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Theorem nulmbl2 23523
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 12038 . . . . 5 1 ∈ ℝ+
21ne0ii 4069 . . . 4 + ≠ ∅
3 r19.2z 4199 . . . 4 ((ℝ+ ≠ ∅ ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → ∃𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))
42, 3mpan 662 . . 3 (∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ∃𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))
5 simprl 746 . . . . . 6 ((𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴𝑦)
6 mblss 23518 . . . . . . 7 (𝑦 ∈ dom vol → 𝑦 ⊆ ℝ)
76adantr 466 . . . . . 6 ((𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝑦 ⊆ ℝ)
85, 7sstrd 3760 . . . . 5 ((𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥)) → 𝐴 ⊆ ℝ)
98rexlimiva 3175 . . . 4 (∃𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ⊆ ℝ)
109rexlimivw 3176 . . 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 3979 . . . . . . . . . . . . 13 (𝑧𝐴) ⊆ 𝑧
1312a1i 11 . . . . . . . . . . . 12 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (𝑧𝐴) ⊆ 𝑧)
14 elpwi 4305 . . . . . . . . . . . . 13 (𝑧 ∈ 𝒫 ℝ → 𝑧 ⊆ ℝ)
1514adantr 466 . . . . . . . . . . . 12 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → 𝑧 ⊆ ℝ)
16 simpr 471 . . . . . . . . . . . 12 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) ∈ ℝ)
17 ovolsscl 23473 . . . . . . . . . . . 12 (((𝑧𝐴) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
1813, 15, 16, 17syl3anc 1475 . . . . . . . . . . 11 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
19 difssd 3887 . . . . . . . . . . . 12 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (𝑧𝐴) ⊆ 𝑧)
20 ovolsscl 23473 . . . . . . . . . . . 12 (((𝑧𝐴) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
2119, 15, 16, 20syl3anc 1475 . . . . . . . . . . 11 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝐴)) ∈ ℝ)
2218, 21readdcld 10270 . . . . . . . . . 10 ((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ)
2322ad2antrr 697 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ)
2416ad2antrr 697 . . . . . . . . . 10 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑧) ∈ ℝ)
25 difssd 3887 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦𝐴) ⊆ 𝑦)
267adantl 467 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑦 ⊆ ℝ)
27 rpre 12041 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
2827ad2antlr 698 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑥 ∈ ℝ)
29 simprrr 759 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ≤ 𝑥)
30 ovollecl 23470 . . . . . . . . . . . 12 ((𝑦 ⊆ ℝ ∧ 𝑥 ∈ ℝ ∧ (vol*‘𝑦) ≤ 𝑥) → (vol*‘𝑦) ∈ ℝ)
3126, 28, 29, 30syl3anc 1475 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑦) ∈ ℝ)
32 ovolsscl 23473 . . . . . . . . . . 11 (((𝑦𝐴) ⊆ 𝑦𝑦 ⊆ ℝ ∧ (vol*‘𝑦) ∈ ℝ) → (vol*‘(𝑦𝐴)) ∈ ℝ)
3325, 26, 31, 32syl3anc 1475 . . . . . . . . . 10 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦𝐴)) ∈ ℝ)
3424, 33readdcld 10270 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + (vol*‘(𝑦𝐴))) ∈ ℝ)
3524, 28readdcld 10270 . . . . . . . . 9 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘𝑧) + 𝑥) ∈ ℝ)
3618ad2antrr 697 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ∈ ℝ)
3721ad2antrr 697 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ∈ ℝ)
38 inss1 3979 . . . . . . . . . . . . 13 (𝑧𝑦) ⊆ 𝑧
3938a1i 11 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ 𝑧)
4015ad2antrr 697 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑧 ⊆ ℝ)
41 ovolsscl 23473 . . . . . . . . . . . 12 (((𝑧𝑦) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝑦)) ∈ ℝ)
4239, 40, 24, 41syl3anc 1475 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℝ)
