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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 1061 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ*) | |
2 | xrleid 11983 | . . 3 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴) | |
3 | 2 | 3ad2ant1 1082 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐴) |
4 | simp3 1063 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) | |
5 | elicc1 12219 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ∈ (𝐴[,]𝐵) ↔ (𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))) | |
6 | 5 | 3adant3 1081 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐴 ∈ (𝐴[,]𝐵) ↔ (𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))) |
7 | 1, 3, 4, 6 | mpbir3and 1245 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1037 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 ℝ*cxr 10073 ≤ cle 10075 [,]cicc 12178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-icc 12182 |
This theorem is referenced by: icccmplem1 22625 reconnlem2 22630 oprpiece1res1 22750 pcoass 22824 ivthlem1 23220 ivth2 23224 ivthle 23225 ivthle2 23226 evthicc 23228 ovolicc2lem5 23289 dyadmaxlem 23365 rolle 23753 cmvth 23754 mvth 23755 dvlip 23756 c1liplem1 23759 dveq0 23763 dvgt0lem1 23765 lhop1lem 23776 dvcnvrelem1 23780 dvcvx 23783 dvfsumle 23784 dvfsumge 23785 dvfsumabs 23786 dvfsumlem2 23790 ftc2 23807 ftc2ditglem 23808 itgparts 23810 itgsubstlem 23811 taylfval 24113 tayl0 24116 efcvx 24203 pige3 24269 logccv 24409 loglesqrt 24499 eliccioo 29639 ftc2re 30676 cvmliftlem6 31272 cvmliftlem8 31274 cvmliftlem9 31275 cvmliftlem10 31276 cvmliftlem13 31278 ivthALT 32330 ftc2nc 33494 areacirc 33505 itgpowd 37800 iccintsng 39749 icccncfext 40100 cncfiooicclem1 40106 dvbdfbdioolem1 40143 itgsin0pilem1 40165 itgcoscmulx 40185 itgsincmulx 40190 fourierdlem20 40344 fourierdlem51 40374 fourierdlem54 40377 fourierdlem64 40387 fourierdlem73 40396 fourierdlem81 40404 fourierdlem102 40425 fourierdlem103 40426 fourierdlem104 40427 fourierdlem114 40437 etransclem46 40497 hoidmv1lelem1 40805 |
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