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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 11076 | . . . . 5 ⊢ -i ∈ ℂ | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℂ → -i ∈ ℂ) |
3 | id 22 | . . . 4 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
4 | 2, 3 | mulcld 10850 | . . 3 ⊢ (𝐴 ∈ ℂ → (-i · 𝐴) ∈ ℂ) |
5 | absrele 14869 | . . 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 14668 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(-i · 𝐴))) | |
8 | 7 | fveq2d 6718 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(ℑ‘𝐴)) = (abs‘(ℜ‘(-i · 𝐴)))) |
9 | absmul 14855 | . . . 4 ⊢ ((-i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (abs‘(-i · 𝐴)) = ((abs‘-i) · (abs‘𝐴))) | |
10 | 1, 9 | mpan 690 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘(-i · 𝐴)) = ((abs‘-i) · (abs‘𝐴))) |
11 | ax-icn 10785 | . . . . . . 7 ⊢ i ∈ ℂ | |
12 | absneg 14838 | . . . . . . 7 ⊢ (i ∈ ℂ → (abs‘-i) = (abs‘i)) | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 ⊢ (abs‘-i) = (abs‘i) |
14 | absi 14847 | . . . . . 6 ⊢ (abs‘i) = 1 | |
15 | 13, 14 | eqtri 2765 | . . . . 5 ⊢ (abs‘-i) = 1 |
16 | 15 | oveq1i 7220 | . . . 4 ⊢ ((abs‘-i) · (abs‘𝐴)) = (1 · (abs‘𝐴)) |
17 | abscl 14839 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
18 | 17 | recnd 10858 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℂ) |
19 | 18 | mulid2d 10848 | . . . 4 ⊢ (𝐴 ∈ ℂ → (1 · (abs‘𝐴)) = (abs‘𝐴)) |
20 | 16, 19 | syl5eq 2790 | . . 3 ⊢ (𝐴 ∈ ℂ → ((abs‘-i) · (abs‘𝐴)) = (abs‘𝐴)) |
21 | 10, 20 | eqtr2d 2778 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (abs‘(-i · 𝐴))) |
22 | 6, 8, 21 | 3brtr4d 5082 | 1 ⊢ (𝐴 ∈ ℂ → (abs‘(ℑ‘𝐴)) ≤ (abs‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 class class class wbr 5050 ‘cfv 6377 (class class class)co 7210 ℂcc 10724 1c1 10727 ici 10728 · cmul 10731 ≤ cle 10865 -cneg 11060 ℜcre 14657 ℑcim 14658 abscabs 14794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 ax-cnex 10782 ax-resscn 10783 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-addrcl 10787 ax-mulcl 10788 ax-mulrcl 10789 ax-mulcom 10790 ax-addass 10791 ax-mulass 10792 ax-distr 10793 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rnegex 10797 ax-rrecex 10798 ax-cnre 10799 ax-pre-lttri 10800 ax-pre-lttrn 10801 ax-pre-ltadd 10802 ax-pre-mulgt0 10803 ax-pre-sup 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-nel 3044 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-er 8388 df-en 8624 df-dom 8625 df-sdom 8626 df-sup 9055 df-pnf 10866 df-mnf 10867 df-xr 10868 df-ltxr 10869 df-le 10870 df-sub 11061 df-neg 11062 df-div 11487 df-nn 11828 df-2 11890 df-3 11891 df-n0 12088 df-z 12174 df-uz 12436 df-rp 12584 df-seq 13572 df-exp 13633 df-cj 14659 df-re 14660 df-im 14661 df-sqrt 14795 df-abs 14796 |
This theorem is referenced by: rlimrecl 15138 imcn2 15160 caucvgr 15236 sin01bnd 15743 recld2 23708 cnheiborlem 23848 bddiblnc 24736 aaliou2b 25231 efif1olem3 25430 logcnlem3 25529 logcnlem4 25530 efopnlem1 25541 abscxpbnd 25636 cntotbnd 35689 dstregt0 42490 absimlere 42693 |
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