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Mirrors > Home > MPE Home > Th. List > min2 | Structured version Visualization version GIF version |
Description: The minimum of two numbers is less than or equal to the second. (Contributed by NM, 3-Aug-2007.) |
Ref | Expression |
---|---|
min2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 10676 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 10676 | . 2 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | xrmin2 12559 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵) | |
4 | 1, 2, 3 | syl2an 598 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ifcif 4425 class class class wbr 5030 ℝcr 10525 ℝ*cxr 10663 ≤ cle 10665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 |
This theorem is referenced by: ssfzunsnext 12947 reccn2 14945 ssblex 23035 nlmvscnlem1 23292 nrginvrcnlem 23297 icccmplem2 23428 xlebnum 23570 ipcnlem1 23849 ivthlem2 24056 ovolicc2lem5 24125 ioombl1lem1 24162 mbfi1fseqlem4 24322 mbfi1fseqlem5 24323 aalioulem5 24932 aalioulem6 24933 cxpcn3lem 25336 ftalem5 25662 chtdif 25743 ppidif 25748 chebbnd1lem1 26053 itg2addnc 35111 min2d 42112 mullimc 42258 mullimcf 42265 limcleqr 42286 addlimc 42290 0ellimcdiv 42291 limclner 42293 stoweidlem5 42647 fourierdlem104 42852 ioorrnopnlem 42946 hoidmv1lelem2 43231 smfmullem1 43423 |
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