Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > imre | Structured version Visualization version GIF version |
Description: The imaginary part of a complex number in terms of the real part function. (Contributed by NM, 12-May-2005.) (Revised by Mario Carneiro, 6-Nov-2013.) |
Ref | Expression |
---|---|
imre | ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(-i · 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imval 14454 | . 2 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(𝐴 / i))) | |
2 | ax-icn 10584 | . . . . 5 ⊢ i ∈ ℂ | |
3 | ine0 11063 | . . . . 5 ⊢ i ≠ 0 | |
4 | divrec2 11303 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ i ∈ ℂ ∧ i ≠ 0) → (𝐴 / i) = ((1 / i) · 𝐴)) | |
5 | 2, 3, 4 | mp3an23 1444 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 / i) = ((1 / i) · 𝐴)) |
6 | irec 13552 | . . . . 5 ⊢ (1 / i) = -i | |
7 | 6 | oveq1i 7155 | . . . 4 ⊢ ((1 / i) · 𝐴) = (-i · 𝐴) |
8 | 5, 7 | syl6eq 2869 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 / i) = (-i · 𝐴)) |
9 | 8 | fveq2d 6667 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘(𝐴 / i)) = (ℜ‘(-i · 𝐴))) |
10 | 1, 9 | eqtrd 2853 | 1 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(-i · 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 ici 10527 · cmul 10530 -cneg 10859 / cdiv 11285 ℜcre 14444 ℑcim 14445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 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-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-im 14448 |
This theorem is referenced by: imcl 14458 cnpart 14587 sqrtneglem 14614 absimle 14657 recan 14684 tanregt0 25050 asinlem3a 25375 asinsinlem 25396 asinsin 25397 asinbnd 25404 atanbndlem 25430 ftc1anclem6 34853 |
Copyright terms: Public domain | W3C validator |