Theorem nulmbl2 23703

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 12117	. . . . 5 ⊢ 1 ∈ ℝ+
2	1	ne0ii 4154	. . . 4 ⊢ ℝ+ ≠ ∅
3		r19.2z 4283	. . . 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 683	. . 3 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ∃𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))
5		simprl 789	. . . . . 6 ⊢ ((𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴 ⊆ 𝑦)
6		mblss 23698	. . . . . . 7 ⊢ (𝑦 ∈ dom vol → 𝑦 ⊆ ℝ)
7	6	adantr 474	. . . . . 6 ⊢ ((𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝑦 ⊆ ℝ)
8	5, 7	sstrd 3838	. . . . 5 ⊢ ((𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴 ⊆ ℝ)
9	8	rexlimiva 3238	. . . 4 ⊢ (∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ⊆ ℝ)
10	9	rexlimivw 3239	. . 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 4058	. . . . . . . . . . . . 13 ⊢ (𝑧 ∩ 𝐴) ⊆ 𝑧
13	12	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (𝑧 ∩ 𝐴) ⊆ 𝑧)
14		elpwi 4389	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝒫 ℝ → 𝑧 ⊆ ℝ)
15	14	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → 𝑧 ⊆ ℝ)
16		simpr 479	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) ∈ ℝ)
17		ovolsscl 23653	. . . . . . . . . . . 12 ⊢ (((𝑧 ∩ 𝐴) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∩ 𝐴)) ∈ ℝ)
18	13, 15, 16, 17	syl3anc 1496	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∩ 𝐴)) ∈ ℝ)
19		difssd 3966	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (𝑧 ∖ 𝐴) ⊆ 𝑧)
20		ovolsscl 23653	. . . . . . . . . . . 12 ⊢ (((𝑧 ∖ 𝐴) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∖ 𝐴)) ∈ ℝ)
21	19, 15, 16, 20	syl3anc 1496	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∖ 𝐴)) ∈ ℝ)
22	18, 21	readdcld 10387	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ∈ ℝ)
23	22	ad2antrr 719	. . . . . . . . 9 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ∈ ℝ)
24	16	ad2antrr 719	. . . . . . . . . 10 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑧) ∈ ℝ)
25		difssd 3966	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦 ∖ 𝐴) ⊆ 𝑦)
26	7	adantl 475	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑦 ⊆ ℝ)
27		rpre 12121	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ)
28	27	ad2antlr 720	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑥 ∈ ℝ)
29		simprrr 802	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ≤ 𝑥)
30		ovollecl 23650	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊆ ℝ ∧ 𝑥 ∈ ℝ ∧ (vol*‘𝑦) ≤ 𝑥) → (vol*‘𝑦) ∈ ℝ)
31	26, 28, 29, 30	syl3anc 1496	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ∈ ℝ)
32		ovolsscl 23653	. . . . . . . . . . 11 ⊢ (((𝑦 ∖ 𝐴) ⊆ 𝑦 ∧ 𝑦 ⊆ ℝ ∧ (vol*‘𝑦) ∈ ℝ) → (vol*‘(𝑦 ∖ 𝐴)) ∈ ℝ)
33	25, 26, 31, 32	syl3anc 1496	. . . . . . . . . 10 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦 ∖ 𝐴)) ∈ ℝ)
34	24, 33	readdcld 10387	. . . . . . . . 9 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + (vol*‘(𝑦 ∖ 𝐴))) ∈ ℝ)
35	24, 28	readdcld 10387	. . . . . . . . 9 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + 𝑥) ∈ ℝ)
36	18	ad2antrr 719	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∩ 𝐴)) ∈ ℝ)
37	21	ad2antrr 719	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∖ 𝐴)) ∈ ℝ)
38		inss1 4058	. . . . . . . . . . . . 13 ⊢ (𝑧 ∩ 𝑦) ⊆ 𝑧
39	38	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∩ 𝑦) ⊆ 𝑧)
40	15	ad2antrr 719	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑧 ⊆ ℝ)
41		ovolsscl 23653	. . . . . . . . . . . 12 ⊢ (((𝑧 ∩ 𝑦) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∩ 𝑦)) ∈ ℝ)
42	39, 40, 24, 41	syl3anc 1496	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∩ 𝑦)) ∈ ℝ)
43		difssd 3966	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ 𝑦) ⊆ 𝑧)
44		ovolsscl 23653	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∖ 𝑦) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∖ 𝑦)) ∈ ℝ)
