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Mirrors > Home > MPE Home > Th. List > iccgelb | Structured version Visualization version GIF version |
Description: An element of a closed interval is more than or equal to its lower bound. (Contributed by Thierry Arnoux, 23-Dec-2016.) |
Ref | Expression |
---|---|
iccgelb | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elicc1 13314 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
2 | 1 | biimpa 478 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
3 | 2 | simp2d 1144 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶) |
4 | 3 | 3impa 1111 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 class class class wbr 5106 (class class class)co 7358 ℝ*cxr 11193 ≤ cle 11195 [,]cicc 13273 |
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-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-xr 11198 df-icc 13277 |
This theorem is referenced by: xrge0neqmnf 13375 supicc 13424 ttgcontlem1 27875 xrge0infss 31712 xrge0addgt0 31931 xrge0adddir 31932 esumcst 32719 esumpinfval 32729 oms0 32954 probmeasb 33087 broucube 36158 areaquad 41593 lefldiveq 43613 xadd0ge 43641 xrge0nemnfd 43653 eliccelioc 43845 iccintsng 43847 eliccnelico 43853 eliccelicod 43854 ge0xrre 43855 inficc 43858 iccdificc 43863 iccgelbd 43867 cncfiooiccre 44222 iblspltprt 44300 itgioocnicc 44304 itgspltprt 44306 itgiccshift 44307 fourierdlem1 44435 fourierdlem20 44454 fourierdlem24 44458 fourierdlem25 44459 fourierdlem27 44461 fourierdlem43 44477 fourierdlem44 44478 fourierdlem50 44483 fourierdlem51 44484 fourierdlem52 44485 fourierdlem64 44497 fourierdlem73 44506 fourierdlem76 44509 fourierdlem81 44514 fourierdlem92 44525 fourierdlem102 44535 fourierdlem103 44536 fourierdlem104 44537 fourierdlem114 44547 rrxsnicc 44627 salgencntex 44670 fge0iccico 44697 gsumge0cl 44698 sge0sn 44706 sge0tsms 44707 sge0cl 44708 sge0ge0 44711 sge0fsum 44714 sge0pr 44721 sge0prle 44728 sge0p1 44741 sge0rernmpt 44749 meage0 44802 omessre 44837 omeiunltfirp 44846 carageniuncllem2 44849 omege0 44860 ovnlerp 44889 ovn0lem 44892 hoidmvlelem1 44922 hoidmvlelem2 44923 hoidmvlelem3 44924 |
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