43 difssd 3887 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ 𝑧)
44 ovolsscl 23473 . . . . . . . . . . . . 13 (((𝑧𝑦) ⊆ 𝑧𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘(𝑧𝑦)) ∈ ℝ)
4543, 40, 24, 44syl3anc 1475 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℝ)
4645, 33readdcld 10270 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))) ∈ ℝ)
47 simprrl 758 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝐴𝑦)
48 sslin 3985 . . . . . . . . . . . . 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 3761 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ ℝ)
51 ovolss 23472 . . . . . . . . . . . 12 (((𝑧𝐴) ⊆ (𝑧𝑦) ∧ (𝑧𝑦) ⊆ ℝ) → (vol*‘(𝑧𝐴)) ≤ (vol*‘(𝑧𝑦)))
5249, 50, 51syl2anc 565 . . . . . . . . . . 11 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝐴)) ≤ (vol*‘(𝑧𝑦)))
5340ssdifssd 3897 . . . . . . . . . . . . . 14 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝑦) ⊆ ℝ)
5426ssdifssd 3897 . . . . . . . . . . . . . 14 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦𝐴) ⊆ ℝ)
5553, 54unssd 3938 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((𝑧𝑦) ∪ (𝑦𝐴)) ⊆ ℝ)
56 ovolun 23486 . . . . . . . . . . . . . 14 ((((𝑧𝑦) ⊆ ℝ ∧ (vol*‘(𝑧𝑦)) ∈ ℝ) ∧ ((𝑦𝐴) ⊆ ℝ ∧ (vol*‘(𝑦𝐴)) ∈ ℝ)) → (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ≤ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))))
5753, 45, 54, 33, 56syl22anc 1476 . . . . . . . . . . . . 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 23470 . . . . . . . . . . . . 13 ((((𝑧𝑦) ∪ (𝑦𝐴)) ⊆ ℝ ∧ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴))) ∈ ℝ ∧ (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ≤ ((vol*‘(𝑧𝑦)) + (vol*‘(𝑦𝐴)))) → (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ∈ ℝ)
5955, 46, 57, 58syl3anc 1475 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))) ∈ ℝ)
60 ssun1 3925 . . . . . . . . . . . . . . . . 17 𝑧 ⊆ (𝑧𝑦)
61 undif1 4183 . . . . . . . . . . . . . . . . 17 ((𝑧𝑦) ∪ 𝑦) = (𝑧𝑦)
6260, 61sseqtr4i 3785 . . . . . . . . . . . . . . . 16 𝑧 ⊆ ((𝑧𝑦) ∪ 𝑦)
63 ssdif 3894 . . . . . . . . . . . . . . . 16 (𝑧 ⊆ ((𝑧𝑦) ∪ 𝑦) → (𝑧𝐴) ⊆ (((𝑧𝑦) ∪ 𝑦) ∖ 𝐴))
6462, 63ax-mp 5 . . . . . . . . . . . . . . 15 (𝑧𝐴) ⊆ (((𝑧𝑦) ∪ 𝑦) ∖ 𝐴)
65 difundir 4027 . . . . . . . . . . . . . . 15 (((𝑧𝑦) ∪ 𝑦) ∖ 𝐴) = (((𝑧𝑦) ∖ 𝐴) ∪ (𝑦𝐴))
6664, 65sseqtri 3784 . . . . . . . . . . . . . 14 (𝑧𝐴) ⊆ (((𝑧𝑦) ∖ 𝐴) ∪ (𝑦𝐴))
67 difun1 4034 . . . . . . . . . . . . . . . 16 (𝑧 ∖ (𝑦𝐴)) = ((𝑧𝑦) ∖ 𝐴)
68 ssequn2 3935 . . . . . . . . . . . . . . . . . 18 (𝐴𝑦 ↔ (𝑦𝐴) = 𝑦)
6947, 68sylib 208 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑦𝐴) = 𝑦)
7069difeq2d 3877 . . . . . . . . . . . . . . . 16 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧 ∖ (𝑦𝐴)) = (𝑧𝑦))
7167, 70syl5eqr 2818 . . . . . . . . . . . . . . 15 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((𝑧𝑦) ∖ 𝐴) = (𝑧𝑦))
7271uneq1d 3915 . . . . . . . . . . . . . 14 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (((𝑧𝑦) ∖ 𝐴) ∪ (𝑦𝐴)) = ((𝑧𝑦) ∪ (𝑦𝐴)))
7366, 72syl5sseq 3800 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (𝑧𝐴) ⊆ ((𝑧𝑦) ∪ (𝑦𝐴)))
74 ovolss 23472 . . . . . . . . . . . . 13 (((𝑧𝐴) ⊆ ((𝑧𝑦) ∪ (𝑦𝐴)) ∧ ((𝑧𝑦) ∪ (𝑦𝐴)) ⊆ ℝ) → (vol*‘(𝑧𝐴)) ≤ (vol*‘((𝑧𝑦) ∪ (𝑦𝐴))))
7573, 55, 74syl2anc 565 . . . . . . . . . . . 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 10395 . . . . . . . . . . 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 10847 . . . . . . . . . 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 746 . . . . . . . . . . . . 13 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → 𝑦 ∈ dom vol)
79 mblsplit 23519 . . . . . . . . . . . . 13 ((𝑦 ∈ dom vol ∧ 𝑧 ⊆ ℝ ∧ (vol*‘𝑧) ∈ ℝ) → (vol*‘𝑧) = ((vol*‘(𝑧𝑦)) + (vol*‘(𝑧𝑦))))