45	43, 40, 24, 44	syl3anc 1496	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∖ 𝑦)) ∈ ℝ)
46	45, 33	readdcld 10387	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧 ∖ 𝑦)) + (vol*‘(𝑦 ∖ 𝐴))) ∈ ℝ)
47		simprrl 801	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝐴 ⊆ 𝑦)
48		sslin 4064	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑦 → (𝑧 ∩ 𝐴) ⊆ (𝑧 ∩ 𝑦))
49	47, 48	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∩ 𝐴) ⊆ (𝑧 ∩ 𝑦))
50	38, 40	syl5ss 3839	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∩ 𝑦) ⊆ ℝ)
51		ovolss 23652	. . . . . . . . . . . 12 ⊢ (((𝑧 ∩ 𝐴) ⊆ (𝑧 ∩ 𝑦) ∧ (𝑧 ∩ 𝑦) ⊆ ℝ) → (vol*‘(𝑧 ∩ 𝐴)) ≤ (vol*‘(𝑧 ∩ 𝑦)))
52	49, 50, 51	syl2anc 581	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∩ 𝐴)) ≤ (vol*‘(𝑧 ∩ 𝑦)))
53	40	ssdifssd 3976	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ 𝑦) ⊆ ℝ)
54	26	ssdifssd 3976	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦 ∖ 𝐴) ⊆ ℝ)
55	53, 54	unssd 4017	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴)) ⊆ ℝ)
56		ovolun 23666	. . . . . . . . . . . . . 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))))
57	53, 45, 54, 33, 56	syl22anc 874	. . . . . . . . . . . . 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))))
58		ovollecl 23650	. . . . . . . . . . . . 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)
59	55, 46, 57, 58	syl3anc 1496	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴))) ∈ ℝ)
60		ssun1 4004	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ⊆ (𝑧 ∪ 𝑦)
61		undif1 4267	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∖ 𝑦) ∪ 𝑦) = (𝑧 ∪ 𝑦)
62	60, 61	sseqtr4i 3864	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ⊆ ((𝑧 ∖ 𝑦) ∪ 𝑦)
63		ssdif 3973	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ ((𝑧 ∖ 𝑦) ∪ 𝑦) → (𝑧 ∖ 𝐴) ⊆ (((𝑧 ∖ 𝑦) ∪ 𝑦) ∖ 𝐴))
64	62, 63	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∖ 𝐴) ⊆ (((𝑧 ∖ 𝑦) ∪ 𝑦) ∖ 𝐴)
65		difundir 4111	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∖ 𝑦) ∪ 𝑦) ∖ 𝐴) = (((𝑧 ∖ 𝑦) ∖ 𝐴) ∪ (𝑦 ∖ 𝐴))
66	64, 65	sseqtri 3863	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∖ 𝐴) ⊆ (((𝑧 ∖ 𝑦) ∖ 𝐴) ∪ (𝑦 ∖ 𝐴))
67		difun1 4118	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∖ (𝑦 ∪ 𝐴)) = ((𝑧 ∖ 𝑦) ∖ 𝐴)
68		ssequn2 4014	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ⊆ 𝑦 ↔ (𝑦 ∪ 𝐴) = 𝑦)
69	47, 68	sylib 210	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦 ∪ 𝐴) = 𝑦)
70	69	difeq2d 3956	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ (𝑦 ∪ 𝐴)) = (𝑧 ∖ 𝑦))
71	67, 70	syl5eqr 2876	. . . . . . . . . . . . . . 15 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((𝑧 ∖ 𝑦) ∖ 𝐴) = (𝑧 ∖ 𝑦))
72	71	uneq1d 3994	. . . . . . . . . . . . . 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)))
73	66, 72	syl5sseq 3879	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ 𝐴) ⊆ ((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴)))
74		ovolss 23652	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∖ 𝐴) ⊆ ((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴)) ∧ ((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴)) ⊆ ℝ) → (vol*‘(𝑧 ∖ 𝐴)) ≤ (vol*‘((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴))))
75	73, 55, 74	syl2anc 581	. . . . . . . . . . . 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))))
76	37, 59, 46, 75, 57	letrd 10514	. . . . . . . . . . 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))))
77	36, 37, 42, 46, 52, 76	le2addd 10972	. . . . . . . . . 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)))))
78		simprl 789	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑦 ∈ dom vol)
79		mblsplit 23699	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ dom vol ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) = ((vol*‘(𝑧 ∩ 𝑦)) + (vol*‘(𝑧 ∖ 𝑦))))
80	78, 40, 24, 79	syl3anc 1496	. . . . . . . . . . . 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))))
