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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 | ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absval 14599 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) | |
2 | cjmulrcl 14505 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 · (∗‘𝐴)) ∈ ℝ) | |
3 | cjmulge0 14507 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ≤ (𝐴 · (∗‘𝐴))) | |
4 | resqrtcl 14615 | . . 3 ⊢ (((𝐴 · (∗‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 · (∗‘𝐴))) → (√‘(𝐴 · (∗‘𝐴))) ∈ ℝ) | |
5 | 2, 3, 4 | syl2anc 586 | . 2 ⊢ (𝐴 ∈ ℂ → (√‘(𝐴 · (∗‘𝐴))) ∈ ℝ) |
6 | 1, 5 | eqeltrd 2915 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 · cmul 10544 ≤ cle 10678 ∗ccj 14457 √csqrt 14594 abscabs 14595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 |
This theorem is referenced by: absreim 14655 absdiv 14657 leabs 14661 absexp 14666 absexpz 14667 sqabs 14669 absimle 14671 abslt 14676 absle 14677 abssubne0 14678 lenegsq 14682 releabs 14683 recval 14684 absidm 14685 absgt0 14686 abstri 14692 abs2dif 14694 abs2difabs 14696 abs1m 14697 absf 14699 abs3lem 14700 abslem2 14701 absrdbnd 14703 caubnd2 14719 caubnd 14720 sqreulem 14721 sqreu 14722 abscli 14757 abscld 14798 mulcn2 14954 seqabs 15171 cvgcmpce 15175 divrcnv 15209 geomulcvg 15234 efcllem 15433 cnbl0 23384 cnblcld 23385 cncmet 23927 iblmulc2 24433 bddmulibl 24441 dveflem 24578 abelth 25031 efiarg 25192 argregt0 25195 argimgt0 25197 tanarg 25204 logtayllem 25244 bndatandm 25509 atantayl 25517 efrlim 25549 ftalem2 25653 lgslem3 25877 smcnlem 28476 cncph 28598 nmophmi 29810 bdophmi 29811 zrhnm 31212 absfico 41488 |
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