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

Theorem nulmbl2 23028
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 11668 . . . . 5 1 ∈ ℝ+
21ne0ii 3881 . . . 4 + ≠ ∅
3 r19.2z 4011 . . . 4 ((ℝ+ ≠ ∅ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → ∃𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))
42, 3mpan 701 . . 3 (∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ∃𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))
5 simprl 789 . . . . . 6 ((𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴𝑦)
6 mblss 23023 . . . . . . 7 (𝑦 ∈ dom vol → 𝑦 ⊆ ℝ)
76adantr 479 . . . . . 6 ((𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝑦 ⊆ ℝ)
85, 7sstrd 3577 . . . . 5 ((𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴 ⊆ ℝ)
98rexlimiva 3009 . . . 4 (∃𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ⊆ ℝ)
109rexlimivw 3010 . . 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 3794 . . . . . . . . . . . . 13 (𝑧𝐴) ⊆ 𝑧
1312a1i 11 . . . . . . . . . . . 12 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (𝑧𝐴) ⊆ 𝑧)
14 elpwi 4116 . . . . . . . . . . . . 13 (𝑧 ∈ 𝒫 ℝ → 𝑧 ⊆ ℝ)
1514adantr 479 . . . . . . . . . . . 12 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → 𝑧 ⊆ ℝ)
16 simpr 475 . . . . . . . . . . . 12 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) ∈ ℝ)
17 ovolsscl 22978 . . . . . . . . . . . 12 (((𝑧𝐴) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
1813, 15, 16, 17syl3anc 1317 . . . . . . . . . . 11 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
19 difssd 3699 . . . . . . . . . . . 12 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (𝑧𝐴) ⊆ 𝑧)
20 ovolsscl 22978 . . . . . . . . . . . 12 (((𝑧𝐴) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
2119, 15, 16, 20syl3anc 1317 . . . . . . . . . . 11 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
2218, 21readdcld 9925 . . . . . . . . . 10 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ)
2322ad2antrr 757 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ)
2416ad2antrr 757 . . . . . . . . . 10 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑧) ∈ ℝ)
25 difssd 3699 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦𝐴) ⊆ 𝑦)
267adantl 480 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑦 ⊆ ℝ)
27 rpre 11671 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
2827ad2antlr 758 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑥 ∈ ℝ)
29 simprrr 800 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ≤ 𝑥)
30 ovollecl 22975 . . . . . . . . . . . 12 ((𝑦 ⊆ ℝ ∧ 𝑥 ∈ ℝ ∧ (vol*‘𝑦) ≤ 𝑥) → (vol*‘𝑦) ∈ ℝ)
3126, 28, 29, 30syl3anc 1317 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ∈ ℝ)
32 ovolsscl 22978 . . . . . . . . . . 11 (((𝑦𝐴) ⊆ 𝑦𝑦 ⊆ ℝ ∧ (vol*‘𝑦) ∈ ℝ) → (vol*‘(𝑦𝐴)) ∈ ℝ)
3325, 26, 31, 32syl3anc 1317 . . . . . . . . . 10 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦𝐴)) ∈ ℝ)
3424, 33readdcld 9925 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + (vol*‘(𝑦𝐴))) ∈ ℝ)
3524, 28readdcld 9925 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + 𝑥) ∈ ℝ)
3618ad2antrr 757 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ∈ ℝ)
3721ad2antrr 757 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ∈ ℝ)
38 inss1 3794 . . . . . . . . . . . . 13 (𝑧𝑦) ⊆ 𝑧
3938a1i 11 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ 𝑧)
4015ad2antrr 757 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑧 ⊆ ℝ)
41 ovolsscl 22978 . . . . . . . . . . . 12 (((𝑧𝑦) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝑦)) ∈ ℝ)
4239, 40, 24, 41syl3anc 1317 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℝ)
