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Mirrors > Home > MPE Home > Th. List > ovolsca | Structured version Visualization version GIF version |
Description: The Lebesgue outer measure function respects scaling of sets by positive reals. (Contributed by Mario Carneiro, 6-Apr-2015.) |
Ref | Expression |
---|---|
ovolsca.1 | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
ovolsca.2 | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
ovolsca.3 | ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴}) |
ovolsca.4 | ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ) |
Ref | Expression |
---|---|
ovolsca | ⊢ (𝜑 → (vol*‘𝐵) = ((vol*‘𝐴) / 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovolsca.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
2 | ovolsca.2 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
3 | ovolsca.3 | . . 3 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴}) | |
4 | ovolsca.4 | . . 3 ⊢ (𝜑 → (vol*‘𝐴) ∈ ℝ) | |
5 | 1, 2, 3, 4 | ovolscalem2 25562 | . 2 ⊢ (𝜑 → (vol*‘𝐵) ≤ ((vol*‘𝐴) / 𝐶)) |
6 | 4 | recnd 11286 | . . . 4 ⊢ (𝜑 → (vol*‘𝐴) ∈ ℂ) |
7 | 2 | rpcnd 13076 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
8 | 2 | rpne0d 13079 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ 0) |
9 | 6, 7, 8 | divrecd 12043 | . . 3 ⊢ (𝜑 → ((vol*‘𝐴) / 𝐶) = ((vol*‘𝐴) · (1 / 𝐶))) |
10 | ssrab2 4089 | . . . . . 6 ⊢ {𝑥 ∈ ℝ ∣ (𝐶 · 𝑥) ∈ 𝐴} ⊆ ℝ | |
11 | 3, 10 | eqsstrdi 4049 | . . . . 5 ⊢ (𝜑 → 𝐵 ⊆ ℝ) |
12 | 2 | rpreccld 13084 | . . . . 5 ⊢ (𝜑 → (1 / 𝐶) ∈ ℝ+) |
13 | 1, 2, 3 | sca2rab 25560 | . . . . 5 ⊢ (𝜑 → 𝐴 = {𝑦 ∈ ℝ ∣ ((1 / 𝐶) · 𝑦) ∈ 𝐵}) |
14 | 4, 2 | rerpdivcld 13105 | . . . . . 6 ⊢ (𝜑 → ((vol*‘𝐴) / 𝐶) ∈ ℝ) |
15 | ovollecl 25531 | . . . . . 6 ⊢ ((𝐵 ⊆ ℝ ∧ ((vol*‘𝐴) / 𝐶) ∈ ℝ ∧ (vol*‘𝐵) ≤ ((vol*‘𝐴) / 𝐶)) → (vol*‘𝐵) ∈ ℝ) | |
16 | 11, 14, 5, 15 | syl3anc 1370 | . . . . 5 ⊢ (𝜑 → (vol*‘𝐵) ∈ ℝ) |
17 | 11, 12, 13, 16 | ovolscalem2 25562 | . . . 4 ⊢ (𝜑 → (vol*‘𝐴) ≤ ((vol*‘𝐵) / (1 / 𝐶))) |
18 | 4, 16, 12 | lemuldivd 13123 | . . . 4 ⊢ (𝜑 → (((vol*‘𝐴) · (1 / 𝐶)) ≤ (vol*‘𝐵) ↔ (vol*‘𝐴) ≤ ((vol*‘𝐵) / (1 / 𝐶)))) |
19 | 17, 18 | mpbird 257 | . . 3 ⊢ (𝜑 → ((vol*‘𝐴) · (1 / 𝐶)) ≤ (vol*‘𝐵)) |
20 | 9, 19 | eqbrtrd 5169 | . 2 ⊢ (𝜑 → ((vol*‘𝐴) / 𝐶) ≤ (vol*‘𝐵)) |
21 | 16, 14 | letri3d 11400 | . 2 ⊢ (𝜑 → ((vol*‘𝐵) = ((vol*‘𝐴) / 𝐶) ↔ ((vol*‘𝐵) ≤ ((vol*‘𝐴) / 𝐶) ∧ ((vol*‘𝐴) / 𝐶) ≤ (vol*‘𝐵)))) |
22 | 5, 20, 21 | mpbir2and 713 | 1 ⊢ (𝜑 → (vol*‘𝐵) = ((vol*‘𝐴) / 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 {crab 3432 ⊆ wss 3962 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 1c1 11153 · cmul 11157 ≤ cle 11293 / cdiv 11917 ℝ+crp 13031 vol*covol 25510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-q 12988 df-rp 13032 df-ioo 13387 df-ico 13389 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719 df-ovol 25512 |
This theorem is referenced by: (None) |
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