MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nulmbl2 Structured version   Visualization version   GIF version

Theorem nulmbl2 24916
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(𝐴𝑦 ∧ (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.
StepHypRef Expression
1 1rp 12924 . . . . 5 1 ∈ ℝ+
21ne0ii 4298 . . . 4 + ≠ ∅
3 r19.2z 4453 . . . 4 ((ℝ+ ≠ ∅ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → ∃𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))
42, 3mpan 689 . . 3 (∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ∃𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))
5 simprl 770 . . . . . 6 ((𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴𝑦)
6 mblss 24911 . . . . . . 7 (𝑦 ∈ dom vol → 𝑦 ⊆ ℝ)
76adantr 482 . . . . . 6 ((𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝑦 ⊆ ℝ)
85, 7sstrd 3955 . . . . 5 ((𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴 ⊆ ℝ)
98rexlimiva 3141 . . . 4 (∃𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ⊆ ℝ)
109rexlimivw 3145 . . 3 (∃𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ⊆ ℝ)
114, 10syl 17 . 2 (∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ⊆ ℝ)
12 inss1 4189 . . . . . . . . . . . 12 (𝑧𝐴) ⊆ 𝑧
13 elpwi 4568 . . . . . . . . . . . . 13 (𝑧 ∈ 𝒫 ℝ → 𝑧 ⊆ ℝ)
1413adantr 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 24866 . . . . . . . . . . . 12 (((𝑧𝐴) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
1712, 14, 15, 16mp3an2i 1467 . . . . . . . . . . 11 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
18 difssd 4093 . . . . . . . . . . . 12 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (𝑧𝐴) ⊆ 𝑧)
19 ovolsscl 24866 . . . . . . . . . . . 12 (((𝑧𝐴) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
2018, 14, 15, 19syl3anc 1372 . . . . . . . . . . 11 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
2117, 20readdcld 11189 . . . . . . . . . 10 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ)
2221ad2antrr 725 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ)
2315ad2antrr 725 . . . . . . . . . 10 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑧) ∈ ℝ)
24 difssd 4093 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦𝐴) ⊆ 𝑦)
257adantl 483 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑦 ⊆ ℝ)
26 rpre 12928 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
2726ad2antlr 726 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑥 ∈ ℝ)
28 simprrr 781 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ≤ 𝑥)
29 ovollecl 24863 . . . . . . . . . . . 12 ((𝑦 ⊆ ℝ ∧ 𝑥 ∈ ℝ ∧ (vol*‘𝑦) ≤ 𝑥) → (vol*‘𝑦) ∈ ℝ)
3025, 27, 28, 29syl3anc 1372 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ∈ ℝ)
31 ovolsscl 24866 . . . . . . . . . . 11 (((𝑦𝐴) ⊆ 𝑦𝑦 ⊆ ℝ ∧ (vol*‘𝑦) ∈ ℝ) → (vol*‘(𝑦𝐴)) ∈ ℝ)
3224, 25, 30, 31syl3anc 1372 . . . . . . . . . 10 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦𝐴)) ∈ ℝ)
3323, 32readdcld 11189 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + (vol*‘(𝑦𝐴))) ∈ ℝ)
3423, 27readdcld 11189 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + 𝑥) ∈ ℝ)
3517ad2antrr 725 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ∈ ℝ)
3620ad2antrr 725 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ∈ ℝ)
37 inss1 4189 . . . . . . . . . . . 12 (𝑧𝑦) ⊆ 𝑧
3814ad2antrr 725 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑧 ⊆ ℝ)
39 ovolsscl 24866 . . . . . . . . . . . 12 (((𝑧𝑦) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝑦)) ∈ ℝ)
4037, 38, 23, 39mp3an2i 1467 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℝ)
41 difssd 4093 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ 𝑧)
42 ovolsscl 24866 . . . . . . . . . . . . 13 (((𝑧𝑦) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝑦)) ∈ ℝ)
4341, 38, 23, 42syl3anc 1372 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℝ)
