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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 13183 | . . . 4 ⊢ < Or ℝ* | |
| 2 | 1 | infex 9533 | . . 3 ⊢ inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ) ∈ V |
| 3 | df-ovol 25499 | . . 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 6711 | . 2 ⊢ vol* Fn 𝒫 ℝ |
| 5 | elpwi 4607 | . . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ) | |
| 6 | ovolcl 25513 | . . . . 5 ⊢ (𝑥 ⊆ ℝ → (vol*‘𝑥) ∈ ℝ*) | |
| 7 | ovolge0 25516 | . . . . 5 ⊢ (𝑥 ⊆ ℝ → 0 ≤ (vol*‘𝑥)) | |
| 8 | pnfge 13172 | . . . . . 6 ⊢ ((vol*‘𝑥) ∈ ℝ* → (vol*‘𝑥) ≤ +∞) | |
| 9 | 6, 8 | syl 17 | . . . . 5 ⊢ (𝑥 ⊆ ℝ → (vol*‘𝑥) ≤ +∞) |
| 10 | 0xr 11308 | . . . . . 6 ⊢ 0 ∈ ℝ* | |
| 11 | pnfxr 11315 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
| 12 | elicc1 13431 | . . . . . 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 3063 | . 2 ⊢ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) ∈ (0[,]+∞) |
| 17 | ffnfv 7139 | . 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 {crab 3436 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 class class class wbr 5143 × cxp 5683 ran crn 5686 ∘ ccom 5689 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 supcsup 9480 infcinf 9481 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 +∞cpnf 11292 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 − cmin 11492 ℕcn 12266 (,)cioo 13387 [,]cicc 13390 seqcseq 14042 abscabs 15273 vol*covol 25497 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-ico 13393 df-icc 13394 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-ovol 25499 |
| This theorem is referenced by: ismbl 25561 volf 25564 ovolfs2 25606 ismbl3 46001 ovolsplit 46003 |
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