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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 13127 | . . . 4 ⊢ < Or ℝ* | |
2 | 1 | infex 9494 | . . 3 ⊢ inf({𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))}, ℝ*, < ) ∈ V |
3 | df-ovol 25313 | . . 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 6693 | . 2 ⊢ vol* Fn 𝒫 ℝ |
5 | elpwi 4609 | . . . 4 ⊢ (𝑥 ∈ 𝒫 ℝ → 𝑥 ⊆ ℝ) | |
6 | ovolcl 25327 | . . . . 5 ⊢ (𝑥 ⊆ ℝ → (vol*‘𝑥) ∈ ℝ*) | |
7 | ovolge0 25330 | . . . . 5 ⊢ (𝑥 ⊆ ℝ → 0 ≤ (vol*‘𝑥)) | |
8 | pnfge 13117 | . . . . . 6 ⊢ ((vol*‘𝑥) ∈ ℝ* → (vol*‘𝑥) ≤ +∞) | |
9 | 6, 8 | syl 17 | . . . . 5 ⊢ (𝑥 ⊆ ℝ → (vol*‘𝑥) ≤ +∞) |
10 | 0xr 11268 | . . . . . 6 ⊢ 0 ∈ ℝ* | |
11 | pnfxr 11275 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
12 | elicc1 13375 | . . . . . 6 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((vol*‘𝑥) ∈ (0[,]+∞) ↔ ((vol*‘𝑥) ∈ ℝ* ∧ 0 ≤ (vol*‘𝑥) ∧ (vol*‘𝑥) ≤ +∞))) | |
13 | 10, 11, 12 | mp2an 689 | . . . . 5 ⊢ ((vol*‘𝑥) ∈ (0[,]+∞) ↔ ((vol*‘𝑥) ∈ ℝ* ∧ 0 ≤ (vol*‘𝑥) ∧ (vol*‘𝑥) ≤ +∞)) |
14 | 6, 7, 9, 13 | syl3anbrc 1342 | . . . 4 ⊢ (𝑥 ⊆ ℝ → (vol*‘𝑥) ∈ (0[,]+∞)) |
15 | 5, 14 | syl 17 | . . 3 ⊢ (𝑥 ∈ 𝒫 ℝ → (vol*‘𝑥) ∈ (0[,]+∞)) |
16 | 15 | rgen 3062 | . 2 ⊢ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) ∈ (0[,]+∞) |
17 | ffnfv 7120 | . 2 ⊢ (vol*:𝒫 ℝ⟶(0[,]+∞) ↔ (vol* Fn 𝒫 ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) ∈ (0[,]+∞))) | |
18 | 4, 16, 17 | mpbir2an 708 | 1 ⊢ vol*:𝒫 ℝ⟶(0[,]+∞) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∃wrex 3069 {crab 3431 ∩ cin 3947 ⊆ wss 3948 𝒫 cpw 4602 ∪ cuni 4908 class class class wbr 5148 × cxp 5674 ran crn 5677 ∘ ccom 5680 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ↑m cmap 8826 supcsup 9441 infcinf 9442 ℝcr 11115 0cc0 11116 1c1 11117 + caddc 11119 +∞cpnf 11252 ℝ*cxr 11254 < clt 11255 ≤ cle 11256 − cmin 11451 ℕcn 12219 (,)cioo 13331 [,]cicc 13334 seqcseq 13973 abscabs 15188 vol*covol 25311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-ico 13337 df-icc 13338 df-fz 13492 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-ovol 25313 |
This theorem is referenced by: ismbl 25375 volf 25378 ovolfs2 25420 ismbl3 45163 ovolsplit 45165 |
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