![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 11301 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 11301 | . 2 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | xrmin1 13204 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴) | |
4 | 1, 2, 3 | syl2an 594 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2099 ifcif 4523 class class class wbr 5145 ℝcr 11148 ℝ*cxr 11288 ≤ cle 11290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-pre-lttri 11223 ax-pre-lttrn 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 |
This theorem is referenced by: ssfzunsnext 13594 reccn2 15594 setsstruct2 17171 ssblex 24422 nlmvscnlem1 24691 nrginvrcnlem 24696 icccmplem2 24827 xlebnum 24979 ipcnlem1 25261 ivthlem2 25469 ioombl1lem4 25578 mbfi1fseqlem5 25737 aalioulem5 26361 aalioulem6 26362 logcnlem3 26668 cxpcn3lem 26772 ftalem5 27102 chtdif 27183 ppidif 27188 chebbnd1lem1 27495 itg2addnc 37388 min1d 45123 mullimc 45273 mullimcf 45280 limcleqr 45301 addlimc 45305 0ellimcdiv 45306 limclner 45308 stoweidlem5 45662 fourierdlem103 45866 fourierdlem104 45867 ioorrnopnlem 45961 hsphoidmvle 46243 hoidmv1lelem1 46248 hoidmv1lelem2 46249 hoidmv1lelem3 46250 smfmullem1 46448 |
Copyright terms: Public domain | W3C validator |