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Mirrors > Home > MPE Home > Th. List > icombl1 | Structured version Visualization version GIF version |
Description: A closed unbounded-above interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.) |
Ref | Expression |
---|---|
icombl1 | ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ dom vol) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 11285 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | pnfxr 11293 | . . . 4 ⊢ +∞ ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℝ → +∞ ∈ ℝ*) |
4 | ltpnf 13127 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
5 | snunioo 13482 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 < +∞) → ({𝐴} ∪ (𝐴(,)+∞)) = (𝐴[,)+∞)) | |
6 | 1, 3, 4, 5 | syl3anc 1369 | . 2 ⊢ (𝐴 ∈ ℝ → ({𝐴} ∪ (𝐴(,)+∞)) = (𝐴[,)+∞)) |
7 | snssi 4808 | . . . 4 ⊢ (𝐴 ∈ ℝ → {𝐴} ⊆ ℝ) | |
8 | ovolsn 25418 | . . . 4 ⊢ (𝐴 ∈ ℝ → (vol*‘{𝐴}) = 0) | |
9 | nulmbl 25458 | . . . 4 ⊢ (({𝐴} ⊆ ℝ ∧ (vol*‘{𝐴}) = 0) → {𝐴} ∈ dom vol) | |
10 | 7, 8, 9 | syl2anc 583 | . . 3 ⊢ (𝐴 ∈ ℝ → {𝐴} ∈ dom vol) |
11 | ioombl1 25485 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐴(,)+∞) ∈ dom vol) | |
12 | 1, 11 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴(,)+∞) ∈ dom vol) |
13 | unmbl 25460 | . . 3 ⊢ (({𝐴} ∈ dom vol ∧ (𝐴(,)+∞) ∈ dom vol) → ({𝐴} ∪ (𝐴(,)+∞)) ∈ dom vol) | |
14 | 10, 12, 13 | syl2anc 583 | . 2 ⊢ (𝐴 ∈ ℝ → ({𝐴} ∪ (𝐴(,)+∞)) ∈ dom vol) |
15 | 6, 14 | eqeltrrd 2830 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∪ cun 3943 ⊆ wss 3945 {csn 4625 class class class wbr 5143 dom cdm 5673 ‘cfv 6543 (class class class)co 7415 ℝcr 11132 0cc0 11133 +∞cpnf 11270 ℝ*cxr 11272 < clt 11273 (,)cioo 13351 [,)cico 13353 vol*covol 25385 volcvol 25386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 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-isom 6552 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7680 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-2o 8482 df-er 8719 df-map 8841 df-pm 8842 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-sup 9460 df-inf 9461 df-oi 9528 df-dju 9919 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-q 12958 df-rp 13002 df-xadd 13120 df-ioo 13355 df-ico 13357 df-icc 13358 df-fz 13512 df-fzo 13655 df-fl 13784 df-seq 13994 df-exp 14054 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-clim 15459 df-rlim 15460 df-sum 15660 df-xmet 21266 df-met 21267 df-ovol 25387 df-vol 25388 |
This theorem is referenced by: icombl 25487 ioombl 25488 |
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