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Mirrors > Home > MPE Home > Th. List > imf | Structured version Visualization version GIF version |
Description: Domain and codomain of the imaginary part function. (Contributed by Paul Chapman, 22-Oct-2007.) (Revised by Mario Carneiro, 6-Nov-2013.) |
Ref | Expression |
---|---|
imf | ⊢ ℑ:ℂ⟶ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-im 14032 | . 2 ⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i))) | |
2 | imval 14038 | . . 3 ⊢ (𝑥 ∈ ℂ → (ℑ‘𝑥) = (ℜ‘(𝑥 / i))) | |
3 | imcl 14042 | . . 3 ⊢ (𝑥 ∈ ℂ → (ℑ‘𝑥) ∈ ℝ) | |
4 | 2, 3 | eqeltrrd 2832 | . 2 ⊢ (𝑥 ∈ ℂ → (ℜ‘(𝑥 / i)) ∈ ℝ) |
5 | 1, 4 | fmpti 6538 | 1 ⊢ ℑ:ℂ⟶ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2131 ⟶wf 6037 ‘cfv 6041 (class class class)co 6805 ℂcc 10118 ℝcr 10119 ici 10122 / cdiv 10868 ℜcre 14028 ℑcim 14029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-op 4320 df-uni 4581 df-br 4797 df-opab 4857 df-mpt 4874 df-id 5166 df-po 5179 df-so 5180 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-er 7903 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-div 10869 df-2 11263 df-cj 14030 df-re 14031 df-im 14032 |
This theorem is referenced by: imcn2 14523 climim 14528 rlimim 14533 caucvgr 14597 fsumim 14732 imcncf 22899 cnrehmeo 22945 ismbf 23588 ismbfcn 23589 mbfconst 23593 ismbfcn2 23597 mbfres 23602 mbfimaopnlem 23613 eff1olem 24485 ellogrn 24497 dvloglem 24585 logf1o2 24587 dvlog 24588 efopnlem2 24594 asinneg 24804 mbfresfi 33761 itgaddnc 33775 itgmulc2nc 33783 mbfres2cn 40669 |
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