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

Theorem nulmbl2 24119
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 12372 . . . . 5 1 ∈ ℝ+
21ne0ii 4279 . . . 4 + ≠ ∅
3 r19.2z 4416 . . . 4 ((ℝ+ ≠ ∅ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → ∃𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))
42, 3mpan 688 . . 3 (∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ∃𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))
5 simprl 769 . . . . . 6 ((𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴𝑦)
6 mblss 24114 . . . . . . 7 (𝑦 ∈ dom vol → 𝑦 ⊆ ℝ)
76adantr 483 . . . . . 6 ((𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝑦 ⊆ ℝ)
85, 7sstrd 3956 . . . . 5 ((𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴 ⊆ ℝ)
98rexlimiva 3268 . . . 4 (∃𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ⊆ ℝ)
109rexlimivw 3269 . . 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 4183 . . . . . . . . . . . 12 (𝑧𝐴) ⊆ 𝑧
13 elpwi 4524 . . . . . . . . . . . . 13 (𝑧 ∈ 𝒫 ℝ → 𝑧 ⊆ ℝ)
1413adantr 483 . . . . . . . . . . . 12 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → 𝑧 ⊆ ℝ)
15 simpr 487 . . . . . . . . . . . 12 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) ∈ ℝ)
16 ovolsscl 24069 . . . . . . . . . . . 12 (((𝑧𝐴) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
1712, 14, 15, 16mp3an2i 1462 . . . . . . . . . . 11 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
18 difssd 4088 . . . . . . . . . . . 12 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (𝑧𝐴) ⊆ 𝑧)
19 ovolsscl 24069 . . . . . . . . . . . 12 (((𝑧𝐴) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
2018, 14, 15, 19syl3anc 1367 . . . . . . . . . . 11 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
2117, 20readdcld 10648 . . . . . . . . . 10 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ)
2221ad2antrr 724 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ)
2315ad2antrr 724 . . . . . . . . . 10 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑧) ∈ ℝ)
24 difssd 4088 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦𝐴) ⊆ 𝑦)
257adantl 484 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑦 ⊆ ℝ)
26 rpre 12376 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
2726ad2antlr 725 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑥 ∈ ℝ)
28 simprrr 780 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ≤ 𝑥)
29 ovollecl 24066 . . . . . . . . . . . 12 ((𝑦 ⊆ ℝ ∧ 𝑥 ∈ ℝ ∧ (vol*‘𝑦) ≤ 𝑥) → (vol*‘𝑦) ∈ ℝ)
3025, 27, 28, 29syl3anc 1367 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ∈ ℝ)
31 ovolsscl 24069 . . . . . . . . . . 11 (((𝑦𝐴) ⊆ 𝑦𝑦 ⊆ ℝ ∧ (vol*‘𝑦) ∈ ℝ) → (vol*‘(𝑦𝐴)) ∈ ℝ)
3224, 25, 30, 31syl3anc 1367 . . . . . . . . . 10 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦𝐴)) ∈ ℝ)
3323, 32readdcld 10648 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + (vol*‘(𝑦𝐴))) ∈ ℝ)
3423, 27readdcld 10648 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + 𝑥) ∈ ℝ)
3517ad2antrr 724 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ∈ ℝ)
3620ad2antrr 724 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ∈ ℝ)
37 inss1 4183 . . . . . . . . . . . 12 (𝑧𝑦) ⊆ 𝑧
3814ad2antrr 724 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑧 ⊆ ℝ)
39 ovolsscl 24069 . . . . . . . . . . . 12 (((𝑧𝑦) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝑦)) ∈ ℝ)
4037, 38, 23, 39mp3an2i 1462 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℝ)
