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Mirrors > Home > MPE Home > Th. List > ovolun | Structured version Visualization version GIF version |
Description: The Lebesgue outer measure function is finitely sub-additive. (Unlike the stronger ovoliun 25378, this does not require any choice principles.) (Contributed by Mario Carneiro, 12-Jun-2014.) |
Ref | Expression |
---|---|
ovolun | ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 764 | . . . 4 ⊢ ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ 𝑥 ∈ ℝ+) → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)) | |
2 | simplr 766 | . . . 4 ⊢ ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ 𝑥 ∈ ℝ+) → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) | |
3 | simpr 484 | . . . 4 ⊢ ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+) | |
4 | 1, 2, 3 | ovolunlem2 25371 | . . 3 ⊢ ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ 𝑥 ∈ ℝ+) → (vol*‘(𝐴 ∪ 𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝑥)) |
5 | 4 | ralrimiva 3138 | . 2 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → ∀𝑥 ∈ ℝ+ (vol*‘(𝐴 ∪ 𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝑥)) |
6 | unss 4177 | . . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ↔ (𝐴 ∪ 𝐵) ⊆ ℝ) | |
7 | 6 | biimpi 215 | . . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) → (𝐴 ∪ 𝐵) ⊆ ℝ) |
8 | 7 | ad2ant2r 744 | . . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (𝐴 ∪ 𝐵) ⊆ ℝ) |
9 | ovolcl 25351 | . . . 4 ⊢ ((𝐴 ∪ 𝐵) ⊆ ℝ → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ*) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ*) |
11 | readdcl 11190 | . . . 4 ⊢ (((vol*‘𝐴) ∈ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ) | |
12 | 11 | ad2ant2l 743 | . . 3 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ) |
13 | xralrple 13185 | . . 3 ⊢ (((vol*‘(𝐴 ∪ 𝐵)) ∈ ℝ* ∧ ((vol*‘𝐴) + (vol*‘𝐵)) ∈ ℝ) → ((vol*‘(𝐴 ∪ 𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)) ↔ ∀𝑥 ∈ ℝ+ (vol*‘(𝐴 ∪ 𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝑥))) | |
14 | 10, 12, 13 | syl2anc 583 | . 2 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → ((vol*‘(𝐴 ∪ 𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵)) ↔ ∀𝑥 ∈ ℝ+ (vol*‘(𝐴 ∪ 𝐵)) ≤ (((vol*‘𝐴) + (vol*‘𝐵)) + 𝑥))) |
15 | 5, 14 | mpbird 257 | 1 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘(𝐴 ∪ 𝐵)) ≤ ((vol*‘𝐴) + (vol*‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2098 ∀wral 3053 ∪ cun 3939 ⊆ wss 3941 class class class wbr 5139 ‘cfv 6534 (class class class)co 7402 ℝcr 11106 + caddc 11110 ℝ*cxr 11246 ≤ cle 11248 ℝ+crp 12975 vol*covol 25335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-inf 9435 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-q 12932 df-rp 12976 df-ioo 13329 df-ico 13331 df-fz 13486 df-fl 13758 df-seq 13968 df-exp 14029 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-ovol 25337 |
This theorem is referenced by: ovolunnul 25373 ovolfiniun 25374 ismbl2 25400 nulmbl2 25409 unmbl 25410 volun 25418 voliunlem2 25424 uniioombllem3 25458 uniioombllem4 25459 volcn 25479 mblfinlem3 37031 mblfinlem4 37032 ovolsplit 45250 |
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