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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 15187 | . 2 โข (๐ด โ โ โ (absโ๐ด) = (โโ(๐ด ยท (โโ๐ด)))) | |
2 | cjmulrcl 15093 | . . 3 โข (๐ด โ โ โ (๐ด ยท (โโ๐ด)) โ โ) | |
3 | cjmulge0 15095 | . . 3 โข (๐ด โ โ โ 0 โค (๐ด ยท (โโ๐ด))) | |
4 | resqrtcl 15202 | . . 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 2825 | 1 โข (๐ด โ โ โ (absโ๐ด) โ โ) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โ wcel 2098 class class class wbr 5139 โcfv 6534 (class class class)co 7402 โcc 11105 โcr 11106 0cc0 11107 ยท cmul 11112 โค cle 11248 โccj 15045 โcsqrt 15182 abscabs 15183 |
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-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
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-rmo 3368 df-reu 3369 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-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 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-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 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-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12976 df-seq 13968 df-exp 14029 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 |
This theorem is referenced by: absreim 15242 absdiv 15244 leabs 15248 absexp 15253 absexpz 15254 sqabs 15256 absimle 15258 abslt 15263 absle 15264 abssubne0 15265 lenegsq 15269 releabs 15270 recval 15271 absidm 15272 absgt0 15273 abstri 15279 abs2dif 15281 abs2difabs 15283 abs1m 15284 absf 15286 abs3lem 15287 abslem2 15288 absrdbnd 15290 caubnd2 15306 caubnd 15307 sqreulem 15308 sqreu 15309 abscli 15344 abscld 15385 mulcn2 15542 seqabs 15762 cvgcmpce 15766 divrcnv 15800 geomulcvg 15824 efcllem 16023 cnbl0 24634 cnblcld 24635 cncmet 25194 iblmulc2 25704 bddmulibl 25712 dveflem 25855 abelth 26319 efiarg 26482 argregt0 26485 argimgt0 26487 tanarg 26494 logtayllem 26534 bndatandm 26802 atantayl 26810 efrlim 26842 efrlimOLD 26843 ftalem2 26947 lgslem3 27173 smcnlem 30445 cncph 30567 nmophmi 31779 bdophmi 31780 zrhnm 33469 sqrtcvallem2 42938 sqrtcvallem3 42939 sqrtcvallem4 42940 sqrtcvallem5 42941 sqrtcval 42942 absfico 44463 |
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