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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 10687 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | pnfxr 10695 | . . . 4 ⊢ +∞ ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℝ → +∞ ∈ ℝ*) |
4 | ltpnf 12516 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
5 | snunioo 12865 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 < +∞) → ({𝐴} ∪ (𝐴(,)+∞)) = (𝐴[,)+∞)) | |
6 | 1, 3, 4, 5 | syl3anc 1367 | . 2 ⊢ (𝐴 ∈ ℝ → ({𝐴} ∪ (𝐴(,)+∞)) = (𝐴[,)+∞)) |
7 | snssi 4741 | . . . 4 ⊢ (𝐴 ∈ ℝ → {𝐴} ⊆ ℝ) | |
8 | ovolsn 24096 | . . . 4 ⊢ (𝐴 ∈ ℝ → (vol*‘{𝐴}) = 0) | |
9 | nulmbl 24136 | . . . 4 ⊢ (({𝐴} ⊆ ℝ ∧ (vol*‘{𝐴}) = 0) → {𝐴} ∈ dom vol) | |
10 | 7, 8, 9 | syl2anc 586 | . . 3 ⊢ (𝐴 ∈ ℝ → {𝐴} ∈ dom vol) |
11 | ioombl1 24163 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐴(,)+∞) ∈ dom vol) | |
12 | 1, 11 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴(,)+∞) ∈ dom vol) |
13 | unmbl 24138 | . . 3 ⊢ (({𝐴} ∈ dom vol ∧ (𝐴(,)+∞) ∈ dom vol) → ({𝐴} ∪ (𝐴(,)+∞)) ∈ dom vol) | |
14 | 10, 12, 13 | syl2anc 586 | . 2 ⊢ (𝐴 ∈ ℝ → ({𝐴} ∪ (𝐴(,)+∞)) ∈ dom vol) |
15 | 6, 14 | eqeltrrd 2914 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∪ cun 3934 ⊆ wss 3936 {csn 4567 class class class wbr 5066 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 +∞cpnf 10672 ℝ*cxr 10674 < clt 10675 (,)cioo 12739 [,)cico 12741 vol*covol 24063 volcvol 24064 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-xadd 12509 df-ioo 12743 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-rlim 14846 df-sum 15043 df-xmet 20538 df-met 20539 df-ovol 24065 df-vol 24066 |
This theorem is referenced by: icombl 24165 ioombl 24166 |
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