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Mirrors > Home > MPE Home > Th. List > absor | Structured version Visualization version GIF version |
Description: The absolute value of a real number is either that number or its negative. (Contributed by NM, 27-Feb-2005.) |
Ref | Expression |
---|---|
absor | ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10240 | . . 3 ⊢ 0 ∈ ℝ | |
2 | letric 10337 | . . 3 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0)) | |
3 | 1, 2 | mpan 670 | . 2 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0)) |
4 | absid 14237 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (abs‘𝐴) = 𝐴) | |
5 | 4 | ex 397 | . . 3 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 → (abs‘𝐴) = 𝐴)) |
6 | absnid 14239 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≤ 0) → (abs‘𝐴) = -𝐴) | |
7 | 6 | ex 397 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 ≤ 0 → (abs‘𝐴) = -𝐴)) |
8 | 5, 7 | orim12d 949 | . 2 ⊢ (𝐴 ∈ ℝ → ((0 ≤ 𝐴 ∨ 𝐴 ≤ 0) → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴))) |
9 | 3, 8 | mpd 15 | 1 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴) = 𝐴 ∨ (abs‘𝐴) = -𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 836 = wceq 1631 ∈ wcel 2145 class class class wbr 4786 ‘cfv 6029 ℝcr 10135 0cc0 10136 ≤ cle 10275 -cneg 10467 abscabs 14175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 ax-pre-sup 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-2nd 7314 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-er 7894 df-en 8108 df-dom 8109 df-sdom 8110 df-sup 8502 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-div 10885 df-nn 11221 df-2 11279 df-3 11280 df-n0 11493 df-z 11578 df-uz 11887 df-rp 12029 df-seq 13002 df-exp 13061 df-cj 14040 df-re 14041 df-im 14042 df-sqrt 14176 df-abs 14177 |
This theorem is referenced by: absmod0 14244 absz 14252 abslt 14255 absle 14256 absori 14321 absord 14355 gcdabs 15451 bezoutlem1 15457 lcmabs 15519 |
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