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Mirrors > Home > ILE Home > Th. List > absimle | 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 8090 | . . . . 5 ⊢ -i ∈ ℂ | |
2 | 1 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ ℂ → -i ∈ ℂ) |
3 | id 19 | . . . 4 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
4 | 2, 3 | mulcld 7910 | . . 3 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ) |
5 | absrele 11011 | . . 3 ⊢ ((-i · 𝐴) ∈ ℂ → (abs‘(ℜ‘(-i · 𝐴))) ≤ (abs‘(-i · 𝐴))) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(ℜ‘(-i · 𝐴))) ≤ (abs‘(-i · 𝐴))) |
7 | imre 10779 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(-i · 𝐴))) | |
8 | 7 | fveq2d 5484 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(ℑ‘𝐴)) = (abs‘(ℜ‘(-i · 𝐴)))) |
9 | absmul 10997 | . . . 4 ⊢ ((-i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (abs‘(-i · 𝐴)) = ((abs‘-i) · (abs‘𝐴))) | |
10 | 1, 9 | mpan 421 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(-i · 𝐴)) = ((abs‘-i) · (abs‘𝐴))) |
11 | ax-icn 7839 | . . . . . . 7 ⊢ i ∈ ℂ | |
12 | absneg 10978 | . . . . . . 7 ⊢ (i ∈ ℂ → (abs‘-i) = (abs‘i)) | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 ⊢ (abs‘-i) = (abs‘i) |
14 | absi 10987 | . . . . . 6 ⊢ (abs‘i) = 1 | |
15 | 13, 14 | eqtri 2185 | . . . . 5 ⊢ (abs‘-i) = 1 |
16 | 15 | oveq1i 5846 | . . . 4 ⊢ ((abs‘-i) · (abs‘𝐴)) = (1 · (abs‘𝐴)) |
17 | abscl 10979 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
18 | 17 | recnd 7918 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℂ) |
19 | 18 | mulid2d 7908 | . . . 4 ⊢ (𝐴 ∈ ℂ → (1 · (abs‘𝐴)) = (abs‘𝐴)) |
20 | 16, 19 | syl5eq 2209 | . . 3 ⊢ (𝐴 ∈ ℂ → ((abs‘-i) · (abs‘𝐴)) = (abs‘𝐴)) |
21 | 10, 20 | eqtr2d 2198 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (abs‘(-i · 𝐴))) |
22 | 6, 8, 21 | 3brtr4d 4008 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘(ℑ‘𝐴)) ≤ (abs‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 ℂcc 7742 1c1 7745 ici 7746 · cmul 7749 ≤ cle 7925 -cneg 8061 ℜcre 10768 ℑcim 10769 abscabs 10925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-rp 9581 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 |
This theorem is referenced by: imcn2 11245 climcvg1nlem 11276 sin01bnd 11684 rpabscxpbnd 13406 |
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