81	80	oveq1d 6921	. . . . . . . . . . 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))))
82	42	recnd 10386	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∩ 𝑦)) ∈ ℂ)
83	45	recnd 10386	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∖ 𝑦)) ∈ ℂ)
84	33	recnd 10386	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦 ∖ 𝐴)) ∈ ℂ)
85	82, 83, 84	addassd 10380	. . . . . . . . . . 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)))))
86	81, 85	eqtrd 2862	. . . . . . . . . 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)))))
87	77, 86	breqtrrd 4902	. . . . . . . . 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))))
88		difss 3965	. . . . . . . . . . . 12 ⊢ (𝑦 ∖ 𝐴) ⊆ 𝑦
89		ovolss 23652	. . . . . . . . . . . 12 ⊢ (((𝑦 ∖ 𝐴) ⊆ 𝑦 ∧ 𝑦 ⊆ ℝ) → (vol*‘(𝑦 ∖ 𝐴)) ≤ (vol*‘𝑦))
90	88, 26, 89	sylancr 583	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦 ∖ 𝐴)) ≤ (vol*‘𝑦))
91	33, 31, 28, 90, 29	letrd 10514	. . . . . . . . . 10 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦 ∖ 𝐴)) ≤ 𝑥)
92	33, 28, 24, 91	leadd2dd 10968	. . . . . . . . 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))
93	23, 34, 35, 87, 92	letrd 10514	. . . . . . . 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))
94	93	rexlimdvaa 3242	. . . . . . 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)))
95	94	ralimdva 3172	. . . . . 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)))
96	95	impcom 398	. . . . 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))
97	22	adantl 475	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ∈ ℝ)
98	97	rexrd 10407	. . . . . 6 ⊢ ((∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ∈ ℝ*)
99		simprr 791	. . . . . 6 ⊢ ((∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → (vol*‘𝑧) ∈ ℝ)
100		xralrple 12325	. . . . . 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)))
101	98, 99, 100	syl2anc 581	. . . . 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)))
102	96, 101	mpbird 249	. . . 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))
103	102	expr 450	. . 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)))
104	103	ralrimiva 3176	. 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)))
105		ismbl2 23694	. 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ≤ (vol*‘𝑧))))
106	11, 104, 105	sylanbrc 580	1 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ∈ dom vol)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166   ≠ wne 3000  ∀wral 3118  ∃wrex 3119   ∖ cdif 3796   ∪ cun 3797   ∩ cin 3798   ⊆ wss 3799  ∅c0 4145  𝒫 cpw 4379   class class class wbr 4874  dom cdm 5343  ‘cfv 6124  (class class class)co 6906  ℝcr 10252  1c1 10254   + caddc 10256  ℝ^*cxr 10391   ≤ cle 10393  ℝ⁺crp 12113  vol*covol 23629  volcvol 23630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310  ax-1cn 10311  ax-icn 10312  ax-addcl 10313  ax-addrcl 10314  ax-mulcl 10315  ax-mulrcl 10316  ax-mulcom 10317  ax-addass 10318  ax-mulass 10319  ax-distr 10320  ax-i2m1 10321  ax-1ne0 10322  ax-1rid 10323  ax-rnegex 10324  ax-rrecex 10325  ax-cnre 10326  ax-pre-lttri 10327  ax-pre-lttrn 10328  ax-pre-ltadd 10329  ax-pre-mulgt0 10330  ax-pre-sup 10331

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-1st 7429  df-2nd 7430  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-er 8010  df-map 8125  df-en 8224  df-dom 8225  df-sdom 8226  df-sup 8618  df-inf 8619  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-sub 10588  df-neg 10589  df-div 11011  df-nn 11352  df-2 11415  df-3 11416  df-n0 11620  df-z 11706  df-uz 11970  df-q 12073  df-rp 12114  df-ioo 12468  df-ico 12470  df-icc 12471  df-fz 12621  df-fl 12889  df-seq 13097  df-exp 13156  df-cj 14217  df-re 14218  df-im 14219  df-sqrt 14353  df-abs 14354  df-ovol 23631  df-vol 23632

This theorem is referenced by: (None)