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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 nonnegative real number. Lemma for sqrtcval 42263. See resqrtval 42265. (Contributed by RP, 11-May-2024.) |
Ref | Expression |
---|---|
sqrtcvallem4 | ⊢ (𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) + (ℜ‘𝐴)) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abscl 15212 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
2 | recl 15044 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
3 | 1, 2 | readdcld 11230 | . 2 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) + (ℜ‘𝐴)) ∈ ℝ) |
4 | 2rp 12966 | . . 3 ⊢ 2 ∈ ℝ+ | |
5 | 4 | a1i 11 | . 2 ⊢ (𝐴 ∈ ℂ → 2 ∈ ℝ+) |
6 | negcl 11447 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
7 | 6 | releabsd 15385 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) ≤ (abs‘-𝐴)) |
8 | 6 | abscld 15370 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) ∈ ℝ) |
9 | 6 | recld 15128 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) ∈ ℝ) |
10 | 8, 9 | subge0d 11791 | . . . 4 ⊢ (𝐴 ∈ ℂ → (0 ≤ ((abs‘-𝐴) − (ℜ‘-𝐴)) ↔ (ℜ‘-𝐴) ≤ (abs‘-𝐴))) |
11 | 7, 10 | mpbird 257 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ≤ ((abs‘-𝐴) − (ℜ‘-𝐴))) |
12 | absneg 15211 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) = (abs‘𝐴)) | |
13 | reneg 15059 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) = -(ℜ‘𝐴)) | |
14 | 12, 13 | oveq12d 7414 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘-𝐴) − (ℜ‘-𝐴)) = ((abs‘𝐴) − -(ℜ‘𝐴))) |
15 | 1 | recnd 11229 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℂ) |
16 | 2 | recnd 11229 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
17 | 15, 16 | subnegd 11565 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) − -(ℜ‘𝐴)) = ((abs‘𝐴) + (ℜ‘𝐴))) |
18 | 14, 17 | eqtrd 2773 | . . 3 ⊢ (𝐴 ∈ ℂ → ((abs‘-𝐴) − (ℜ‘-𝐴)) = ((abs‘𝐴) + (ℜ‘𝐴))) |
19 | 11, 18 | breqtrd 5170 | . 2 ⊢ (𝐴 ∈ ℂ → 0 ≤ ((abs‘𝐴) + (ℜ‘𝐴))) |
20 | 3, 5, 19 | divge0d 13043 | 1 ⊢ (𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) + (ℜ‘𝐴)) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 class class class wbr 5144 ‘cfv 6535 (class class class)co 7396 ℂcc 11095 0cc0 11097 + caddc 11100 ≤ cle 11236 − cmin 11431 -cneg 11432 / cdiv 11858 2c2 12254 ℝ+crp 12961 ℜcre 15031 abscabs 15168 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-sup 9424 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-3 12263 df-n0 12460 df-z 12546 df-uz 12810 df-rp 12962 df-seq 13954 df-exp 14015 df-cj 15033 df-re 15034 df-im 15035 df-sqrt 15169 df-abs 15170 |
This theorem is referenced by: sqrtcvallem5 42262 sqrtcval 42263 |
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