩ 𝑦)) ∈ ℝ)
41		difssd 4060	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ 𝑦) ⊆ 𝑧)
42		ovolsscl 24090	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∖ 𝑦) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∖ 𝑦)) ∈ ℝ)
43	41, 38, 23, 42	syl3anc 1368	. . . . . . . . . . . 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 10659	. . . . . . . . . . 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 4161	. . . . . . . . . . . . 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 3926	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∩ 𝑦) ⊆ ℝ)
49		ovolss 24089	. . . . . . . . . . . 12 ⊢ (((𝑧 ∩ 𝐴) ⊆ (𝑧 ∩ 𝑦) ∧ (𝑧 ∩ 𝑦) ⊆ ℝ) → (vol*‘(𝑧 ∩ 𝐴)) ≤ (vol*‘(𝑧 ∩ 𝑦)))
50	47, 48, 49	syl2anc 587	. . . . . . . . . . 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 4070	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ 𝑦) ⊆ ℝ)
52	25	ssdifssd 4070	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦 ∖ 𝐴) ⊆ ℝ)
53	51, 52	unssd 4113	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴)) ⊆ ℝ)
54		ovolun 24103	. . . . . . . . . . . . . 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 837	. . . . . . . . . . . . 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 24087	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 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 4099	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ⊆ (𝑧 ∪ 𝑦)
59		undif1 4382	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∖ 𝑦) ∪ 𝑦) = (𝑧 ∪ 𝑦)
60	58, 59	sseqtrri 3952	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ⊆ ((𝑧 ∖ 𝑦) ∪ 𝑦)
61		ssdif 4067	. . . . . . . . . . . . . . . 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 4207	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∖ 𝑦) ∪ 𝑦) ∖ 𝐴) = (((𝑧 ∖ 𝑦) ∖ 𝐴) ∪ (𝑦 ∖ 𝐴))
64	62, 63	sseqtri 3951	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∖ 𝐴) ⊆ (((𝑧 ∖ 𝑦) ∖ 𝐴) ∪ (𝑦 ∖ 𝐴))
65		difun1 4214	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∖ (𝑦 ∪ 𝐴)) = ((𝑧 ∖ 𝑦) ∖ 𝐴)
66		ssequn2 4110	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ⊆ 𝑦 ↔ (𝑦 ∪ 𝐴) = 𝑦)
67	45, 66	sylib 221	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦 ∪ 𝐴) = 𝑦)
68	67	difeq2d 4050	. . . . . . . . . . . . . . . 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	syl5eqr 2847	. . . . . . . . . . . . . . 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 4089	. . . . . . . . . . . . . 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 3967	. . . . . . . . . . . . 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 24089	. . . . . . . . . . . . 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 587	. . . . . . . . . . . 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 10786	. . . . . . . . . . 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 11248	. . . . . . . . . 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 24136	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ dom vol ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) = ((vol*‘(𝑧 ∩ 𝑦)) + (vol*‘(𝑧 ∖ 𝑦))))
78	76, 38, 23, 77	syl3anc 1368	. . . . . . . . . . . 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 7150	. . . . . . . . . . 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 10658	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∩ 𝑦)) ∈ ℂ)
81	43	recnd 10658	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∖ 𝑦)) ∈ ℂ)
82	32	recnd 10658	. . . . . . . . . . . 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 10652	. . . . . . . . . . 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 2833	. . . . . . . . . 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 5058	. . . . . . . . 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 4059	. . . . . . . . . . . 12 ⊢ (𝑦 ∖ 𝐴) ⊆ 𝑦
87		ovolss 24089	. . . . . . . . . . . 12 ⊢ (((𝑦 ∖ 𝐴) ⊆ 𝑦 ∧ 𝑦 ⊆ ℝ) → (vol*‘(𝑦 ∖ 𝐴)) ≤ (vol*‘𝑦))
88	86, 25, 87	sylancr 590	. . . . . . . . . . 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 10786	. . . . . . . . . 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 11244	. . . . . . . . 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 10786	. . . . . . . 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 3244	. . . . . . 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 3144	. . . . . 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 411	. . . . 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 485	. . . . . . 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 10680	. . . . . 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 12586	. . . . . 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 587	. . . . 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 260	. . . 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 460	. . 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 3149	. 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 24131	. 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 586	1 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ∈ dom vol)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107   ∖ cdif 3878   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497   class class class wbr 5030  dom cdm 5519  ‘cfv 6324  (class class class)co 7135  ℝcr 10525  1c1 10527   + caddc 10529  ℝ^*cxr 10663   ≤ cle 10665  ℝ⁺crp 12377  vol*covol 24066  volcvol 24067

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-sup 8890  df-inf 8891  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-ioo 12730  df-ico 12732  df-icc 12733  df-fz 12886  df-fl 13157  df-seq 13365  df-exp 13426  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-ovol 24068  df-vol 24069

This theorem is referenced by: (None)