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Mirrors > Home > MPE Home > Th. List > absreim | Structured version Visualization version GIF version |
Description: Absolute value of a number that has been decomposed into real and imaginary parts. (Contributed by NM, 14-Jan-2006.) |
Ref | Expression |
---|---|
absreim | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (abs‘(𝐴 + (i · 𝐵))) = (√‘((𝐴↑2) + (𝐵↑2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 10362 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | ax-icn 10331 | . . . . . 6 ⊢ i ∈ ℂ | |
3 | recn 10362 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
4 | mulcl 10356 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ 𝐵 ∈ ℂ) → (i · 𝐵) ∈ ℂ) | |
5 | 2, 3, 4 | sylancr 581 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (i · 𝐵) ∈ ℂ) |
6 | addcl 10354 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (i · 𝐵) ∈ ℂ) → (𝐴 + (i · 𝐵)) ∈ ℂ) | |
7 | 1, 5, 6 | syl2an 589 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + (i · 𝐵)) ∈ ℂ) |
8 | abscl 14425 | . . . 4 ⊢ ((𝐴 + (i · 𝐵)) ∈ ℂ → (abs‘(𝐴 + (i · 𝐵))) ∈ ℝ) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (abs‘(𝐴 + (i · 𝐵))) ∈ ℝ) |
10 | absge0 14434 | . . . 4 ⊢ ((𝐴 + (i · 𝐵)) ∈ ℂ → 0 ≤ (abs‘(𝐴 + (i · 𝐵)))) | |
11 | 7, 10 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ (abs‘(𝐴 + (i · 𝐵)))) |
12 | sqrtsq 14417 | . . 3 ⊢ (((abs‘(𝐴 + (i · 𝐵))) ∈ ℝ ∧ 0 ≤ (abs‘(𝐴 + (i · 𝐵)))) → (√‘((abs‘(𝐴 + (i · 𝐵)))↑2)) = (abs‘(𝐴 + (i · 𝐵)))) | |
13 | 9, 11, 12 | syl2anc 579 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (√‘((abs‘(𝐴 + (i · 𝐵)))↑2)) = (abs‘(𝐴 + (i · 𝐵)))) |
14 | absreimsq 14439 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((abs‘(𝐴 + (i · 𝐵)))↑2) = ((𝐴↑2) + (𝐵↑2))) | |
15 | 14 | fveq2d 6450 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (√‘((abs‘(𝐴 + (i · 𝐵)))↑2)) = (√‘((𝐴↑2) + (𝐵↑2)))) |
16 | 13, 15 | eqtr3d 2815 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (abs‘(𝐴 + (i · 𝐵))) = (√‘((𝐴↑2) + (𝐵↑2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 ℝcr 10271 0cc0 10272 ici 10274 + caddc 10275 · cmul 10277 ≤ cle 10412 2c2 11430 ↑cexp 13178 √csqrt 14380 abscabs 14381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-sup 8636 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 |
This theorem is referenced by: absefi 15328 ipidsq 28137 ftc1anclem3 34107 |
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