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Mirrors > Home > MPE Home > Th. List > absresq | Structured version Visualization version GIF version |
Description: Square of the absolute value of a real number. (Contributed by NM, 16-Jan-2006.) |
Ref | Expression |
---|---|
absresq | ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴)↑2) = (𝐴↑2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cjre 14083 | . . 3 ⊢ (𝐴 ∈ ℝ → (∗‘𝐴) = 𝐴) | |
2 | 1 | oveq2d 6808 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 · (∗‘𝐴)) = (𝐴 · 𝐴)) |
3 | recn 10228 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
4 | absvalsq 14224 | . . 3 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))) |
6 | 3 | sqvald 13208 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) = (𝐴 · 𝐴)) |
7 | 2, 5, 6 | 3eqtr4d 2815 | 1 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴)↑2) = (𝐴↑2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ‘cfv 6029 (class class class)co 6792 ℂcc 10136 ℝcr 10137 · cmul 10143 2c2 11272 ↑cexp 13063 ∗ccj 14040 abscabs 14178 |
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 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 |
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 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7213 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-sup 8504 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-n0 11496 df-z 11581 df-uz 11890 df-rp 12032 df-seq 13005 df-exp 13064 df-cj 14043 df-re 14044 df-im 14045 df-sqrt 14179 df-abs 14180 |
This theorem is referenced by: lenegsq 14264 sqnprm 15617 zgcdsq 15664 zringunit 20047 trirn 23398 rrxdstprj1 23407 tanregt0 24502 chordthmlem4 24779 heron 24782 lgsne0 25277 lgssq 25279 lgssq2 25280 lgsqr 25293 lgsquad3 25329 2sqblem 25373 sqsscirc2 30291 sinccvglem 31900 dvasin 33824 areacirclem1 33828 areacirclem2 33829 areacirclem4 33831 areacirclem5 33832 areacirc 33833 cntotbnd 33923 rrndstprj1 33957 rrndstprj2 33958 ismrer1 33965 pellexlem6 37921 pell14qrgt0 37946 abslt2sqd 40089 rrndistlt 41024 |
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