4443, 32readdcld 11189 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))) ∈ ℝ)
45 simprrl 780 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝐴𝑦)
46 sslin 4195 . . . . . . . . . . . . 13 (𝐴𝑦 → (𝑧𝐴) ⊆ (𝑧𝑦))
4745, 46syl 17 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝐴) ⊆ (𝑧𝑦))
4837, 38sstrid 3956 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ ℝ)
49 ovolss 24865 . . . . . . . . . . . 12 (((𝑧𝐴) ⊆ (𝑧𝑦) ∧ (𝑧𝑦) ⊆ ℝ) → (vol*‘(𝑧𝐴)) ≤ (vol*‘(𝑧𝑦)))
5047, 48, 49syl2anc 585 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ≤ (vol*‘(𝑧𝑦)))
5138ssdifssd 4103 . . . . . . . . . . . . . 14 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ ℝ)
5225ssdifssd 4103 . . . . . . . . . . . . . 14 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦𝐴) ⊆ ℝ)
5351, 52unssd 4147 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((𝑧𝑦) ∪ (𝑦𝐴)) ⊆ ℝ)
54 ovolun 24879 . . . . . . . . . . . . . 14 ((((𝑧𝑦) ⊆ ℝ ∧ (vol*‘(𝑧𝑦)) ∈ ℝ) ∧ ((𝑦𝐴) ⊆ ℝ ∧ (vol*‘(𝑦𝐴)) ∈ ℝ)) → (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ≤ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))))
5551, 43, 52, 32, 54syl22anc 838 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ≤ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))))
56 ovollecl 24863 . . . . . . . . . . . . 13 ((((𝑧𝑦) ∪ (𝑦𝐴)) ⊆ ℝ ∧ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))) ∈ ℝ ∧ (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ≤ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴)))) → (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ∈ ℝ)
5753, 44, 55, 56syl3anc 1372 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ∈ ℝ)
58 ssun1 4133 . . . . . . . . . . . . . . . . 17 𝑧 ⊆ (𝑧𝑦)
59 undif1 4436 . . . . . . . . . . . . . . . . 17 ((𝑧𝑦) ∪ 𝑦) = (𝑧𝑦)
6058, 59sseqtrri 3982 . . . . . . . . . . . . . . . 16 𝑧 ⊆ ((𝑧𝑦) ∪ 𝑦)
61 ssdif 4100 . . . . . . . . . . . . . . . 16 (𝑧 ⊆ ((𝑧𝑦) ∪ 𝑦) → (𝑧𝐴) ⊆ (((𝑧𝑦) ∪ 𝑦) ∖ 𝐴))
6260, 61ax-mp 5 . . . . . . . . . . . . . . 15 (𝑧𝐴) ⊆ (((𝑧𝑦) ∪ 𝑦) ∖ 𝐴)
63 difundir 4241 . . . . . . . . . . . . . . 15 (((𝑧𝑦) ∪ 𝑦) ∖ 𝐴) = (((𝑧𝑦) ∖ 𝐴) ∪ (𝑦𝐴))
6462, 63sseqtri 3981 . . . . . . . . . . . . . 14 (𝑧𝐴) ⊆ (((𝑧𝑦) ∖ 𝐴) ∪ (𝑦𝐴))
65 difun1 4250 . . . . . . . . . . . . . . . 16 (𝑧 ∖ (𝑦𝐴)) = ((𝑧𝑦) ∖ 𝐴)
66 ssequn2 4144 . . . . . . . . . . . . . . . . . 18 (𝐴𝑦 ↔ (𝑦𝐴) = 𝑦)
6745, 66sylib 217 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦𝐴) = 𝑦)
6867difeq2d 4083 . . . . . . . . . . . . . . . 16 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ (𝑦𝐴)) = (𝑧𝑦))
6965, 68eqtr3id 2787 . . . . . . . . . . . . . . 15 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((𝑧𝑦) ∖ 𝐴) = (𝑧𝑦))
7069uneq1d 4123 . . . . . . . . . . . . . 14 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (((𝑧𝑦) ∖ 𝐴) ∪ (𝑦𝐴)) = ((𝑧𝑦) ∪ (𝑦𝐴)))
7164, 70sseqtrid 3997 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝐴) ⊆ ((𝑧𝑦) ∪ (𝑦𝐴)))
72 ovolss 24865 . . . . . . . . . . . . 13 (((𝑧𝐴) ⊆ ((𝑧𝑦) ∪ (𝑦𝐴)) ∧ ((𝑧𝑦) ∪ (𝑦𝐴)) ⊆ ℝ) → (vol*‘(𝑧𝐴)) ≤ (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))))
7371, 53, 72syl2anc 585 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ≤ (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))))
7436, 57, 44, 73, 55letrd 11317 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ≤ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))))
7535, 36, 40, 44, 50, 74le2addd 11779 . . . . . . . . . 10 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘(𝑧𝑦)) + ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴)))))
76 simprl 770 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑦 ∈ dom vol)
77 mblsplit 24912 . . . . . . . . . . . . 13 ((𝑦 ∈ dom vol ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) = ((vol*‘(𝑧𝑦)) + (vol*‘(𝑧𝑦))))
7876, 38, 23, 77syl3anc 1372 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑧) = ((vol*‘(𝑧𝑦)) + (vol*‘(𝑧𝑦))))
