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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 13163. (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 11258 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 11258 | . 2 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | xrmax1 13152 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | |
4 | 1, 2, 3 | syl2an 595 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 ifcif 4521 class class class wbr 5139 ℝcr 11106 ℝ*cxr 11245 ≤ cle 11247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-pre-lttri 11181 ax-pre-lttrn 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-po 5579 df-so 5580 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 |
This theorem is referenced by: z2ge 13175 ssfzunsnext 13544 uzsup 13826 expmulnbnd 14196 discr1 14200 rexuzre 15297 rexico 15298 caubnd 15303 limsupgre 15423 limsupbnd2 15425 rlim3 15440 lo1bdd2 15466 o1lo1 15479 rlimclim1 15487 lo1mul 15570 rlimno1 15598 cvgrat 15827 ruclem10 16181 bitsfzo 16375 1arith 16861 setsstruct2 17108 evth 24809 ioombl1lem1 25411 mbfi1flimlem 25576 itg2monolem3 25606 iblre 25647 itgreval 25650 iblss 25658 i1fibl 25661 itgitg1 25662 itgle 25663 itgeqa 25667 iblconst 25671 itgconst 25672 ibladdlem 25673 itgaddlem2 25677 iblabslem 25681 iblabsr 25683 iblmulc2 25684 itgmulc2lem2 25686 itgsplit 25689 plyaddlem1 26069 coeaddlem 26105 o1cxp 26826 cxp2lim 26828 cxploglim2 26830 ftalem1 26924 ftalem2 26925 chtppilim 27327 dchrisumlem3 27343 ostth2lem2 27486 ostth3 27490 knoppndvlem18 35896 ibladdnclem 37038 itgaddnclem2 37041 iblabsnclem 37045 iblmulc2nc 37047 itgmulc2nclem2 37049 ftc1anclem5 37059 irrapxlem4 42077 irrapxlem5 42078 rexabslelem 44638 uzublem 44650 max1d 44670 uzubioo 44790 climsuse 44834 limsupubuzlem 44938 limsupmnfuzlem 44952 limsupequzmptlem 44954 limsupre3uzlem 44961 liminflelimsuplem 45001 ioodvbdlimc1lem2 45158 ioodvbdlimc2lem 45160 |
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