43 difssd 3699 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ 𝑧)
44 ovolsscl 22978 . . . . . . . . . . . . 13 (((𝑧𝑦) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝑦)) ∈ ℝ)
4543, 40, 24, 44syl3anc 1317 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℝ)
4645, 33readdcld 9925 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))) ∈ ℝ)
47 simprrl 799 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝐴𝑦)
48 sslin 3800 . . . . . . . . . . . . 13 (𝐴𝑦 → (𝑧𝐴) ⊆ (𝑧𝑦))
4947, 48syl 17 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝐴) ⊆ (𝑧𝑦))
5038, 40syl5ss 3578 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ ℝ)
51 ovolss 22977 . . . . . . . . . . . 12 (((𝑧𝐴) ⊆ (𝑧𝑦) ∧ (𝑧𝑦) ⊆ ℝ) → (vol*‘(𝑧𝐴)) ≤ (vol*‘(𝑧𝑦)))
5249, 50, 51syl2anc 690 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ≤ (vol*‘(𝑧𝑦)))
5340ssdifssd 3709 . . . . . . . . . . . . . 14 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ ℝ)
5426ssdifssd 3709 . . . . . . . . . . . . . 14 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦𝐴) ⊆ ℝ)
5553, 54unssd 3750 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((𝑧𝑦) ∪ (𝑦𝐴)) ⊆ ℝ)
56 ovolun 22991 . . . . . . . . . . . . . 14 ((((𝑧𝑦) ⊆ ℝ ∧ (vol*‘(𝑧𝑦)) ∈ ℝ) ∧ ((𝑦𝐴) ⊆ ℝ ∧ (vol*‘(𝑦𝐴)) ∈ ℝ)) → (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ≤ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))))
5753, 45, 54, 33, 56syl22anc 1318 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ≤ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))))
58 ovollecl 22975 . . . . . . . . . . . . 13 ((((𝑧𝑦) ∪ (𝑦𝐴)) ⊆ ℝ ∧ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))) ∈ ℝ ∧ (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ≤ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴)))) → (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ∈ ℝ)
5955, 46, 57, 58syl3anc 1317 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ∈ ℝ)
60 ssun1 3737 . . . . . . . . . . . . . . . . 17 𝑧 ⊆ (𝑧𝑦)
61 undif1 3994 . . . . . . . . . . . . . . . . 17 ((𝑧𝑦) ∪ 𝑦) = (𝑧𝑦)
6260, 61sseqtr4i 3600 . . . . . . . . . . . . . . . 16 𝑧 ⊆ ((𝑧𝑦) ∪ 𝑦)
63 ssdif 3706 . . . . . . . . . . . . . . . 16 (𝑧 ⊆ ((𝑧𝑦) ∪ 𝑦) → (𝑧𝐴) ⊆ (((𝑧𝑦) ∪ 𝑦) ∖ 𝐴))
6462, 63ax-mp 5 . . . . . . . . . . . . . . 15 (𝑧𝐴) ⊆ (((𝑧𝑦) ∪ 𝑦) ∖ 𝐴)
65 difundir 3838 . . . . . . . . . . . . . . 15 (((𝑧𝑦) ∪ 𝑦) ∖ 𝐴) = (((𝑧𝑦) ∖ 𝐴) ∪ (𝑦𝐴))
6664, 65sseqtri 3599 . . . . . . . . . . . . . 14 (𝑧𝐴) ⊆ (((𝑧𝑦) ∖ 𝐴) ∪ (𝑦𝐴))
67 difun1 3845 . . . . . . . . . . . . . . . 16 (𝑧 ∖ (𝑦𝐴)) = ((𝑧𝑦) ∖ 𝐴)
68 ssequn2 3747 . . . . . . . . . . . . . . . . . 18 (𝐴𝑦 ↔ (𝑦𝐴) = 𝑦)
6947, 68sylib 206 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦𝐴) = 𝑦)
7069difeq2d 3689 . . . . . . . . . . . . . . . 16 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ (𝑦𝐴)) = (𝑧𝑦))
7167, 70syl5eqr 2657 . . . . . . . . . . . . . . 15 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((𝑧𝑦) ∖ 𝐴) = (𝑧𝑦))
7271uneq1d 3727 . . . . . . . . . . . . . 14 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (((𝑧𝑦) ∖ 𝐴) ∪ (𝑦𝐴)) = ((𝑧𝑦) ∪ (𝑦𝐴)))
7366, 72syl5sseq 3615 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝐴) ⊆ ((𝑧𝑦) ∪ (𝑦𝐴)))
74 ovolss 22977 . . . . . . . . . . . . 13 (((𝑧𝐴) ⊆ ((𝑧𝑦) ∪ (𝑦𝐴)) ∧ ((𝑧𝑦) ∪ (𝑦𝐴)) ⊆ ℝ) → (vol*‘(𝑧𝐴)) ≤ (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))))
7573, 55, 74syl2anc 690 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ≤ (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))))
7637, 59, 46, 75, 57letrd 10045 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ≤ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))))
7736, 37, 42, 46, 52, 76le2addd 10495 . . . . . . . . . 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)))))
