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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 12770 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
2 | 1 | biimpa 477 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
3 | 2 | simp2d 1135 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶) |
4 | 3 | 3impa 1102 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 ℝ*cxr 10662 ≤ cle 10664 [,]cicc 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-xr 10667 df-icc 12733 |
This theorem is referenced by: xrge0neqmnf 12828 supicc 12874 ttgcontlem1 26598 xrge0infss 30410 xrge0addgt0 30605 xrge0adddir 30606 esumcst 31221 esumpinfval 31231 oms0 31454 probmeasb 31587 broucube 34807 areaquad 39701 lefldiveq 41435 xadd0ge 41464 xrge0nemnfd 41476 eliccelioc 41673 iccintsng 41675 eliccnelico 41681 eliccelicod 41682 ge0xrre 41683 inficc 41686 iccdificc 41691 iccgelbd 41695 cncfiooiccre 42054 iblspltprt 42134 itgioocnicc 42138 itgspltprt 42140 itgiccshift 42141 fourierdlem1 42270 fourierdlem20 42289 fourierdlem24 42293 fourierdlem25 42294 fourierdlem27 42296 fourierdlem43 42312 fourierdlem44 42313 fourierdlem50 42318 fourierdlem51 42319 fourierdlem52 42320 fourierdlem64 42332 fourierdlem73 42341 fourierdlem76 42344 fourierdlem81 42349 fourierdlem92 42360 fourierdlem102 42370 fourierdlem103 42371 fourierdlem104 42372 fourierdlem114 42382 rrxsnicc 42462 salgencntex 42503 fge0iccico 42529 gsumge0cl 42530 sge0sn 42538 sge0tsms 42539 sge0cl 42540 sge0ge0 42543 sge0fsum 42546 sge0pr 42553 sge0prle 42560 sge0p1 42573 sge0rernmpt 42581 meage0 42634 omessre 42669 omeiunltfirp 42678 carageniuncllem2 42681 omege0 42692 ovnlerp 42721 ovn0lem 42724 hoidmvlelem1 42754 hoidmvlelem2 42755 hoidmvlelem3 42756 |
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