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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 15016 | . . 3 ⊢ (𝐴 ∈ ℝ → (∗‘𝐴) = 𝐴) | |
2 | 1 | oveq2d 7369 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 · (∗‘𝐴)) = (𝐴 · 𝐴)) |
3 | recn 11137 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
4 | absvalsq 15157 | . . 3 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))) |
6 | 3 | sqvald 14040 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) = (𝐴 · 𝐴)) |
7 | 2, 5, 6 | 3eqtr4d 2786 | 1 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴)↑2) = (𝐴↑2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7353 ℂcc 11045 ℝcr 11046 · cmul 11052 2c2 12204 ↑cexp 13959 ∗ccj 14973 abscabs 15111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9374 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-3 12213 df-n0 12410 df-z 12496 df-uz 12760 df-rp 12908 df-seq 13899 df-exp 13960 df-cj 14976 df-re 14977 df-im 14978 df-sqrt 15112 df-abs 15113 |
This theorem is referenced by: lenegsq 15197 sqnprm 16570 zgcdsq 16620 zringunit 20872 trirn 24748 rrxdstprj1 24757 tanregt0 25879 chordthmlem4 26169 heron 26172 lgsne0 26667 lgssq 26669 lgssq2 26670 lgsqr 26683 lgsquad3 26719 2sqblem 26763 sqsscirc2 32359 sinccvglem 34129 dvasin 36129 areacirclem1 36133 areacirclem2 36134 areacirclem4 36136 areacirclem5 36137 areacirc 36138 cntotbnd 36222 rrndstprj1 36256 rrndstprj2 36257 ismrer1 36264 dffltz 40910 pellexlem6 41095 pell14qrgt0 41120 abslt2sqd 43530 rrndistlt 44463 |
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