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Mirrors > Home > MPE Home > Th. List > ovolre | Structured version Visualization version GIF version |
Description: The measure of the real numbers. (Contributed by Mario Carneiro, 14-Jun-2014.) |
Ref | Expression |
---|---|
ovolre | ⊢ (vol*‘ℝ) = +∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 4004 | . . . 4 ⊢ ℝ ⊆ ℝ | |
2 | ovolcl 24987 | . . . 4 ⊢ (ℝ ⊆ ℝ → (vol*‘ℝ) ∈ ℝ*) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (vol*‘ℝ) ∈ ℝ* |
4 | pnfge 13107 | . . 3 ⊢ ((vol*‘ℝ) ∈ ℝ* → (vol*‘ℝ) ≤ +∞) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ (vol*‘ℝ) ≤ +∞ |
6 | 0re 11213 | . . . 4 ⊢ 0 ∈ ℝ | |
7 | ovolicopnf 25033 | . . . 4 ⊢ (0 ∈ ℝ → (vol*‘(0[,)+∞)) = +∞) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ (vol*‘(0[,)+∞)) = +∞ |
9 | rge0ssre 13430 | . . . 4 ⊢ (0[,)+∞) ⊆ ℝ | |
10 | ovolss 24994 | . . . 4 ⊢ (((0[,)+∞) ⊆ ℝ ∧ ℝ ⊆ ℝ) → (vol*‘(0[,)+∞)) ≤ (vol*‘ℝ)) | |
11 | 9, 1, 10 | mp2an 691 | . . 3 ⊢ (vol*‘(0[,)+∞)) ≤ (vol*‘ℝ) |
12 | 8, 11 | eqbrtrri 5171 | . 2 ⊢ +∞ ≤ (vol*‘ℝ) |
13 | pnfxr 11265 | . . 3 ⊢ +∞ ∈ ℝ* | |
14 | xrletri3 13130 | . . 3 ⊢ (((vol*‘ℝ) ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((vol*‘ℝ) = +∞ ↔ ((vol*‘ℝ) ≤ +∞ ∧ +∞ ≤ (vol*‘ℝ)))) | |
15 | 3, 13, 14 | mp2an 691 | . 2 ⊢ ((vol*‘ℝ) = +∞ ↔ ((vol*‘ℝ) ≤ +∞ ∧ +∞ ≤ (vol*‘ℝ))) |
16 | 5, 12, 15 | mpbir2an 710 | 1 ⊢ (vol*‘ℝ) = +∞ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3948 class class class wbr 5148 ‘cfv 6541 (class class class)co 7406 ℝcr 11106 0cc0 11107 +∞cpnf 11242 ℝ*cxr 11244 ≤ cle 11246 [,)cico 13323 vol*covol 24971 |
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 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 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-int 4951 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-se 5632 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 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-sum 15630 df-rest 17365 df-topgen 17386 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-top 22388 df-topon 22405 df-bases 22441 df-cmp 22883 df-ovol 24973 |
This theorem is referenced by: i1f0rn 25191 ovoliunnfl 36519 voliunnfl 36521 volsupnfl 36522 |
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