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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 8160 | . . . . 5 ⊢ -i ∈ ℂ | |
2 | 1 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ ℂ → -i ∈ ℂ) |
3 | id 19 | . . . 4 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
4 | 2, 3 | mulcld 7980 | . . 3 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ) |
5 | absrele 11094 | . . 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 10862 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(-i · 𝐴))) | |
8 | 7 | fveq2d 5521 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(ℑ‘𝐴)) = (abs‘(ℜ‘(-i · 𝐴)))) |
9 | absmul 11080 | . . . 4 ⊢ ((-i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (abs‘(-i · 𝐴)) = ((abs‘-i) · (abs‘𝐴))) | |
10 | 1, 9 | mpan 424 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(-i · 𝐴)) = ((abs‘-i) · (abs‘𝐴))) |
11 | ax-icn 7908 | . . . . . . 7 ⊢ i ∈ ℂ | |
12 | absneg 11061 | . . . . . . 7 ⊢ (i ∈ ℂ → (abs‘-i) = (abs‘i)) | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 ⊢ (abs‘-i) = (abs‘i) |
14 | absi 11070 | . . . . . 6 ⊢ (abs‘i) = 1 | |
15 | 13, 14 | eqtri 2198 | . . . . 5 ⊢ (abs‘-i) = 1 |
16 | 15 | oveq1i 5887 | . . . 4 ⊢ ((abs‘-i) · (abs‘𝐴)) = (1 · (abs‘𝐴)) |
17 | abscl 11062 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
18 | 17 | recnd 7988 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℂ) |
19 | 18 | mulid2d 7978 | . . . 4 ⊢ (𝐴 ∈ ℂ → (1 · (abs‘𝐴)) = (abs‘𝐴)) |
20 | 16, 19 | eqtrid 2222 | . . 3 ⊢ (𝐴 ∈ ℂ → ((abs‘-i) · (abs‘𝐴)) = (abs‘𝐴)) |
21 | 10, 20 | eqtr2d 2211 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (abs‘(-i · 𝐴))) |
22 | 6, 8, 21 | 3brtr4d 4037 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘(ℑ‘𝐴)) ≤ (abs‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 class class class wbr 4005 ‘cfv 5218 (class class class)co 5877 ℂcc 7811 1c1 7814 ici 7815 · cmul 7818 ≤ cle 7995 -cneg 8131 ℜcre 10851 ℑcim 10852 abscabs 11008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-frec 6394 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-n0 9179 df-z 9256 df-uz 9531 df-rp 9656 df-seqfrec 10448 df-exp 10522 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 |
This theorem is referenced by: imcn2 11328 climcvg1nlem 11359 sin01bnd 11767 rpabscxpbnd 14444 |
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