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Mirrors > Home > MPE Home > Th. List > absid | Structured version Visualization version GIF version |
Description: A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
absid | ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (abs‘𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 486 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ) | |
2 | 1 | recnd 10826 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℂ) |
3 | absval 14766 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) | |
4 | 2, 3 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) |
5 | 1 | cjred 14754 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (∗‘𝐴) = 𝐴) |
6 | 5 | oveq2d 7207 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 · (∗‘𝐴)) = (𝐴 · 𝐴)) |
7 | 2 | sqvald 13678 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴↑2) = (𝐴 · 𝐴)) |
8 | 6, 7 | eqtr4d 2774 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 · (∗‘𝐴)) = (𝐴↑2)) |
9 | 8 | fveq2d 6699 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘(𝐴 · (∗‘𝐴))) = (√‘(𝐴↑2))) |
10 | sqrtsq 14798 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘(𝐴↑2)) = 𝐴) | |
11 | 4, 9, 10 | 3eqtrd 2775 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (abs‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 ℝcr 10693 0cc0 10694 · cmul 10699 ≤ cle 10833 2c2 11850 ↑cexp 13600 ∗ccj 14624 √csqrt 14761 abscabs 14762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-sup 9036 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-z 12142 df-uz 12404 df-rp 12552 df-seq 13540 df-exp 13601 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 |
This theorem is referenced by: abs1 14826 absnid 14827 leabs 14828 absor 14829 sqabs 14836 max0add 14839 absidm 14852 abssubge0 14856 fzomaxdiflem 14871 absidi 14906 absidd 14951 o1fsum 15340 geo2lim 15402 geoihalfsum 15409 ege2le3 15614 eirrlem 15728 rpnnen2lem3 15740 rpnnen2lem9 15746 6gcd4e2 16061 lcmgcdnn 16131 lcmfun 16165 lcmfass 16166 zringndrg 20409 ncvsge0 24004 iscmet3lem3 24141 minveclem2 24277 mbfi1fseqlem6 24572 dvfsumrlim 24882 aaliou3lem3 25191 pserulm 25268 pige3ALT 25363 efif1olem4 25388 cxpcn3lem 25587 log2cnv 25781 log2tlbnd 25782 cxplim 25808 cxploglim2 25815 divsqrtsumo1 25820 fsumharmonic 25848 zetacvg 25851 logfacrlim 26059 logexprlim 26060 dchrmusum2 26329 dchrvmasumlem3 26334 dchrisum0lem1 26351 dchrisum0lem2a 26352 dchrisum0lem2 26353 mudivsum 26365 mulogsumlem 26366 log2sumbnd 26379 selberglem2 26381 selberg3lem1 26392 pntpbnd2 26422 pntibndlem2 26426 pntlemn 26435 pntlemj 26438 pntlemo 26442 ex-abs 28492 ex-gcd 28494 nvsge0 28699 nmoub2i 28809 minvecolem2 28910 subfacval3 32818 knoppndvlem14 34391 poimir 35496 ftc1anclem5 35540 lcm2un 39705 oddcomabszz 40410 reabsifneg 40857 reabsifnpos 40858 reabsifpos 40859 reabsifnneg 40860 fourierdlem68 43333 itsclc0yqsol 45726 |
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