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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 9964 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 9964 | . 2 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | xrmin2 11883 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵) | |
4 | 1, 2, 3 | syl2an 493 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ifcif 4036 class class class wbr 4583 ℝcr 9814 ℝ*cxr 9952 ≤ cle 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 |
This theorem is referenced by: reccn2 14175 ssblex 22043 nlmvscnlem1 22300 nrginvrcnlem 22305 icccmplem2 22434 xlebnum 22572 ipcnlem1 22852 ivthlem2 23028 ovolicc2lem5 23096 ioombl1lem1 23133 mbfi1fseqlem4 23291 mbfi1fseqlem5 23292 aalioulem5 23895 aalioulem6 23896 cxpcn3lem 24288 ftalem5 24603 chtdif 24684 ppidif 24689 chebbnd1lem1 24958 itg2addnc 32634 mullimc 38683 mullimcf 38690 limcleqr 38711 addlimc 38715 0ellimcdiv 38716 limclner 38718 stoweidlem5 38898 fourierdlem104 39103 ioorrnopnlem 39200 hoidmv1lelem2 39482 smfmullem1 39676 |
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