8078, 40, 24, 79syl3anc 1475 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘𝑧) = ((vol*‘(𝑧𝑦)) + (vol*‘(𝑧𝑦))))
8180oveq1d 6807 . . . . . . . . . . 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 10269 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℂ)
8345recnd 10269 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑧𝑦)) ∈ ℂ)
8433recnd 10269 . . . . . . . . . . . 12 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦𝐴)) ∈ ℂ)
8582, 83, 84addassd 10263 . . . . . . . . . . 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 2804 . . . . . . . . . 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 4812 . . . . . . . . 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 3886 . . . . . . . . . . . 12 (𝑦𝐴) ⊆ 𝑦
89 ovolss 23472 . . . . . . . . . . . 12 (((𝑦𝐴) ⊆ 𝑦𝑦 ⊆ ℝ) → (vol*‘(𝑦𝐴)) ≤ (vol*‘𝑦))
9088, 26, 89sylancr 567 . . . . . . . . . . 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 10395 . . . . . . . . . 10 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → (vol*‘(𝑦𝐴)) ≤ 𝑥)
9233, 28, 24, 91leadd2dd 10843 . . . . . . . . 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 10395 . . . . . . . 8 ((((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ (𝑦 ∈ dom vol ∧ (𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥))) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘𝑧) + 𝑥))
9493rexlimdvaa 3179 . . . . . . 7 (((𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (∃𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ ((vol*‘𝑧) + 𝑥)))
9594ralimdva 3110 . . . . . 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 394 . . . . 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 467 . . . . . . 7 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ)
9897rexrd 10290 . . . . . 6 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ∈ ℝ*)
99 simprr 748 . . . . . 6 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → (vol*‘𝑧) ∈ ℝ)
100 xralrple 12240 . . . . . 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 565 . . . . 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 247 . . . 4 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ (𝑧 ∈ 𝒫 ℝ ∧ (vol*‘𝑧) ∈ ℝ)) → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧))
103102expr 444 . . 3 ((∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) ∧ 𝑧 ∈ 𝒫 ℝ) → ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧)))
104103ralrimiva 3114 . 2 (∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → ∀𝑧 ∈ 𝒫 ℝ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧)))
105 ismbl2 23514 . 2 (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ((vol*‘𝑧) ∈ ℝ → ((vol*‘(𝑧𝐴)) + (vol*‘(𝑧𝐴))) ≤ (vol*‘𝑧))))
10611, 104, 105sylanbrc 564 1 (∀𝑥 ∈ ℝ+𝑦 ∈ dom vol(𝐴𝑦 ∧ (vol*‘𝑦) ≤ 𝑥) → 𝐴 ∈ dom vol)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wne 2942  wral 3060  wrex 3061  cdif 3718  cun 3719  cin 3720  wss 3721  c0 4061  𝒫 cpw 4295   class class class wbr 4784  dom cdm 5249  cfv 6031  (class class class)co 6792  cr 10136  1c1 10138   + caddc 10140  *cxr 10274  cle 10276  +crp 12034  vol*covol 23449  volcvol 23450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-pre-sup 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-sup 8503  df-inf 8504  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-div 10886  df-nn 11222  df-2 11280  df-3 11281  df-n0 11494  df-z 11579  df-uz 11888  df-q 11991  df-rp 12035  df-ioo 12383  df-ico 12385  df-icc 12386  df-fz 12533  df-fl 12800  df-seq 13008  df-exp 13067  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-ovol 23451  df-vol 23452
This theorem is referenced by: (None)
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