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Mirrors > Home > MPE Home > Th. List > min1 | Structured version Visualization version GIF version |
Description: The minimum of two numbers is less than or equal to the first. (Contributed by NM, 3-Aug-2007.) |
Ref | Expression |
---|---|
min1 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 10409 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 10409 | . 2 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | xrmin1 12303 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴) | |
4 | 1, 2, 3 | syl2an 589 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2164 ifcif 4308 class class class wbr 4875 ℝcr 10258 ℝ*cxr 10397 ≤ cle 10399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-pre-lttri 10333 ax-pre-lttrn 10334 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 |
This theorem is referenced by: ssfzunsnext 12686 reccn2 14711 setsstruct2 16267 ssblex 22610 nlmvscnlem1 22867 nrginvrcnlem 22872 icccmplem2 23003 xlebnum 23141 ipcnlem1 23420 ivthlem2 23625 ioombl1lem4 23734 mbfi1fseqlem5 23892 aalioulem5 24497 aalioulem6 24498 logcnlem3 24796 cxpcn3lem 24897 ftalem5 25223 chtdif 25304 ppidif 25309 chebbnd1lem1 25578 itg2addnc 34002 min1d 40490 mullimc 40637 mullimcf 40644 limcleqr 40665 addlimc 40669 0ellimcdiv 40670 limclner 40672 stoweidlem5 41010 fourierdlem103 41214 fourierdlem104 41215 ioorrnopnlem 41309 hsphoidmvle 41588 hoidmv1lelem1 41593 hoidmv1lelem2 41594 hoidmv1lelem3 41595 smfmullem1 41786 |
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