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Mirrors > Home > MPE Home > Th. List > max2 | Structured version Visualization version GIF version |
Description: A number is less than or equal to the maximum of it and another. (Contributed by NM, 3-Apr-2005.) |
Ref | Expression |
---|---|
max2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 11159 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 11159 | . 2 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | xrmax2 13049 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | |
4 | 1, 2, 3 | syl2an 596 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ifcif 4484 class class class wbr 5103 ℝcr 11008 ℝ*cxr 11146 ≤ cle 11148 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-pre-lttri 11083 ax-pre-lttrn 11084 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 |
This theorem is referenced by: lemaxle 13068 z2ge 13071 ssfzunsnext 13440 uzsup 13722 expmulnbnd 14092 discr1 14096 rexuzre 15197 caubnd 15203 limsupgre 15323 limsupbnd2 15325 rlim3 15340 lo1bdd2 15366 o1lo1 15379 rlimclim1 15387 lo1mul 15470 rlimno1 15498 cvgrat 15728 ruclem10 16081 bitsfzo 16275 1arith 16759 evth 24274 ioombl1lem4 24877 itg2monolem3 25069 itgle 25126 ibladdlem 25136 plyaddlem1 25526 coeaddlem 25562 o1cxp 26276 cxp2lim 26278 cxploglim2 26280 ftalem1 26374 ftalem2 26375 chtppilim 26775 dchrisumlem3 26791 ostth2lem2 26934 ostth2lem3 26935 ostth2lem4 26936 ostth3 26938 knoppndvlem18 34930 ibladdnclem 36072 ftc1anclem5 36093 irrapxlem4 41057 irrapxlem5 41058 rexabslelem 43558 uzublem 43570 max2d 43598 climsuse 43750 limsupubuzlem 43854 limsupmnfuzlem 43868 limsupequzmptlem 43870 limsupre3uzlem 43877 liminflelimsuplem 43917 ioodvbdlimc1lem2 44074 ioodvbdlimc2lem 44076 hoidifhspdmvle 44762 |
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