Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 14949 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) | |
2 | cjmulrcl 14855 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 · (∗‘𝐴)) ∈ ℝ) | |
3 | cjmulge0 14857 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ≤ (𝐴 · (∗‘𝐴))) | |
4 | resqrtcl 14965 | . . 3 ⊢ (((𝐴 · (∗‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 · (∗‘𝐴))) → (√‘(𝐴 · (∗‘𝐴))) ∈ ℝ) | |
5 | 2, 3, 4 | syl2anc 584 | . 2 ⊢ (𝐴 ∈ ℂ → (√‘(𝐴 · (∗‘𝐴))) ∈ ℝ) |
6 | 1, 5 | eqeltrd 2839 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 · cmul 10876 ≤ cle 11010 ∗ccj 14807 √csqrt 14944 abscabs 14945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 |
This theorem is referenced by: absreim 15005 absdiv 15007 leabs 15011 absexp 15016 absexpz 15017 sqabs 15019 absimle 15021 abslt 15026 absle 15027 abssubne0 15028 lenegsq 15032 releabs 15033 recval 15034 absidm 15035 absgt0 15036 abstri 15042 abs2dif 15044 abs2difabs 15046 abs1m 15047 absf 15049 abs3lem 15050 abslem2 15051 absrdbnd 15053 caubnd2 15069 caubnd 15070 sqreulem 15071 sqreu 15072 abscli 15107 abscld 15148 mulcn2 15305 seqabs 15526 cvgcmpce 15530 divrcnv 15564 geomulcvg 15588 efcllem 15787 cnbl0 23937 cnblcld 23938 cncmet 24486 iblmulc2 24995 bddmulibl 25003 dveflem 25143 abelth 25600 efiarg 25762 argregt0 25765 argimgt0 25767 tanarg 25774 logtayllem 25814 bndatandm 26079 atantayl 26087 efrlim 26119 ftalem2 26223 lgslem3 26447 smcnlem 29059 cncph 29181 nmophmi 30393 bdophmi 30394 zrhnm 31919 sqrtcvallem2 41245 sqrtcvallem3 41246 sqrtcvallem4 41247 sqrtcvallem5 41248 sqrtcval 41249 absfico 42758 |
Copyright terms: Public domain | W3C validator |