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Mirrors > Home > MPE Home > Th. List > Mathboxes > sqrtcvallem4 | Structured version Visualization version GIF version |
Description: Equivalent to saying that the square of the real component of the square root of a complex number is a non-negative real number. Lemma for sqrtcval 40699. See resqrtval 40701. (Contributed by RP, 11-May-2024.) |
Ref | Expression |
---|---|
sqrtcvallem4 | ⊢ (𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) + (ℜ‘𝐴)) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abscl 14671 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
2 | recl 14502 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
3 | 1, 2 | readdcld 10693 | . 2 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) + (ℜ‘𝐴)) ∈ ℝ) |
4 | 2rp 12420 | . . 3 ⊢ 2 ∈ ℝ+ | |
5 | 4 | a1i 11 | . 2 ⊢ (𝐴 ∈ ℂ → 2 ∈ ℝ+) |
6 | negcl 10909 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
7 | 6 | releabsd 14844 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) ≤ (abs‘-𝐴)) |
8 | 6 | abscld 14829 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) ∈ ℝ) |
9 | 6 | recld 14586 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) ∈ ℝ) |
10 | 8, 9 | subge0d 11253 | . . . 4 ⊢ (𝐴 ∈ ℂ → (0 ≤ ((abs‘-𝐴) − (ℜ‘-𝐴)) ↔ (ℜ‘-𝐴) ≤ (abs‘-𝐴))) |
11 | 7, 10 | mpbird 260 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ≤ ((abs‘-𝐴) − (ℜ‘-𝐴))) |
12 | absneg 14670 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) = (abs‘𝐴)) | |
13 | reneg 14517 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) = -(ℜ‘𝐴)) | |
14 | 12, 13 | oveq12d 7161 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘-𝐴) − (ℜ‘-𝐴)) = ((abs‘𝐴) − -(ℜ‘𝐴))) |
15 | 1 | recnd 10692 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℂ) |
16 | 2 | recnd 10692 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
17 | 15, 16 | subnegd 11027 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) − -(ℜ‘𝐴)) = ((abs‘𝐴) + (ℜ‘𝐴))) |
18 | 14, 17 | eqtrd 2794 | . . 3 ⊢ (𝐴 ∈ ℂ → ((abs‘-𝐴) − (ℜ‘-𝐴)) = ((abs‘𝐴) + (ℜ‘𝐴))) |
19 | 11, 18 | breqtrd 5051 | . 2 ⊢ (𝐴 ∈ ℂ → 0 ≤ ((abs‘𝐴) + (ℜ‘𝐴))) |
20 | 3, 5, 19 | divge0d 12497 | 1 ⊢ (𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) + (ℜ‘𝐴)) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 class class class wbr 5025 ‘cfv 6328 (class class class)co 7143 ℂcc 10558 0cc0 10560 + caddc 10563 ≤ cle 10699 − cmin 10893 -cneg 10894 / cdiv 11320 2c2 11714 ℝ+crp 12415 ℜcre 14489 abscabs 14626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 ax-pre-sup 10638 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-sup 8924 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-div 11321 df-nn 11660 df-2 11722 df-3 11723 df-n0 11920 df-z 12006 df-uz 12268 df-rp 12416 df-seq 13404 df-exp 13465 df-cj 14491 df-re 14492 df-im 14493 df-sqrt 14627 df-abs 14628 |
This theorem is referenced by: sqrtcvallem5 40698 sqrtcval 40699 |
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