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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 14687 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) | |
2 | cjmulrcl 14593 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 · (∗‘𝐴)) ∈ ℝ) | |
3 | cjmulge0 14595 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ≤ (𝐴 · (∗‘𝐴))) | |
4 | resqrtcl 14703 | . . 3 ⊢ (((𝐴 · (∗‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 · (∗‘𝐴))) → (√‘(𝐴 · (∗‘𝐴))) ∈ ℝ) | |
5 | 2, 3, 4 | syl2anc 587 | . 2 ⊢ (𝐴 ∈ ℂ → (√‘(𝐴 · (∗‘𝐴))) ∈ ℝ) |
6 | 1, 5 | eqeltrd 2833 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 class class class wbr 5030 ‘cfv 6339 (class class class)co 7170 ℂcc 10613 ℝcr 10614 0cc0 10615 · cmul 10620 ≤ cle 10754 ∗ccj 14545 √csqrt 14682 abscabs 14683 |
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 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
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 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-sup 8979 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-z 12063 df-uz 12325 df-rp 12473 df-seq 13461 df-exp 13522 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 |
This theorem is referenced by: absreim 14743 absdiv 14745 leabs 14749 absexp 14754 absexpz 14755 sqabs 14757 absimle 14759 abslt 14764 absle 14765 abssubne0 14766 lenegsq 14770 releabs 14771 recval 14772 absidm 14773 absgt0 14774 abstri 14780 abs2dif 14782 abs2difabs 14784 abs1m 14785 absf 14787 abs3lem 14788 abslem2 14789 absrdbnd 14791 caubnd2 14807 caubnd 14808 sqreulem 14809 sqreu 14810 abscli 14845 abscld 14886 mulcn2 15043 seqabs 15262 cvgcmpce 15266 divrcnv 15300 geomulcvg 15324 efcllem 15523 cnbl0 23526 cnblcld 23527 cncmet 24074 iblmulc2 24583 bddmulibl 24591 dveflem 24731 abelth 25188 efiarg 25350 argregt0 25353 argimgt0 25355 tanarg 25362 logtayllem 25402 bndatandm 25667 atantayl 25675 efrlim 25707 ftalem2 25811 lgslem3 26035 smcnlem 28632 cncph 28754 nmophmi 29966 bdophmi 29967 zrhnm 31489 sqrtcvallem2 40790 sqrtcvallem3 40791 sqrtcvallem4 40792 sqrtcvallem5 40793 sqrtcval 40794 absfico 42296 |
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