7978oveq1d 7373 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + (vol*‘(𝑦𝐴))) = (((vol*‘(𝑧𝑦)) + (vol*‘(𝑧𝑦))) + (vol*‘(𝑦𝐴))))
8040recnd 11188 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℂ)
8143recnd 11188 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℂ)
8232recnd 11188 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦𝐴)) ∈ ℂ)
8380, 81, 82addassd 11182 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (((vol*‘(𝑧𝑦)) + (vol*‘(𝑧𝑦))) + (vol*‘(𝑦𝐴))) = ((vol*‘(𝑧𝑦)) + ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴)))))
8479, 83eqtrd 2773 . . . . . . . . . 10 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + (vol*‘(𝑦𝐴))) = ((vol*‘(𝑧𝑦)) + ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴)))))
8575, 84breqtrrd 5134 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘𝑧) + (vol*‘(𝑦𝐴))))
86 difss 4092 . . . . . . . . . . . 12 (𝑦𝐴) ⊆ 𝑦
87 ovolss 24865 . . . . . . . . . . . 12 (((𝑦𝐴) ⊆ 𝑦𝑦 ⊆ ℝ) → (vol*‘(𝑦𝐴)) ≤ (vol*‘𝑦))
8886, 25, 87sylancr 588 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦𝐴)) ≤ (vol*‘𝑦))
8932, 30, 27, 88, 28letrd 11317 . . . . . . . . . 10 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦𝐴)) ≤ 𝑥)
9032, 27, 23, 89leadd2dd 11775 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + (vol*‘(𝑦𝐴))) ≤ ((vol*‘𝑧) + 𝑥))
9122, 33, 34, 85, 90letrd 11317 . . . . . . . 8 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘𝑧) + 𝑥))
9291rexlimdvaa 3150 . . . . . . 7 (((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (∃𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘𝑧) + 𝑥)))
9392ralimdva 3161 . . . . . 6 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ∀𝑥 ∈ ℝ+ ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘𝑧) + 𝑥)))
9493impcom 409 . . . . 5 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ∀𝑥 ∈ ℝ+ ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘𝑧) + 𝑥))
9521adantl 483 . . . . . . 7 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ)
9695rexrd 11210 . . . . . 6 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ*)
97 simprr 772 . . . . . 6 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → (vol*‘𝑧) ∈ ℝ)
98 xralrple 13130 . . . . . 6 ((((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ* ∧ (vol*‘𝑧) ∈ ℝ) → (((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧) ↔ ∀𝑥 ∈ ℝ+ ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘𝑧) + 𝑥)))
9996, 97, 98syl2anc 585 . . . . 5 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → (((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧) ↔ ∀𝑥 ∈ ℝ+ ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘𝑧) + 𝑥)))
10094, 99mpbird 257 . . . 4 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧))
101100expr 458 . . 3 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ 𝑧 ∈ 𝒫 ℝ) → ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧)))
102101ralrimiva 3140 . 2 (∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ∀𝑧 ∈ 𝒫 ℝ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧)))
103 ismbl2 24907 . 2 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧))))
10411, 102, 103sylanbrc 584 1 (∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2940  wral 3061  wrex 3070  cdif 3908  cun 3909  cin 3910  wss 3911  c0 4283  𝒫 cpw 4561   class class class wbr 5106  dom cdm 5634  cfv 6497  (class class class)co 7358  cr 11055  1c1 11057   + caddc 11059  *cxr 11193  cle 11195  +crp 12920  vol*covol 24842  volcvol 24843
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9383  df-inf 9384  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-n0 12419  df-z 12505  df-uz 12769  df-q 12879  df-rp 12921  df-ioo 13274  df-ico 13276  df-icc 13277  df-fz 13431  df-fl 13703  df-seq 13913  df-exp 13974  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-ovol 24844  df-vol 24845
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator