| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 7454 | . 2 ⊢ (𝑥 = 𝐴 → (ℜ‘(𝑥 / i)) = (ℜ‘(𝐴 / i))) | |
| 2 | df-im 15140 | . 2 ⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i))) | |
| 3 | fvex 6919 | . 2 ⊢ (ℜ‘(𝐴 / i)) ∈ V | |
| 4 | 1, 2, 3 | fvmpt 7016 | 1 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = (ℜ‘(𝐴 / i))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ici 11157 / cdiv 11920 ℜcre 15136 ℑcim 15137 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-im 15140 |
| This theorem is referenced by: imre 15147 reim 15148 imf 15152 crim 15154 iblcnlem1 25823 itgcnlem 25825 tanregt0 26581 cxpsqrtlem 26744 ang180lem2 26853 cnre2csqima 33910 ftc1anclem6 37705 |
| Copyright terms: Public domain | W3C validator |