41 difssd 4088 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ 𝑧)
42 ovolsscl 24069 . . . . . . . . . . . . 13 (((𝑧𝑦) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝑦)) ∈ ℝ)
4341, 38, 23, 42syl3anc 1367 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℝ)
4443, 32readdcld 10648 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))) ∈ ℝ)
45 simprrl 779 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝐴𝑦)
46 sslin 4189 . . . . . . . . . . . . 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 3957 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ ℝ)
49 ovolss 24068 . . . . . . . . . . . 12 (((𝑧𝐴) ⊆ (𝑧𝑦) ∧ (𝑧𝑦) ⊆ ℝ) → (vol*‘(𝑧𝐴)) ≤ (vol*‘(𝑧𝑦)))
5047, 48, 49syl2anc 586 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ≤ (vol*‘(𝑧𝑦)))
5138ssdifssd 4098 . . . . . . . . . . . . . 14 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ ℝ)
5225ssdifssd 4098 . . . . . . . . . . . . . 14 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦𝐴) ⊆ ℝ)
5351, 52unssd 4141 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((𝑧𝑦) ∪ (𝑦𝐴)) ⊆ ℝ)
54 ovolun 24082 . . . . . . . . . . . . . 14 ((((𝑧𝑦) ⊆ ℝ ∧ (vol*‘(𝑧𝑦)) ∈ ℝ) ∧ ((𝑦𝐴) ⊆ ℝ ∧ (vol*‘(𝑦𝐴)) ∈ ℝ)) → (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ≤ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))))
5551, 43, 52, 32, 54syl22anc 836 . . . . . . . . . . . . 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 24066 . . . . . . . . . . . . 13 ((((𝑧𝑦) ∪ (𝑦𝐴)) ⊆ ℝ ∧ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))) ∈ ℝ ∧ (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ≤ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴)))) → (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ∈ ℝ)
5753, 44, 55, 56syl3anc 1367 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ∈ ℝ)
58 ssun1 4127 . . . . . . . . . . . . . . . . 17 𝑧 ⊆ (𝑧𝑦)
59 undif1 4400 . . . . . . . . . . . . . . . . 17 ((𝑧𝑦) ∪ 𝑦) = (𝑧𝑦)
6058, 59sseqtrri 3983 . . . . . . . . . . . . . . . 16 𝑧 ⊆ ((𝑧𝑦) ∪ 𝑦)
61 ssdif 4095 . . . . . . . . . . . . . . . 16 (𝑧 ⊆ ((𝑧𝑦) ∪ 𝑦) → (𝑧𝐴) ⊆ (((𝑧𝑦) ∪ 𝑦) ∖ 𝐴))
6260, 61ax-mp 5 . . . . . . . . . . . . . . 15 (𝑧𝐴) ⊆ (((𝑧𝑦) ∪ 𝑦) ∖ 𝐴)
63 difundir 4235 . . . . . . . . . . . . . . 15 (((𝑧𝑦) ∪ 𝑦) ∖ 𝐴) = (((𝑧𝑦) ∖ 𝐴) ∪ (𝑦𝐴))
6462, 63sseqtri 3982 . . . . . . . . . . . . . 14 (𝑧𝐴) ⊆ (((𝑧𝑦) ∖ 𝐴) ∪ (𝑦𝐴))
65 difun1 4242 . . . . . . . . . . . . . . . 16 (𝑧 ∖ (𝑦𝐴)) = ((𝑧𝑦) ∖ 𝐴)
66 ssequn2 4138 . . . . . . . . . . . . . . . . . 18 (𝐴𝑦 ↔ (𝑦𝐴) = 𝑦)
6745, 66sylib 220 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦𝐴) = 𝑦)
6867difeq2d 4078 . . . . . . . . . . . . . . . 16 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ (𝑦𝐴)) = (𝑧𝑦))
6965, 68syl5eqr 2869 . . . . . . . . . . . . . . 15 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((𝑧𝑦) ∖ 𝐴) = (𝑧𝑦))
7069uneq1d 4117 . . . . . . . . . . . . . 14 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (((𝑧𝑦) ∖ 𝐴) ∪ (𝑦𝐴)) = ((𝑧𝑦) ∪ (𝑦𝐴)))
7164, 70sseqtrid 3998 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝐴) ⊆ ((𝑧𝑦) ∪ (𝑦𝐴)))
72 ovolss 24068 . . . . . . . . . . . . 13 (((𝑧𝐴) ⊆ ((𝑧𝑦) ∪ (𝑦𝐴)) ∧ ((𝑧𝑦) ∪ (𝑦𝐴)) ⊆ ℝ) → (vol*‘(𝑧𝐴)) ≤ (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))))
7371, 53, 72syl2anc 586 . . . . . . . . . . . 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 10775 . . . . . . . . . . 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 11237 . . . . . . . . . 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 769 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑦 ∈ dom vol)
