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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 13431 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
| 2 | 1 | biimpa 476 | . . 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 1110 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝ*cxr 11294 ≤ cle 11296 [,]cicc 13390 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-xr 11299 df-icc 13394 |
| This theorem is referenced by: xrge0neqmnf 13492 supicc 13541 ttgcontlem1 28899 xrge0infss 32764 xrge0addgt0 33022 xrge0adddir 33023 esumcst 34064 esumpinfval 34074 oms0 34299 probmeasb 34432 broucube 37661 areaquad 43228 lefldiveq 45304 xadd0ge 45332 xrge0nemnfd 45343 eliccelioc 45534 iccintsng 45536 eliccnelico 45542 eliccelicod 45543 ge0xrre 45544 inficc 45547 iccdificc 45552 iccgelbd 45556 cncfiooiccre 45910 iblspltprt 45988 itgioocnicc 45992 itgspltprt 45994 itgiccshift 45995 fourierdlem1 46123 fourierdlem20 46142 fourierdlem24 46146 fourierdlem25 46147 fourierdlem27 46149 fourierdlem43 46165 fourierdlem44 46166 fourierdlem50 46171 fourierdlem51 46172 fourierdlem52 46173 fourierdlem64 46185 fourierdlem73 46194 fourierdlem76 46197 fourierdlem81 46202 fourierdlem92 46213 fourierdlem102 46223 fourierdlem103 46224 fourierdlem104 46225 fourierdlem114 46235 rrxsnicc 46315 salgencntex 46358 fge0iccico 46385 gsumge0cl 46386 sge0sn 46394 sge0tsms 46395 sge0cl 46396 sge0ge0 46399 sge0fsum 46402 sge0pr 46409 sge0prle 46416 sge0p1 46429 sge0rernmpt 46437 meage0 46490 omessre 46525 omeiunltfirp 46534 carageniuncllem2 46537 omege0 46548 ovnlerp 46577 ovn0lem 46580 hoidmvlelem1 46610 hoidmvlelem2 46611 hoidmvlelem3 46612 |
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