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Mirrors > Home > MPE Home > Th. List > ovolf | Structured version Visualization version GIF version |
Description: The domain and codomain of the outer volume function. (Contributed by Mario Carneiro, 16-Mar-2014.) (Proof shortened by AV, 17-Sep-2020.) |
Ref | Expression |
---|---|
ovolf | ⊢ vol*:𝒫 ℝ⟶(0[,]+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrltso 13071 | . . . 4 ⊢ < Or ℝ* | |
2 | 1 | infex 9439 | . . 3 ⊢ inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ) ∈ V |
3 | df-ovol 24866 | . . 3 ⊢ vol* = (𝑥 ∈ 𝒫 ℝ ↦ inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < )) | |
4 | 2, 3 | fnmpti 6650 | . 2 ⊢ vol* Fn 𝒫 ℝ |
5 | elpwi 4573 | . . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ) | |
6 | ovolcl 24880 | . . . . 5 ⊢ (𝑥 ⊆ ℝ → (vol*‘𝑥) ∈ ℝ*) | |
7 | ovolge0 24883 | . . . . 5 ⊢ (𝑥 ⊆ ℝ → 0 ≤ (vol*‘𝑥)) | |
8 | pnfge 13061 | . . . . . 6 ⊢ ((vol*‘𝑥) ∈ ℝ* → (vol*‘𝑥) ≤ +∞) | |
9 | 6, 8 | syl 17 | . . . . 5 ⊢ (𝑥 ⊆ ℝ → (vol*‘𝑥) ≤ +∞) |
10 | 0xr 11212 | . . . . . 6 ⊢ 0 ∈ ℝ* | |
11 | pnfxr 11219 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
12 | elicc1 13319 | . . . . . 6 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((vol*‘𝑥) ∈ (0[,]+∞) ↔ ((vol*‘𝑥) ∈ ℝ* ∧ 0 ≤ (vol*‘𝑥) ∧ (vol*‘𝑥) ≤ +∞))) | |
13 | 10, 11, 12 | mp2an 691 | . . . . 5 ⊢ ((vol*‘𝑥) ∈ (0[,]+∞) ↔ ((vol*‘𝑥) ∈ ℝ* ∧ 0 ≤ (vol*‘𝑥) ∧ (vol*‘𝑥) ≤ +∞)) |
14 | 6, 7, 9, 13 | syl3anbrc 1344 | . . . 4 ⊢ (𝑥 ⊆ ℝ → (vol*‘𝑥) ∈ (0[,]+∞)) |
15 | 5, 14 | syl 17 | . . 3 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ∈ (0[,]+∞)) |
16 | 15 | rgen 3063 | . 2 ⊢ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) ∈ (0[,]+∞) |
17 | ffnfv 7072 | . 2 ⊢ (vol*:𝒫 ℝ⟶(0[,]+∞) ↔ (vol* Fn 𝒫 ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) ∈ (0[,]+∞))) | |
18 | 4, 16, 17 | mpbir2an 710 | 1 ⊢ vol*:𝒫 ℝ⟶(0[,]+∞) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ∃wrex 3070 {crab 3406 ∩ cin 3913 ⊆ wss 3914 𝒫 cpw 4566 ∪ cuni 4871 class class class wbr 5111 × cxp 5637 ran crn 5640 ∘ ccom 5643 Fn wfn 6497 ⟶wf 6498 ‘cfv 6502 (class class class)co 7363 ↑m cmap 8773 supcsup 9386 infcinf 9387 ℝcr 11060 0cc0 11061 1c1 11062 + caddc 11064 +∞cpnf 11196 ℝ*cxr 11198 < clt 11199 ≤ cle 11200 − cmin 11395 ℕcn 12163 (,)cioo 13275 [,]cicc 13278 seqcseq 13917 abscabs 15132 vol*covol 24864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2703 ax-sep 5262 ax-nul 5269 ax-pow 5326 ax-pr 5390 ax-un 7678 ax-cnex 11117 ax-resscn 11118 ax-1cn 11119 ax-icn 11120 ax-addcl 11121 ax-addrcl 11122 ax-mulcl 11123 ax-mulrcl 11124 ax-mulcom 11125 ax-addass 11126 ax-mulass 11127 ax-distr 11128 ax-i2m1 11129 ax-1ne0 11130 ax-1rid 11131 ax-rnegex 11132 ax-rrecex 11133 ax-cnre 11134 ax-pre-lttri 11135 ax-pre-lttrn 11136 ax-pre-ltadd 11137 ax-pre-mulgt0 11138 ax-pre-sup 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4289 df-if 4493 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4872 df-iun 4962 df-br 5112 df-opab 5174 df-mpt 5195 df-tr 5229 df-id 5537 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5594 df-we 5596 df-xp 5645 df-rel 5646 df-cnv 5647 df-co 5648 df-dm 5649 df-rn 5650 df-res 5651 df-ima 5652 df-pred 6259 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7319 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7809 df-1st 7927 df-2nd 7928 df-frecs 8218 df-wrecs 8249 df-recs 8323 df-rdg 8362 df-er 8656 df-map 8775 df-en 8892 df-dom 8893 df-sdom 8894 df-sup 9388 df-inf 9389 df-pnf 11201 df-mnf 11202 df-xr 11203 df-ltxr 11204 df-le 11205 df-sub 11397 df-neg 11398 df-div 11823 df-nn 12164 df-2 12226 df-3 12227 df-n0 12424 df-z 12510 df-uz 12774 df-rp 12926 df-ico 13281 df-icc 13282 df-fz 13436 df-seq 13918 df-exp 13979 df-cj 14997 df-re 14998 df-im 14999 df-sqrt 15133 df-abs 15134 df-ovol 24866 |
This theorem is referenced by: ismbl 24928 volf 24931 ovolfs2 24973 ismbl3 44329 ovolsplit 44331 |
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