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Mirrors > Home > MPE Home > Th. List > leabsd | Structured version Visualization version GIF version |
Description: A real number is less than or equal to its absolute value. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
resqrcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
leabsd | ⊢ (𝜑 → 𝐴 ≤ (abs‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resqrcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | leabs 14752 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (abs‘𝐴)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 𝐴 ≤ (abs‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 class class class wbr 5031 ‘cfv 6340 ℝcr 10617 ≤ cle 10757 abscabs 14686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-pre-sup 10696 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-sup 8982 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 df-nn 11720 df-2 11782 df-3 11783 df-n0 11980 df-z 12066 df-uz 12328 df-rp 12476 df-seq 13464 df-exp 13525 df-cj 14551 df-re 14552 df-im 14553 df-sqrt 14687 df-abs 14688 |
This theorem is referenced by: lo1bddrp 14975 o1rlimmul 15069 nm2dif 23381 trirn 24155 bddiblnc 24597 mtestbdd 25155 abscxpbnd 25497 cxploglim2 25719 logexprlim 25964 rplogsumlem2 26224 dchrvmasumlem2 26237 dchrvmasumlem3 26238 dchrisum0flblem1 26247 dchrisum0fno1 26250 dchrisum0lem1 26255 mulog2sumlem2 26274 selberglem2 26285 chpdifbndlem1 26292 selberg3lem1 26296 pntrsumo1 26304 pntrlog2bndlem2 26317 pntrlog2bndlem3 26318 leopnmid 30076 dnibndlem7 34310 dnibndlem8 34311 dnibndlem12 34315 geomcau 35563 radcnvrat 41493 rexabslelem 42519 climleltrp 42782 ioodvbdlimc1lem1 43037 ioodvbdlimc1lem2 43038 ioodvbdlimc2lem 43040 fourierdlem77 43289 ioorrnopnlem 43410 sge0isum 43530 hoicvr 43651 smflimlem4 43871 smfmullem1 43887 |
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