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 10684 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 10684 | . 2 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | xrmin1 12568 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴) | |
4 | 1, 2, 3 | syl2an 597 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2113 ifcif 4464 class class class wbr 5063 ℝcr 10533 ℝ*cxr 10671 ≤ cle 10673 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-pre-lttri 10608 ax-pre-lttrn 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4836 df-br 5064 df-opab 5126 df-mpt 5144 df-id 5457 df-po 5471 df-so 5472 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-er 8286 df-en 8507 df-dom 8508 df-sdom 8509 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 |
This theorem is referenced by: ssfzunsnext 12950 reccn2 14949 setsstruct2 16517 ssblex 23034 nlmvscnlem1 23291 nrginvrcnlem 23296 icccmplem2 23427 xlebnum 23565 ipcnlem1 23844 ivthlem2 24049 ioombl1lem4 24158 mbfi1fseqlem5 24316 aalioulem5 24923 aalioulem6 24924 logcnlem3 25225 cxpcn3lem 25326 ftalem5 25652 chtdif 25733 ppidif 25738 chebbnd1lem1 26043 itg2addnc 34984 min1d 41822 mullimc 41971 mullimcf 41978 limcleqr 41999 addlimc 42003 0ellimcdiv 42004 limclner 42006 stoweidlem5 42364 fourierdlem103 42568 fourierdlem104 42569 ioorrnopnlem 42663 hsphoidmvle 42942 hoidmv1lelem1 42947 hoidmv1lelem2 42948 hoidmv1lelem3 42949 smfmullem1 43140 |
Copyright terms: Public domain | W3C validator |