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Mirrors > Home > MPE Home > Th. List > lbicc2 | Structured version Visualization version GIF version |
Description: The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Sep-2015.) |
Ref | Expression |
---|---|
lbicc2 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ*) | |
2 | xrleid 13127 | . . 3 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴) | |
3 | 2 | 3ad2ant1 1130 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐴) |
4 | simp3 1135 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) | |
5 | elicc1 13365 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ∈ (𝐴[,]𝐵) ↔ (𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))) | |
6 | 5 | 3adant3 1129 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐴 ∈ (𝐴[,]𝐵) ↔ (𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))) |
7 | 1, 3, 4, 6 | mpbir3and 1339 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5138 (class class class)co 7401 ℝ*cxr 11244 ≤ cle 11246 [,]cicc 13324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-pre-lttri 11180 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-icc 13328 |
This theorem is referenced by: icccmplem1 24660 reconnlem2 24665 oprpiece1res1 24798 pcoass 24873 ivthlem1 25302 ivth2 25306 ivthle 25307 ivthle2 25308 evthicc 25310 ovolicc2lem5 25372 dyadmaxlem 25448 rolle 25844 cmvth 25845 cmvthOLD 25846 mvth 25847 dvlip 25848 c1liplem1 25851 dveq0 25855 dvgt0lem1 25857 lhop1lem 25868 dvcnvrelem1 25872 dvcvx 25875 dvfsumle 25876 dvfsumleOLD 25877 dvfsumge 25878 dvfsumabs 25879 dvfsumlem2 25883 dvfsumlem2OLD 25884 ftc2 25901 ftc2ditglem 25902 itgparts 25904 itgsubstlem 25905 itgpowd 25907 taylfval 26212 tayl0 26215 efcvx 26303 pige3ALT 26371 logccv 26513 loglesqrt 26609 eliccioo 32564 ftc2re 34099 cvmliftlem6 34770 cvmliftlem8 34772 cvmliftlem9 34773 cvmliftlem10 34774 cvmliftlem13 34776 ivthALT 35710 ftc2nc 37060 areacirc 37071 iccintsng 44721 icccncfext 45088 cncfiooicclem1 45094 dvbdfbdioolem1 45129 itgsin0pilem1 45151 itgcoscmulx 45170 itgsincmulx 45175 fourierdlem20 45328 fourierdlem51 45358 fourierdlem54 45361 fourierdlem64 45371 fourierdlem73 45380 fourierdlem81 45388 fourierdlem102 45409 fourierdlem103 45410 fourierdlem104 45411 fourierdlem114 45421 etransclem46 45481 hoidmv1lelem1 45792 |
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