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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(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 12117 . . . . 5 1 ∈ ℝ+
21ne0ii 4154 . . . 4 + ≠ ∅
3 r19.2z 4283 . . . 4 ((ℝ+ ≠ ∅ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → ∃𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))
42, 3mpan 683 . . 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 23698 . . . . . . 7 (𝑦 ∈ dom vol → 𝑦 ⊆ ℝ)
76adantr 474 . . . . . 6 ((𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝑦 ⊆ ℝ)
85, 7sstrd 3838 . . . . 5 ((𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴 ⊆ ℝ)
98rexlimiva 3238 . . . 4 (∃𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ⊆ ℝ)
109rexlimivw 3239 . . 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 4058 . . . . . . . . . . . . 13 (𝑧𝐴) ⊆ 𝑧
1312a1i 11 . . . . . . . . . . . 12 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (𝑧𝐴) ⊆ 𝑧)
14 elpwi 4389 . . . . . . . . . . . . 13 (𝑧 ∈ 𝒫 ℝ → 𝑧 ⊆ ℝ)
1514adantr 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 (((𝑧𝐴) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
1813, 15, 16, 17syl3anc 1496 . . . . . . . . . . 11 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
19 difssd 3966 . . . . . . . . . . . 12 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (𝑧𝐴) ⊆ 𝑧)
20 ovolsscl 23653 . . . . . . . . . . . 12 (((𝑧𝐴) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
2119, 15, 16, 20syl3anc 1496 . . . . . . . . . . 11 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
2218, 21readdcld 10387 . . . . . . . . . 10 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ)
2322ad2antrr 719 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ)
2416ad2antrr 719 . . . . . . . . . 10 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑧) ∈ ℝ)
25 difssd 3966 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦𝐴) ⊆ 𝑦)
267adantl 475 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑦 ⊆ ℝ)
27 rpre 12121 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
2827ad2antlr 720 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑥 ∈ ℝ)
29 simprrr 802 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ≤ 𝑥)
30 ovollecl 23650 . . . . . . . . . . . 12 ((𝑦 ⊆ ℝ ∧ 𝑥 ∈ ℝ ∧ (vol*‘𝑦) ≤ 𝑥) → (vol*‘𝑦) ∈ ℝ)
3126, 28, 29, 30syl3anc 1496 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ∈ ℝ)
32 ovolsscl 23653 . . . . . . . . . . 11 (((𝑦𝐴) ⊆ 𝑦𝑦 ⊆ ℝ ∧ (vol*‘𝑦) ∈ ℝ) → (vol*‘(𝑦𝐴)) ∈ ℝ)
3325, 26, 31, 32syl3anc 1496 . . . . . . . . . 10 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦𝐴)) ∈ ℝ)
3424, 33readdcld 10387 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + (vol*‘(𝑦𝐴))) ∈ ℝ)
3524, 28readdcld 10387 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + 𝑥) ∈ ℝ)
3618ad2antrr 719 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ∈ ℝ)
3721ad2antrr 719 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ∈ ℝ)
38 inss1 4058 . . . . . . . . . . . . 13 (𝑧𝑦) ⊆ 𝑧
3938a1i 11 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ 𝑧)
4015ad2antrr 719 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑧 ⊆ ℝ)
41 ovolsscl 23653 . . . . . . . . . . . 12 (((𝑧𝑦) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝑦)) ∈ ℝ)
4239, 40, 24, 41syl3anc 1496 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℝ)
43 difssd 3966 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ 𝑧)
44 ovolsscl 23653 . . . . . . . . . . . . 13 (((𝑧𝑦) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝑦)) ∈ ℝ)
4543, 40, 24, 44syl3anc 1496 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℝ)
4645, 33readdcld 10387 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))) ∈ ℝ)
47 simprrl 801 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝐴𝑦)
48 sslin 4064 . . . . . . . . . . . . 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 3839 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ ℝ)
51 ovolss 23652 . . . . . . . . . . . 12 (((𝑧𝐴) ⊆ (𝑧𝑦) ∧ (𝑧𝑦) ⊆ ℝ) → (vol*‘(𝑧𝐴)) ≤ (vol*‘(𝑧𝑦)))
5249, 50, 51syl2anc 581 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ≤ (vol*‘(𝑧𝑦)))
5340ssdifssd 3976 . . . . . . . . . . . . . 14 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ ℝ)
5426ssdifssd 3976 . . . . . . . . . . . . . 14 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦𝐴) ⊆ ℝ)
5553, 54unssd 4017 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((𝑧𝑦) ∪ (𝑦𝐴)) ⊆ ℝ)
56 ovolun 23666 . . . . . . . . . . . . . 14 ((((𝑧𝑦) ⊆ ℝ ∧ (vol*‘(𝑧𝑦)) ∈ ℝ) ∧ ((𝑦𝐴) ⊆ ℝ ∧ (vol*‘(𝑦𝐴)) ∈ ℝ)) → (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ≤ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))))
