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Mirrors > Home > MPE Home > Th. List > absimle | Structured version Visualization version GIF version |
Description: The absolute value of a complex number is greater than or equal to the absolute value of its imaginary part. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
absimle | ⊢ (𝐴 ∈ ℂ → (abs‘(ℑ‘𝐴)) ≤ (abs‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negicn 10689 | . . . . 5 ⊢ -i ∈ ℂ | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℂ → -i ∈ ℂ) |
3 | id 22 | . . . 4 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
4 | 2, 3 | mulcld 10462 | . . 3 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ) |
5 | absrele 14532 | . . 3 ⊢ ((-i · 𝐴) ∈ ℂ → (abs‘(ℜ‘(-i · 𝐴))) ≤ (abs‘(-i · 𝐴))) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(ℜ‘(-i · 𝐴))) ≤ (abs‘(-i · 𝐴))) |
7 | imre 14331 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(-i · 𝐴))) | |
8 | 7 | fveq2d 6505 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(ℑ‘𝐴)) = (abs‘(ℜ‘(-i · 𝐴)))) |
9 | absmul 14518 | . . . 4 ⊢ ((-i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (abs‘(-i · 𝐴)) = ((abs‘-i) · (abs‘𝐴))) | |
10 | 1, 9 | mpan 677 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(-i · 𝐴)) = ((abs‘-i) · (abs‘𝐴))) |
11 | ax-icn 10396 | . . . . . . 7 ⊢ i ∈ ℂ | |
12 | absneg 14501 | . . . . . . 7 ⊢ (i ∈ ℂ → (abs‘-i) = (abs‘i)) | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 ⊢ (abs‘-i) = (abs‘i) |
14 | absi 14510 | . . . . . 6 ⊢ (abs‘i) = 1 | |
15 | 13, 14 | eqtri 2802 | . . . . 5 ⊢ (abs‘-i) = 1 |
16 | 15 | oveq1i 6988 | . . . 4 ⊢ ((abs‘-i) · (abs‘𝐴)) = (1 · (abs‘𝐴)) |
17 | abscl 14502 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
18 | 17 | recnd 10470 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℂ) |
19 | 18 | mulid2d 10460 | . . . 4 ⊢ (𝐴 ∈ ℂ → (1 · (abs‘𝐴)) = (abs‘𝐴)) |
20 | 16, 19 | syl5eq 2826 | . . 3 ⊢ (𝐴 ∈ ℂ → ((abs‘-i) · (abs‘𝐴)) = (abs‘𝐴)) |
21 | 10, 20 | eqtr2d 2815 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (abs‘(-i · 𝐴))) |
22 | 6, 8, 21 | 3brtr4d 4962 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘(ℑ‘𝐴)) ≤ (abs‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 class class class wbr 4930 ‘cfv 6190 (class class class)co 6978 ℂcc 10335 1c1 10338 ici 10339 · cmul 10342 ≤ cle 10477 -cneg 10673 ℜcre 14320 ℑcim 14321 abscabs 14457 |
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 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-pre-sup 10415 |
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 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-sup 8703 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-3 11507 df-n0 11711 df-z 11797 df-uz 12062 df-rp 12208 df-seq 13188 df-exp 13248 df-cj 14322 df-re 14323 df-im 14324 df-sqrt 14458 df-abs 14459 |
This theorem is referenced by: rlimrecl 14801 imcn2 14822 caucvgr 14896 sin01bnd 15401 recld2 23128 cnheiborlem 23264 aaliou2b 24636 efif1olem3 24832 logcnlem3 24931 logcnlem4 24932 efopnlem1 24943 abscxpbnd 25038 bddiblnc 34403 cntotbnd 34516 dstregt0 40977 absimlere 41188 |
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