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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 6933 | . 2 ⊢ (𝑥 = 𝐴 → (ℜ‘(𝑥 / i)) = (ℜ‘(𝐴 / i))) | |
2 | df-im 14225 | . 2 ⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i))) | |
3 | fvex 6450 | . 2 ⊢ (ℜ‘(𝐴 / i)) ∈ V | |
4 | 1, 2, 3 | fvmpt 6533 | 1 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(𝐴 / i))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ‘cfv 6127 (class class class)co 6910 ℂcc 10257 ici 10261 / cdiv 11016 ℜcre 14221 ℑcim 14222 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-iota 6090 df-fun 6129 df-fv 6135 df-ov 6913 df-im 14225 |
This theorem is referenced by: imre 14232 reim 14233 imf 14237 crim 14239 iblcnlem1 23960 itgcnlem 23962 tanregt0 24692 cxpsqrtlem 24854 ang180lem2 24957 cnre2csqima 30498 ftc1anclem6 34028 |
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