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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 42881. See resqrtval 42883. (Contributed by RP, 11-May-2024.) |
Ref | Expression |
---|---|
sqrtcvallem4 | ⊢ (𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) + (ℜ‘𝐴)) / 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abscl 15222 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
2 | recl 15054 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
3 | 1, 2 | readdcld 11240 | . 2 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) + (ℜ‘𝐴)) ∈ ℝ) |
4 | 2rp 12976 | . . 3 ⊢ 2 ∈ ℝ+ | |
5 | 4 | a1i 11 | . 2 ⊢ (𝐴 ∈ ℂ → 2 ∈ ℝ+) |
6 | negcl 11457 | . . . . 5 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
7 | 6 | releabsd 15395 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) ≤ (abs‘-𝐴)) |
8 | 6 | abscld 15380 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) ∈ ℝ) |
9 | 6 | recld 15138 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) ∈ ℝ) |
10 | 8, 9 | subge0d 11801 | . . . 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 15221 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) = (abs‘𝐴)) | |
13 | reneg 15069 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) = -(ℜ‘𝐴)) | |
14 | 12, 13 | oveq12d 7419 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘-𝐴) − (ℜ‘-𝐴)) = ((abs‘𝐴) − -(ℜ‘𝐴))) |
15 | 1 | recnd 11239 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℂ) |
16 | 2 | recnd 11239 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
17 | 15, 16 | subnegd 11575 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) − -(ℜ‘𝐴)) = ((abs‘𝐴) + (ℜ‘𝐴))) |
18 | 14, 17 | eqtrd 2764 | . . 3 ⊢ (𝐴 ∈ ℂ → ((abs‘-𝐴) − (ℜ‘-𝐴)) = ((abs‘𝐴) + (ℜ‘𝐴))) |
19 | 11, 18 | breqtrd 5164 | . 2 ⊢ (𝐴 ∈ ℂ → 0 ≤ ((abs‘𝐴) + (ℜ‘𝐴))) |
20 | 3, 5, 19 | divge0d 13053 | 1 ⊢ (𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) + (ℜ‘𝐴)) / 2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 class class class wbr 5138 ‘cfv 6533 (class class class)co 7401 ℂcc 11104 0cc0 11106 + caddc 11109 ≤ cle 11246 − cmin 11441 -cneg 11442 / cdiv 11868 2c2 12264 ℝ+crp 12971 ℜcre 15041 abscabs 15178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 |
This theorem is referenced by: sqrtcvallem5 42880 sqrtcval 42881 |
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