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Mirrors > Home > MPE Home > Th. List > imval | Structured version Visualization version GIF version |
Description: The value of the imaginary part of a complex number. (Contributed by NM, 9-May-1999.) (Revised by Mario Carneiro, 6-Nov-2013.) |
Ref | Expression |
---|---|
imval | ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(𝐴 / i))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvoveq1 7437 | . 2 ⊢ (𝑥 = 𝐴 → (ℜ‘(𝑥 / i)) = (ℜ‘(𝐴 / i))) | |
2 | df-im 15099 | . 2 ⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i))) | |
3 | fvex 6904 | . 2 ⊢ (ℜ‘(𝐴 / i)) ∈ V | |
4 | 1, 2, 3 | fvmpt 6999 | 1 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(𝐴 / i))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6544 (class class class)co 7414 ℂcc 11145 ici 11149 / cdiv 11910 ℜcre 15095 ℑcim 15096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3465 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4324 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7417 df-im 15099 |
This theorem is referenced by: imre 15106 reim 15107 imf 15111 crim 15113 iblcnlem1 25803 itgcnlem 25805 tanregt0 26561 cxpsqrtlem 26724 ang180lem2 26833 cnre2csqima 33737 ftc1anclem6 37410 |
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