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Mirrors > Home > MPE Home > Th. List > abscl | Structured version Visualization version GIF version |
Description: Real closure of absolute value. (Contributed by NM, 3-Oct-1999.) |
Ref | Expression |
---|---|
abscl | โข (๐ด โ โ โ (absโ๐ด) โ โ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absval 15218 | . 2 โข (๐ด โ โ โ (absโ๐ด) = (โโ(๐ด ยท (โโ๐ด)))) | |
2 | cjmulrcl 15124 | . . 3 โข (๐ด โ โ โ (๐ด ยท (โโ๐ด)) โ โ) | |
3 | cjmulge0 15126 | . . 3 โข (๐ด โ โ โ 0 โค (๐ด ยท (โโ๐ด))) | |
4 | resqrtcl 15233 | . . 3 โข (((๐ด ยท (โโ๐ด)) โ โ โง 0 โค (๐ด ยท (โโ๐ด))) โ (โโ(๐ด ยท (โโ๐ด))) โ โ) | |
5 | 2, 3, 4 | syl2anc 583 | . 2 โข (๐ด โ โ โ (โโ(๐ด ยท (โโ๐ด))) โ โ) |
6 | 1, 5 | eqeltrd 2829 | 1 โข (๐ด โ โ โ (absโ๐ด) โ โ) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wcel 2099 class class class wbr 5148 โcfv 6548 (class class class)co 7420 โcc 11137 โcr 11138 0cc0 11139 ยท cmul 11144 โค cle 11280 โccj 15076 โcsqrt 15213 abscabs 15214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 |
This theorem is referenced by: absreim 15273 absdiv 15275 leabs 15279 absexp 15284 absexpz 15285 sqabs 15287 absimle 15289 abslt 15294 absle 15295 abssubne0 15296 lenegsq 15300 releabs 15301 recval 15302 absidm 15303 absgt0 15304 abstri 15310 abs2dif 15312 abs2difabs 15314 abs1m 15315 absf 15317 abs3lem 15318 abslem2 15319 absrdbnd 15321 caubnd2 15337 caubnd 15338 sqreulem 15339 sqreu 15340 abscli 15375 abscld 15416 mulcn2 15573 seqabs 15793 cvgcmpce 15797 divrcnv 15831 geomulcvg 15855 efcllem 16054 cnbl0 24703 cnblcld 24704 cncmet 25263 iblmulc2 25773 bddmulibl 25781 dveflem 25924 abelth 26391 efiarg 26554 argregt0 26557 argimgt0 26559 tanarg 26566 logtayllem 26606 bndatandm 26874 atantayl 26882 efrlim 26914 efrlimOLD 26915 ftalem2 27019 lgslem3 27245 smcnlem 30520 cncph 30642 nmophmi 31854 bdophmi 31855 zrhnm 33570 sqrtcvallem2 43067 sqrtcvallem3 43068 sqrtcvallem4 43069 sqrtcvallem5 43070 sqrtcval 43071 absfico 44591 |
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