77 mblsplit 24115 . . . . . . . . . . . . 13 ((𝑦 ∈ dom vol ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) = ((vol*‘(𝑧𝑦)) + (vol*‘(𝑧𝑦))))
7876, 38, 23, 77syl3anc 1367 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑧) = ((vol*‘(𝑧𝑦)) + (vol*‘(𝑧𝑦))))
7978oveq1d 7148 . . . . . . . . . . 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 10647 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℂ)
8143recnd 10647 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℂ)
8232recnd 10647 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦𝐴)) ∈ ℂ)
8380, 81, 82addassd 10641 . . . . . . . . . . 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 2855 . . . . . . . . . 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 5070 . . . . . . . . 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 4087 . . . . . . . . . . . 12 (𝑦𝐴) ⊆ 𝑦
87 ovolss 24068 . . . . . . . . . . . 12 (((𝑦𝐴) ⊆ 𝑦𝑦 ⊆ ℝ) → (vol*‘(𝑦𝐴)) ≤ (vol*‘𝑦))
8886, 25, 87sylancr 589 . . . . . . . . . . 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 10775 . . . . . . . . . 10 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦𝐴)) ≤ 𝑥)
9032, 27, 23, 89leadd2dd 11233 . . . . . . . . 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 10775 . . . . . . . 8 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘𝑧) + 𝑥))
9291rexlimdvaa 3272 . . . . . . 7 (((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (∃𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘𝑧) + 𝑥)))
9392ralimdva 3164 . . . . . 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 410 . . . . 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 484 . . . . . . 7 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ)
9695rexrd 10669 . . . . . 6 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ*)
97 simprr 771 . . . . . 6 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → (vol*‘𝑧) ∈ ℝ)
98 xralrple 12577 . . . . . 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 586 . . . . 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 259 . . . 4 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧))
101100expr 459 . . 3 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ 𝑧 ∈ 𝒫 ℝ) → ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧)))
102101ralrimiva 3169 . 2 (∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ∀𝑧 ∈ 𝒫 ℝ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧)))
103 ismbl2 24110 . 2 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧))))
10411, 102, 103sylanbrc 585 1 (∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wcel 2114  wne 3006  wral 3125  wrex 3126  cdif 3910  cun 3911  cin 3912  wss 3913  c0 4269  𝒫 cpw 4515   class class class wbr 5042  dom cdm 5531  cfv 6331  (class class class)co 7133  cr 10514  1c1 10516   + caddc 10518  *cxr 10652  cle 10654  +crp 12368  vol*covol 24045  volcvol 24046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592  ax-pre-sup 10593
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-er 8267  df-map 8386  df-en 8488  df-dom 8489  df-sdom 8490  df-sup 8884  df-inf 8885  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-div 11276  df-nn 11617  df-2 11679  df-3 11680  df-n0 11877  df-z 11961  df-uz 12223  df-q 12328  df-rp 12369  df-ioo 12721  df-ico 12723  df-icc 12724  df-fz 12877  df-fl 13146  df-seq 13354  df-exp 13415  df-cj 14438  df-re 14439  df-im 14440  df-sqrt 14574  df-abs 14575  df-ovol 24047  df-vol 24048
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator