Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ovolf | Structured version Visualization version GIF version |
Description: The domain and range 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 12629 | . . . 4 ⊢ < Or ℝ* | |
2 | 1 | infex 9042 | . . 3 ⊢ inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ) ∈ V |
3 | df-ovol 24228 | . . 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 6490 | . 2 ⊢ vol* Fn 𝒫 ℝ |
5 | elpwi 4507 | . . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ) | |
6 | ovolcl 24242 | . . . . 5 ⊢ (𝑥 ⊆ ℝ → (vol*‘𝑥) ∈ ℝ*) | |
7 | ovolge0 24245 | . . . . 5 ⊢ (𝑥 ⊆ ℝ → 0 ≤ (vol*‘𝑥)) | |
8 | pnfge 12620 | . . . . . 6 ⊢ ((vol*‘𝑥) ∈ ℝ* → (vol*‘𝑥) ≤ +∞) | |
9 | 6, 8 | syl 17 | . . . . 5 ⊢ (𝑥 ⊆ ℝ → (vol*‘𝑥) ≤ +∞) |
10 | 0xr 10778 | . . . . . 6 ⊢ 0 ∈ ℝ* | |
11 | pnfxr 10785 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
12 | elicc1 12877 | . . . . . 6 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((vol*‘𝑥) ∈ (0[,]+∞) ↔ ((vol*‘𝑥) ∈ ℝ* ∧ 0 ≤ (vol*‘𝑥) ∧ (vol*‘𝑥) ≤ +∞))) | |
13 | 10, 11, 12 | mp2an 692 | . . . . 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 3064 | . 2 ⊢ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) ∈ (0[,]+∞) |
17 | ffnfv 6904 | . 2 ⊢ (vol*:𝒫 ℝ⟶(0[,]+∞) ↔ (vol* Fn 𝒫 ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) ∈ (0[,]+∞))) | |
18 | 4, 16, 17 | mpbir2an 711 | 1 ⊢ vol*:𝒫 ℝ⟶(0[,]+∞) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3054 ∃wrex 3055 {crab 3058 ∩ cin 3852 ⊆ wss 3853 𝒫 cpw 4498 ∪ cuni 4806 class class class wbr 5040 × cxp 5533 ran crn 5536 ∘ ccom 5539 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7182 ↑m cmap 8449 supcsup 8989 infcinf 8990 ℝcr 10626 0cc0 10627 1c1 10628 + caddc 10630 +∞cpnf 10762 ℝ*cxr 10764 < clt 10765 ≤ cle 10766 − cmin 10960 ℕcn 11728 (,)cioo 12833 [,]cicc 12836 seqcseq 13472 abscabs 14695 vol*covol 24226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-er 8332 df-map 8451 df-en 8568 df-dom 8569 df-sdom 8570 df-sup 8991 df-inf 8992 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-n0 11989 df-z 12075 df-uz 12337 df-rp 12485 df-ico 12839 df-icc 12840 df-fz 12994 df-seq 13473 df-exp 13534 df-cj 14560 df-re 14561 df-im 14562 df-sqrt 14696 df-abs 14697 df-ovol 24228 |
This theorem is referenced by: ismbl 24290 volf 24293 ovolfs2 24335 ismbl3 43109 ovolsplit 43111 |
Copyright terms: Public domain | W3C validator |