Theorem nulmbl2 25571

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 13038	. . . . 5 ⊢ 1 ∈ ℝ+
2	1	ne0ii 4344	. . . 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 690	. . 3 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ∃𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))
5		simprl 771	. . . . . 6 ⊢ ((𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴 ⊆ 𝑦)
6		mblss 25566	. . . . . . 7 ⊢ (𝑦 ∈ dom vol → 𝑦 ⊆ ℝ)
7	6	adantr 480	. . . . . 6 ⊢ ((𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝑦 ⊆ ℝ)
8	5, 7	sstrd 3994	. . . . 5 ⊢ ((𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴 ⊆ ℝ)
9	8	rexlimiva 3147	. . . 4 ⊢ (∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ⊆ ℝ)
10	9	rexlimivw 3151	. . 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 4237	. . . . . . . . . . . 12 ⊢ (𝑧 ∩ 𝐴) ⊆ 𝑧
13		elpwi 4607	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝒫 ℝ → 𝑧 ⊆ ℝ)
14	13	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → 𝑧 ⊆ ℝ)
15		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) ∈ ℝ)
16		ovolsscl 25521	. . . . . . . . . . . 12 ⊢ (((𝑧 ∩ 𝐴) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∩ 𝐴)) ∈ ℝ)
17	12, 14, 15, 16	mp3an2i 1468	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∩ 𝐴)) ∈ ℝ)
18		difssd 4137	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (𝑧 ∖ 𝐴) ⊆ 𝑧)
19		ovolsscl 25521	. . . . . . . . . . . 12 ⊢ (((𝑧 ∖ 𝐴) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∖ 𝐴)) ∈ ℝ)
20	18, 14, 15, 19	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∖ 𝐴)) ∈ ℝ)
21	17, 20	readdcld 11290	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ∈ ℝ)
22	21	ad2antrr 726	. . . . . . . . 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 726	. . . . . . . . . 10 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑧) ∈ ℝ)
24		difssd 4137	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦 ∖ 𝐴) ⊆ 𝑦)
25	7	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑦 ⊆ ℝ)
26		rpre 13043	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ)
27	26	ad2antlr 727	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑥 ∈ ℝ)
28		simprrr 782	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ≤ 𝑥)
29		ovollecl 25518	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊆ ℝ ∧ 𝑥 ∈ ℝ ∧ (vol*‘𝑦) ≤ 𝑥) → (vol*‘𝑦) ∈ ℝ)
30	25, 27, 28, 29	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ∈ ℝ)
31		ovolsscl 25521	. . . . . . . . . . 11 ⊢ (((𝑦 ∖ 𝐴) ⊆ 𝑦 ∧ 𝑦 ⊆ ℝ ∧ (vol*‘𝑦) ∈ ℝ) → (vol*‘(𝑦 ∖ 𝐴)) ∈ ℝ)
32	24, 25, 30, 31	syl3anc 1373	. . . . . . . . . 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 11290	. . . . . . . . 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 11290	. . . . . . . . 9 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + 𝑥) ∈ ℝ)
35	17	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∩ 𝐴)) ∈ ℝ)
36	20	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∖ 𝐴)) ∈ ℝ)
37		inss1 4237	. . . . . . . . . . . 12 ⊢ (𝑧 ∩ 𝑦) ⊆ 𝑧
38	14	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑧 ⊆ ℝ)
39		ovolsscl 25521	. . . . . . . . . . . 12 ⊢ (((𝑧 ∩ 𝑦) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∩ 𝑦)) ∈ ℝ)
40	37, 38, 23, 39	mp3an2i 1468	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∩ 𝑦)) ∈ ℝ)
41		difssd 4137	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ 𝑦) ⊆ 𝑧)
42		ovolsscl 25521	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∖ 𝑦) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∖ 𝑦)) ∈ ℝ)
43	41, 38, 23, 42	syl3anc 1373	. . . . . . . . . . . 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 11290	. . . . . . . . . . 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 781	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝐴 ⊆ 𝑦)
46		sslin 4243	. . . . . . . . . . . . 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 3995	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∩ 𝑦) ⊆ ℝ)
49		ovolss 25520	. . . . . . . . . . . 12 ⊢ (((𝑧 ∩ 𝐴) ⊆ (𝑧 ∩ 𝑦) ∧ (𝑧 ∩ 𝑦) ⊆ ℝ) → (vol*‘(𝑧 ∩ 𝐴)) ≤ (vol*‘(𝑧 ∩ 𝑦)))
50	47, 48, 49	syl2anc 584	. . . . . . . . . . 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 4147	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ 𝑦) ⊆ ℝ)
52	25	ssdifssd 4147	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦 ∖ 𝐴) ⊆ ℝ)
53	51, 52	unssd 4192	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴)) ⊆ ℝ)
54		ovolun 25534	. . . . . . . . . . . . . 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 839	. . . . . . . . . . . . 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 25518	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 4178	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ⊆ (𝑧 ∪ 𝑦)