78 simprl 789 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑦 ∈ dom vol)
79 mblsplit 23024 . . . . . . . . . . . . 13 ((𝑦 ∈ dom vol ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) = ((vol*‘(𝑧𝑦)) + (vol*‘(𝑧𝑦))))
8078, 40, 24, 79syl3anc 1317 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑧) = ((vol*‘(𝑧𝑦)) + (vol*‘(𝑧𝑦))))
8180oveq1d 6542 . . . . . . . . . . 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))))
8242recnd 9924 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℂ)
8345recnd 9924 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℂ)
8433recnd 9924 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦𝐴)) ∈ ℂ)
8582, 83, 84addassd 9918 . . . . . . . . . . 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)))))
8681, 85eqtrd 2643 . . . . . . . . . 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)))))
8777, 86breqtrrd 4605 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘𝑧) + (vol*‘(𝑦𝐴))))
88 difss 3698 . . . . . . . . . . . 12 (𝑦𝐴) ⊆ 𝑦
89 ovolss 22977 . . . . . . . . . . . 12 (((𝑦𝐴) ⊆ 𝑦𝑦 ⊆ ℝ) → (vol*‘(𝑦𝐴)) ≤ (vol*‘𝑦))
9088, 26, 89sylancr 693 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦𝐴)) ≤ (vol*‘𝑦))
9133, 31, 28, 90, 29letrd 10045 . . . . . . . . . 10 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦𝐴)) ≤ 𝑥)
9233, 28, 24, 91leadd2dd 10491 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + (vol*‘(𝑦𝐴))) ≤ ((vol*‘𝑧) + 𝑥))
9323, 34, 35, 87, 92letrd 10045 . . . . . . . 8 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘𝑧) + 𝑥))
9493rexlimdvaa 3013 . . . . . . 7 (((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (∃𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘𝑧) + 𝑥)))
9594ralimdva 2944 . . . . . 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)))
9695impcom 444 . . . . 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))
9722adantl 480 . . . . . . 7 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ)
9897rexrd 9945 . . . . . 6 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ*)
99 simprr 791 . . . . . 6 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → (vol*‘𝑧) ∈ ℝ)
100 xralrple 11869 . . . . . 6 ((((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ* ∧ (vol*‘𝑧) ∈ ℝ) → (((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧) ↔ ∀𝑥 ∈ ℝ+ ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘𝑧) + 𝑥)))
10198, 99, 100syl2anc 690 . . . . 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)))
10296, 101mpbird 245 . . . 4 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧))
103102expr 640 . . 3 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ 𝑧 ∈ 𝒫 ℝ) → ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧)))
104103ralrimiva 2948 . 2 (∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ∀𝑧 ∈ 𝒫 ℝ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧)))
105 ismbl2 23019 . 2 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧))))
10611, 104, 105sylanbrc 694 1 (∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  cdif 3536  cun 3537  cin 3538  wss 3539  c0 3873  𝒫 cpw 4107   class class class wbr 4577  dom cdm 5028  cfv 5790  (class class class)co 6527  cr 9791  1c1 9793   + caddc 9795  *cxr 9929  cle 9931  +crp 11664  vol*covol 22955  volcvol 22956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-sup 8208  df-inf 8209  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-q 11621  df-rp 11665  df-ioo 12006  df-ico 12008  df-icc 12009  df-fz 12153  df-fl 12410  df-seq 12619  df-exp 12678  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-ovol 22957  df-vol 22958
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator