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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 11891. (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 9964 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 9964 | . 2 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | xrmax1 11880 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | |
4 | 1, 2, 3 | syl2an 493 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) |
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: z2ge 11903 ssfzunsn 12257 uzsup 12524 expmulnbnd 12858 discr1 12862 rexuzre 13940 rexico 13941 caubnd 13946 limsupgre 14060 limsupbnd2 14062 rlim3 14077 lo1bdd2 14103 o1lo1 14116 rlimclim1 14124 lo1mul 14206 rlimno1 14232 cvgrat 14454 ruclem10 14807 bitsfzo 14995 1arith 15469 evth 22566 ioombl1lem1 23133 mbfi1flimlem 23295 itg2monolem3 23325 iblre 23366 itgreval 23369 iblss 23377 i1fibl 23380 itgitg1 23381 itgle 23382 itgeqa 23386 iblconst 23390 itgconst 23391 ibladdlem 23392 itgaddlem2 23396 iblabslem 23400 iblabsr 23402 iblmulc2 23403 itgmulc2lem2 23405 itgsplit 23408 plyaddlem1 23773 coeaddlem 23809 o1cxp 24501 cxp2lim 24503 cxploglim2 24505 ftalem1 24599 ftalem2 24600 chtppilim 24964 dchrisumlem3 24980 ostth2lem2 25123 ostth3 25127 knoppndvlem18 31690 ibladdnclem 32636 itgaddnclem2 32639 iblabsnclem 32643 iblmulc2nc 32645 itgmulc2nclem2 32647 ftc1anclem5 32659 irrapxlem4 36407 irrapxlem5 36408 climsuse 38675 ioodvbdlimc1lem2 38822 ioodvbdlimc2lem 38824 |
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