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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 472 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ) | |
2 | 1 | recnd 9947 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℂ) |
3 | absval 13826 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) | |
4 | 2, 3 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) |
5 | 1 | cjred 13814 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (∗‘𝐴) = 𝐴) |
6 | 5 | oveq2d 6565 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 · (∗‘𝐴)) = (𝐴 · 𝐴)) |
7 | 2 | sqvald 12867 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴↑2) = (𝐴 · 𝐴)) |
8 | 6, 7 | eqtr4d 2647 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 · (∗‘𝐴)) = (𝐴↑2)) |
9 | 8 | fveq2d 6107 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘(𝐴 · (∗‘𝐴))) = (√‘(𝐴↑2))) |
10 | sqrtsq 13858 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘(𝐴↑2)) = 𝐴) | |
11 | 4, 9, 10 | 3eqtrd 2648 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (abs‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 · cmul 9820 ≤ cle 9954 2c2 10947 ↑cexp 12722 ∗ccj 13684 √csqrt 13821 abscabs 13822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 |
This theorem is referenced by: abs1 13885 absnid 13886 leabs 13887 absor 13888 sqabs 13895 max0add 13898 absidm 13911 abssubge0 13915 fzomaxdiflem 13930 absidi 13965 absidd 14009 o1fsum 14386 geo2lim 14445 geoihalfsum 14453 ege2le3 14659 eirrlem 14771 rpnnen2lem3 14784 rpnnen2lem9 14790 6gcd4e2 15093 lcmgcdnn 15162 lcmfun 15196 lcmfass 15197 zringndrg 19657 ncvsge0 22761 iscmet3lem3 22896 minveclem2 23005 mbfi1fseqlem6 23293 dvfsumrlim 23598 aaliou3lem3 23903 pserulm 23980 pige3 24073 efif1olem4 24095 cxpcn3lem 24288 log2cnv 24471 log2tlbnd 24472 cxplim 24498 cxploglim2 24505 divsqrtsumo1 24510 fsumharmonic 24538 zetacvg 24541 logfacrlim 24749 logexprlim 24750 dchrmusum2 24983 dchrvmasumlem3 24988 dchrisum0lem1 25005 dchrisum0lem2a 25006 dchrisum0lem2 25007 mudivsum 25019 mulogsumlem 25020 log2sumbnd 25033 selberglem2 25035 selberg3lem1 25046 pntpbnd2 25076 pntibndlem2 25080 pntlemn 25089 pntlemj 25092 pntlemo 25096 ex-abs 26704 ex-gcd 26706 nvsge0 26903 nmoub2i 27013 minvecolem2 27115 subfacval3 30425 knoppndvlem14 31686 poimir 32612 ftc1anclem5 32659 oddcomabszz 36527 fourierdlem68 39067 |
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