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Mirrors > Home > MPE Home > Th. List > max1 | Structured version Visualization version GIF version |
Description: A number is less than or equal to the maximum of it and another. See also max1ALT 12312. (Contributed by NM, 3-Apr-2005.) |
Ref | Expression |
---|---|
max1 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 10409 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 10409 | . 2 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | xrmax1 12301 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | |
4 | 1, 2, 3 | syl2an 589 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2164 ifcif 4308 class class class wbr 4875 ℝcr 10258 ℝ*cxr 10397 ≤ cle 10399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-pre-lttri 10333 ax-pre-lttrn 10334 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 |
This theorem is referenced by: z2ge 12324 ssfzunsnext 12686 uzsup 12964 expmulnbnd 13297 discr1 13301 rexuzre 14476 rexico 14477 caubnd 14482 limsupgre 14596 limsupbnd2 14598 rlim3 14613 lo1bdd2 14639 o1lo1 14652 rlimclim1 14660 lo1mul 14742 rlimno1 14768 cvgrat 14995 ruclem10 15349 bitsfzo 15537 1arith 16009 setsstruct2 16267 evth 23135 ioombl1lem1 23731 mbfi1flimlem 23895 itg2monolem3 23925 iblre 23966 itgreval 23969 iblss 23977 i1fibl 23980 itgitg1 23981 itgle 23982 itgeqa 23986 iblconst 23990 itgconst 23991 ibladdlem 23992 itgaddlem2 23996 iblabslem 24000 iblabsr 24002 iblmulc2 24003 itgmulc2lem2 24005 itgsplit 24008 plyaddlem1 24375 coeaddlem 24411 o1cxp 25121 cxp2lim 25123 cxploglim2 25125 ftalem1 25219 ftalem2 25220 chtppilim 25584 dchrisumlem3 25600 ostth2lem2 25743 ostth3 25747 knoppndvlem18 33047 ibladdnclem 34004 itgaddnclem2 34007 iblabsnclem 34011 iblmulc2nc 34013 itgmulc2nclem2 34015 ftc1anclem5 34027 irrapxlem4 38228 irrapxlem5 38229 rexabslelem 40434 uzublem 40446 max1d 40467 uzubioo 40583 climsuse 40629 limsupubuzlem 40733 limsupmnfuzlem 40747 limsupequzmptlem 40749 limsupre3uzlem 40756 liminflelimsuplem 40796 ioodvbdlimc1lem2 40936 ioodvbdlimc2lem 40938 |
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