5753, 45, 54, 33, 56syl22anc 874 . . . . . . . . . . . . 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 23650 . . . . . . . . . . . . 13 ((((𝑧𝑦) ∪ (𝑦𝐴)) ⊆ ℝ ∧ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))) ∈ ℝ ∧ (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ≤ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴)))) → (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ∈ ℝ)
5955, 46, 57, 58syl3anc 1496 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ∈ ℝ)
60 ssun1 4004 . . . . . . . . . . . . . . . . 17 𝑧 ⊆ (𝑧𝑦)
61 undif1 4267 . . . . . . . . . . . . . . . . 17 ((𝑧𝑦) ∪ 𝑦) = (𝑧𝑦)
6260, 61sseqtr4i 3864 . . . . . . . . . . . . . . . 16 𝑧 ⊆ ((𝑧𝑦) ∪ 𝑦)
63 ssdif 3973 . . . . . . . . . . . . . . . 16 (𝑧 ⊆ ((𝑧𝑦) ∪ 𝑦) → (𝑧𝐴) ⊆ (((𝑧𝑦) ∪ 𝑦) ∖ 𝐴))
6462, 63ax-mp 5 . . . . . . . . . . . . . . 15 (𝑧𝐴) ⊆ (((𝑧𝑦) ∪ 𝑦) ∖ 𝐴)
65 difundir 4111 . . . . . . . . . . . . . . 15 (((𝑧𝑦) ∪ 𝑦) ∖ 𝐴) = (((𝑧𝑦) ∖ 𝐴) ∪ (𝑦𝐴))
6664, 65sseqtri 3863 . . . . . . . . . . . . . 14 (𝑧𝐴) ⊆ (((𝑧𝑦) ∖ 𝐴) ∪ (𝑦𝐴))
67 difun1 4118 . . . . . . . . . . . . . . . 16 (𝑧 ∖ (𝑦𝐴)) = ((𝑧𝑦) ∖ 𝐴)
68 ssequn2 4014 . . . . . . . . . . . . . . . . . 18 (𝐴𝑦 ↔ (𝑦𝐴) = 𝑦)
6947, 68sylib 210 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦𝐴) = 𝑦)
7069difeq2d 3956 . . . . . . . . . . . . . . . 16 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ (𝑦𝐴)) = (𝑧𝑦))
7167, 70syl5eqr 2876 . . . . . . . . . . . . . . 15 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((𝑧𝑦) ∖ 𝐴) = (𝑧𝑦))
7271uneq1d 3994 . . . . . . . . . . . . . 14 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (((𝑧𝑦) ∖ 𝐴) ∪ (𝑦𝐴)) = ((𝑧𝑦) ∪ (𝑦𝐴)))
7366, 72syl5sseq 3879 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝐴) ⊆ ((𝑧𝑦) ∪ (𝑦𝐴)))
74 ovolss 23652 . . . . . . . . . . . . 13 (((𝑧𝐴) ⊆ ((𝑧𝑦) ∪ (𝑦𝐴)) ∧ ((𝑧𝑦) ∪ (𝑦𝐴)) ⊆ ℝ) → (vol*‘(𝑧𝐴)) ≤ (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))))
7573, 55, 74syl2anc 581 . . . . . . . . . . . 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 10514 . . . . . . . . . . 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 10972 . . . . . . . . . 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 23699 . . . . . . . . . . . . 13 ((𝑦 ∈ dom vol ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) = ((vol*‘(𝑧𝑦)) + (vol*‘(𝑧𝑦))))
8078, 40, 24, 79syl3anc 1496 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑧) = ((vol*‘(𝑧𝑦)) + (vol*‘(𝑧𝑦))))
8180oveq1d 6921 . . . . . . . . . . 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 10386 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℂ)
8345recnd 10386 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℂ)
8433recnd 10386 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦𝐴)) ∈ ℂ)
8582, 83, 84addassd 10380 . . . . . . . . . . 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 2862 . . . . . . . . . 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 4902 . . . . . . . . 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 3965 . . . . . . . . . . . 12 (𝑦𝐴) ⊆ 𝑦
89 ovolss 23652 . . . . . . . . . . . 12 (((𝑦𝐴) ⊆ 𝑦𝑦 ⊆ ℝ) → (vol*‘(𝑦𝐴)) ≤ (vol*‘𝑦))
9088, 26, 89sylancr 583 . . . . . . . . . . 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 10514 . . . . . . . . . 10 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦𝐴)) ≤ 𝑥)
9233, 28, 24, 91leadd2dd 10968 . . . . . . . . 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 10514 . . . . . . . 8 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘𝑧) + 𝑥))
9493rexlimdvaa 3242 . . . . . . 7 (((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (∃𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘𝑧) + 𝑥)))
9594ralimdva 3172 . . . . . 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 398 . . . . 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 475 . . . . . . 7 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ)
9897rexrd 10407 . . . . . 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 12325 . . . . . 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 581 . . . . 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 249 . . . 4 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧))
103102expr 450 . . 3 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ 𝑧 ∈ 𝒫 ℝ) → ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧)))
104103ralrimiva 3176 . 2 (∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ∀𝑧 ∈ 𝒫 ℝ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧)))
105 ismbl2 23694 . 2 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧))))
10611, 104, 105sylanbrc 580 1 (∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ∈ dom vol)
Colors of variables: wff setvar class
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  cr 10252  1c1 10254   + caddc 10256  *cxr 10391  cle 10393  +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)
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