Theorem nulmbl2 25732

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 4300	. . . 4 ⊢ ℝ+ ≠ ∅
3		r19.2z 4465	. . . 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 703	. . 3 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ∃𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))
5		simprl 783	. . . . . 6 ⊢ ((𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴 ⊆ 𝑦)
6		mblss 25727	. . . . . . 7 ⊢ (𝑦 ∈ dom vol → 𝑦 ⊆ ℝ)
7	6	adantr 486	. . . . . 6 ⊢ ((𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝑦 ⊆ ℝ)
8	5, 7	sstrd 3950	. . . . 5 ⊢ ((𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴 ⊆ ℝ)
9	8	rexlimiva 3161	. . . 4 ⊢ (∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ⊆ ℝ)
10	9	rexlimivw 3165	. . 3 ⊢ (∃𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ⊆ ℝ)
11	4, 10	syl 18	. 2 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ⊆ ℝ)
12		inss1 4192	. . . . . . . . . . . 12 ⊢ (𝑧 ∩ 𝐴) ⊆ 𝑧
13		elpwi 4574	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝒫 ℝ → 𝑧 ⊆ ℝ)
14	13	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → 𝑧 ⊆ ℝ)
15		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) ∈ ℝ)
16		ovolsscl 25682	. . . . . . . . . . . 12 ⊢ (((𝑧 ∩ 𝐴) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∩ 𝐴)) ∈ ℝ)
17	12, 14, 15, 16	mp3an2i 1495	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∩ 𝐴)) ∈ ℝ)
18		difssd 4094	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (𝑧 ∖ 𝐴) ⊆ 𝑧)
19		ovolsscl 25682	. . . . . . . . . . . 12 ⊢ (((𝑧 ∖ 𝐴) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∖ 𝐴)) ∈ ℝ)
20	18, 14, 15, 19	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∖ 𝐴)) ∈ ℝ)
21	17, 20	readdcld 11256	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → ((vol*‘(𝑧 ∩ 𝐴)) + (vol*‘(𝑧 ∖ 𝐴))) ∈ ℝ)
22	21	ad2antrr 739	. . . . . . . . 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 739	. . . . . . . . . 10 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑧) ∈ ℝ)
24		difssd 4094	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦 ∖ 𝐴) ⊆ 𝑦)
25	7	adantl 487	. . . . . . . . . . 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 740	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑥 ∈ ℝ)
28		simprrr 794	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ≤ 𝑥)
29		ovollecl 25679	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊆ ℝ ∧ 𝑥 ∈ ℝ ∧ (vol*‘𝑦) ≤ 𝑥) → (vol*‘𝑦) ∈ ℝ)
30	25, 27, 28, 29	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ∈ ℝ)
31		ovolsscl 25682	. . . . . . . . . . 11 ⊢ (((𝑦 ∖ 𝐴) ⊆ 𝑦 ∧ 𝑦 ⊆ ℝ ∧ (vol*‘𝑦) ∈ ℝ) → (vol*‘(𝑦 ∖ 𝐴)) ∈ ℝ)
32	24, 25, 30, 31	syl3anc 1398	. . . . . . . . . 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 11256	. . . . . . . . 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 11256	. . . . . . . . 9 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + 𝑥) ∈ ℝ)
35	17	ad2antrr 739	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∩ 𝐴)) ∈ ℝ)
36	20	ad2antrr 739	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∖ 𝐴)) ∈ ℝ)
37		inss1 4192	. . . . . . . . . . . 12 ⊢ (𝑧 ∩ 𝑦) ⊆ 𝑧
38	14	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑧 ⊆ ℝ)
39		ovolsscl 25682	. . . . . . . . . . . 12 ⊢ (((𝑧 ∩ 𝑦) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∩ 𝑦)) ∈ ℝ)
40	37, 38, 23, 39	mp3an2i 1495	. . . . . . . . . . 11 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∩ 𝑦)) ∈ ℝ)
41		difssd 4094	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ 𝑦) ⊆ 𝑧)
42		ovolsscl 25682	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∖ 𝑦) ⊆ 𝑧 ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧 ∖ 𝑦)) ∈ ℝ)
43	41, 38, 23, 42	syl3anc 1398	. . . . . . . . . . . 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 11256	. . . . . . . . . . 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 793	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝐴 ⊆ 𝑦)
46		sslin 4198	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑦 → (𝑧 ∩ 𝐴) ⊆ (𝑧 ∩ 𝑦))
47	45, 46	syl 18	. . . . . . . . . . . 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 3951	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∩ 𝑦) ⊆ ℝ)
49		ovolss 25681	. . . . . . . . . . . 12 ⊢ (((𝑧 ∩ 𝐴) ⊆ (𝑧 ∩ 𝑦) ∧ (𝑧 ∩ 𝑦) ⊆ ℝ) → (vol*‘(𝑧 ∩ 𝐴)) ≤ (vol*‘(𝑧 ∩ 𝑦)))
50	47, 48, 49	syl2anc 596	. . . . . . . . . . 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 4104	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ 𝑦) ⊆ ℝ)
52	25	ssdifssd 4104	. . . . . . . . . . . . . 14 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦 ∖ 𝐴) ⊆ ℝ)
53	51, 52	unssd 4148	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((𝑧 ∖ 𝑦) ∪ (𝑦 ∖ 𝐴)) ⊆ ℝ)
54		ovolun 25695	. . . . . . . . . . . . . 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 852	. . . . . . . . . . . . 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 25679	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 4134	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ⊆ (𝑧 ∪ 𝑦)
59		undif1 4440	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∖ 𝑦) ∪ 𝑦) = (𝑧 ∪ 𝑦)
60	58, 59	sseqtrri 3989	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ⊆ ((𝑧 ∖ 𝑦) ∪ 𝑦)
61		ssdif 4101	. . . . . . . . . . . . . . . 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 4247	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 ∖ 𝑦) ∪ 𝑦) ∖ 𝐴) = (((𝑧 ∖ 𝑦) ∖ 𝐴) ∪ (𝑦 ∖ 𝐴))
64	62, 63	sseqtri 3988	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∖ 𝐴) ⊆ (((𝑧 ∖ 𝑦) ∖ 𝐴) ∪ (𝑦 ∖ 𝐴))
65		difun1 4255	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∖ (𝑦 ∪ 𝐴)) = ((𝑧 ∖ 𝑦) ∖ 𝐴)
66		ssequn2 4145	. . . . . . . . . . . . . . . . . 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 4084	. . . . . . . . . . . . . . . 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 2815	. . . . . . . . . . . . . . 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 4124	. . . . . . . . . . . . . 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 3982	. . . . . . . . . . . . 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 25681	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 11385	. . . . . . . . . . 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 11851	. . . . . . . . . 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 783	. . . . . . . . . . . . 13 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑦 ∈ dom vol)
77		mblsplit 25728	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ dom vol ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) = ((vol*‘(𝑧 ∩ 𝑦)) + (vol*‘(𝑧 ∖ 𝑦))))
78	76, 38, 23, 77	syl3anc 1398	. . . . . . . . . . . 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 7438	. . . . . . . . . . 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 11255	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∩ 𝑦)) ∈ ℂ)
81	43	recnd 11255	. . . . . . . . . . . 12 ⊢ ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧 ∖ 𝑦)) ∈ ℂ)
82	32	recnd 11255	. . . . . . . . . . . 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 11249	. . . . . . . . . . 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 2801	. . . . . . . . . 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 5144	. . . . . . . . 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 4093	. . . . . . . . . . . 12 ⊢ (𝑦 ∖ 𝐴) ⊆ 𝑦
87		ovolss 25681	. . . . . . . . . . . 12 ⊢ (((𝑦 ∖ 𝐴) ⊆ 𝑦 ∧ 𝑦 ⊆ ℝ) → (vol*‘(𝑦 ∖ 𝐴)) ≤ (vol*‘𝑦))
88	86, 25, 87	sylancr 599	. . . . . . . . . . 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 11385	. . . . . . . . . 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 11847	. . . . . . . . 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 11385	. . . . . . . 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 3170	. . . . . . 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 3180	. . . . . 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 413	. . . . 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 487	. . . . . . 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 11277	. . . . . 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 785	. . . . . 6 ⊢ ((∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → (vol*‘𝑧) ∈ ℝ)
98		xralrple 13249	. . . . . 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 596	. . . . 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 462	. . 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 3160	. 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 25723	. 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 595	1 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ dom vol(𝐴 ⊆ 𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ∈ dom vol)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  ∀wral 3082  ∃wrex 3092   ∖ cdif 3905   ∪ cun 3906   ∩ cin 3907   ⊆ wss 3908  ∅c0 4289  𝒫 cpw 4567   class class class wbr 5114  dom cdm 5666  ‘cfv 6543  (class class class)co 7423  ℝcr 11117  1c1 11119   + caddc 11121  ℝ^*cxr 11260   ≤ cle 11262  ℝ⁺crp 13034  vol*covol 25658  volcvol 25659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195  ax-pre-sup 11196

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-er 8703  df-map 8835  df-en 8953  df-dom 8954  df-sdom 8955  df-sup 9412  df-inf 9413  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-div 11890  df-nn 12252  df-2 12321  df-3 12322  df-n0 12523  df-z 12610  df-uz 12881  df-q 12991  df-rp 13035  df-ioo 13394  df-ico 13396  df-icc 13397  df-fz 13554  df-fl 13845  df-seq 14058  df-exp 14118  df-cj 15176  df-re 15177  df-im 15178  df-sqrt 15312  df-abs 15313  df-ovol 25660  df-vol 25661

This theorem is used by: (None)