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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 14349 | . . 3 ⊢ (𝐴 ∈ ℝ → (∗‘𝐴) = 𝐴) | |
2 | 1 | oveq2d 6986 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 · (∗‘𝐴)) = (𝐴 · 𝐴)) |
3 | recn 10417 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
4 | absvalsq 14491 | . . 3 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))) |
6 | 3 | sqvald 13315 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) = (𝐴 · 𝐴)) |
7 | 2, 5, 6 | 3eqtr4d 2818 | 1 ⊢ (𝐴 ∈ ℝ → ((abs‘𝐴)↑2) = (𝐴↑2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2048 ‘cfv 6182 (class class class)co 6970 ℂcc 10325 ℝcr 10326 · cmul 10332 2c2 11488 ↑cexp 13237 ∗ccj 14306 abscabs 14444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-sup 8693 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-n0 11701 df-z 11787 df-uz 12052 df-rp 12198 df-seq 13178 df-exp 13238 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 |
This theorem is referenced by: lenegsq 14531 sqnprm 15892 zgcdsq 15939 zringunit 20327 trirn 23696 rrxdstprj1 23705 tanregt0 24814 chordthmlem4 25104 heron 25107 lgsne0 25603 lgssq 25605 lgssq2 25606 lgsqr 25619 lgsquad3 25655 2sqblem 25699 sqsscirc2 30753 sinccvglem 32375 dvasin 34367 areacirclem1 34371 areacirclem2 34372 areacirclem4 34374 areacirclem5 34375 areacirc 34376 cntotbnd 34464 rrndstprj1 34498 rrndstprj2 34499 ismrer1 34506 dffltz 38623 pellexlem6 38772 pell14qrgt0 38797 abslt2sqd 41003 rrndistlt 41952 |
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