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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 12567. (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 10676 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 10676 | . 2 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | xrmax1 12556 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | |
4 | 1, 2, 3 | syl2an 598 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ifcif 4425 class class class wbr 5030 ℝcr 10525 ℝ*cxr 10663 ≤ cle 10665 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 |
This theorem is referenced by: z2ge 12579 ssfzunsnext 12947 uzsup 13226 expmulnbnd 13592 discr1 13596 rexuzre 14704 rexico 14705 caubnd 14710 limsupgre 14830 limsupbnd2 14832 rlim3 14847 lo1bdd2 14873 o1lo1 14886 rlimclim1 14894 lo1mul 14976 rlimno1 15002 cvgrat 15231 ruclem10 15584 bitsfzo 15774 1arith 16253 setsstruct2 16513 evth 23564 ioombl1lem1 24162 mbfi1flimlem 24326 itg2monolem3 24356 iblre 24397 itgreval 24400 iblss 24408 i1fibl 24411 itgitg1 24412 itgle 24413 itgeqa 24417 iblconst 24421 itgconst 24422 ibladdlem 24423 itgaddlem2 24427 iblabslem 24431 iblabsr 24433 iblmulc2 24434 itgmulc2lem2 24436 itgsplit 24439 plyaddlem1 24810 coeaddlem 24846 o1cxp 25560 cxp2lim 25562 cxploglim2 25564 ftalem1 25658 ftalem2 25659 chtppilim 26059 dchrisumlem3 26075 ostth2lem2 26218 ostth3 26222 knoppndvlem18 33981 ibladdnclem 35113 itgaddnclem2 35116 iblabsnclem 35120 iblmulc2nc 35122 itgmulc2nclem2 35124 ftc1anclem5 35134 irrapxlem4 39766 irrapxlem5 39767 rexabslelem 42055 uzublem 42067 max1d 42088 uzubioo 42204 climsuse 42250 limsupubuzlem 42354 limsupmnfuzlem 42368 limsupequzmptlem 42370 limsupre3uzlem 42377 liminflelimsuplem 42417 ioodvbdlimc1lem2 42574 ioodvbdlimc2lem 42576 |
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