59		undif1 4476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∖ 𝑦) ∪ 𝑦) = (𝑧 ∪ 𝑦)
60	58, 59	sseqtrri 4033	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ⊆ ((𝑧 ∖ 𝑦) ∪ 𝑦)
61		ssdif 4144	. . . . . . . . . . . . . . . 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 4291	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∖ 𝑦) ∪ 𝑦) ∖ 𝐴) = (((𝑧 ∖ 𝑦) ∖ 𝐴) ∪ (𝑦 ∖ 𝐴))
64	62, 63	sseqtri 4032	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∖ 𝐴) ⊆ (((𝑧 ∖ 𝑦) ∖ 𝐴) ∪ (𝑦 ∖ 𝐴))
65		difun1 4299	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∖ (𝑦 ∪ 𝐴)) = ((𝑧 ∖ 𝑦) ∖ 𝐴)
66		ssequn2 4189	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ⊆ 𝑦 ↔ (𝑦 ∪ 𝐴) = 𝑦)
67	45, 66	sylib 218	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦 ∪ 𝐴) = 𝑦)
68	67	difeq2d 4126	. . . . . . . . . . . . . . . 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 2791	. . . . . . . . . . . . . . 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 4167	. . . . . . . . . . . . . 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 4026	. . . . . . . . . . . . 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 25520	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 11418	. . . . . . . . . . 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 11882	. . . . . . . . . 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 771	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑦 ∈ dom vol)
77		mblsplit 25567	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ dom vol ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) = ((vol*‘(𝑧 ∩ 𝑦)) + (vol*‘(𝑧 ∖ 𝑦))))
78	76, 38, 23, 77	syl3anc 1373	. . . . . . . . . . . 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 7446	. . . . . . . . . . 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 11289	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∩ 𝑦)) ∈ ℂ)
81	43	recnd 11289	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∖ 𝑦)) ∈ ℂ)
82	32	recnd 11289	. . . . . . . . . . . 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 11283	. . . . . . . . . . 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 2777	. . . . . . . . . 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 5171	. . . . . . . . 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 4136	. . . . . . . . . . . 12 ⊢ (𝑦 ∖ 𝐴) ⊆ 𝑦
87		ovolss 25520	. . . . . . . . . . . 12 ⊢ (((𝑦 ∖ 𝐴) ⊆ 𝑦 ∧ 𝑦 ⊆ ℝ) → (vol*‘(𝑦 ∖ 𝐴)) ≤ (vol*‘𝑦))
88	86, 25, 87	sylancr 587	. . . . . . . . . . 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 11418	. . . . . . . . . 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 11878	. . . . . . . . 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 11418	. . . . . . . 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 3156	. . . . . . 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 3167	. . . . . 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 407	. . . . 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 481	. . . . . . 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 11311	. . . . . 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 773	. . . . . 6 ⊢ ((∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → (vol*‘𝑧) ∈ ℝ)
98		xralrple 13247	. . . . . 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 584	. . . . 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 456	. . 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 3146	. 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 25562	. 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 583	1 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ∈ dom vol)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ∖ cdif 3948   ∪ cun 3949   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600   class class class wbr 5143  dom cdm 5685  ‘cfv 6561  (class class class)co 7431  ℝcr 11154  1c1 11156   + caddc 11158  ℝ^*cxr 11294   ≤ cle 11296  ℝ⁺crp 13034  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-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  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-iun 4993  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-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-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  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-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fl 13832  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-ovol 25499  df-vol 25500

This theorem is referenced by: (None)