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Mirrors > Home > MPE Home > Th. List > crimd | Structured version Visualization version GIF version |
Description: The imaginary part of a complex number representation. Definition 10-3.1 of [Gleason] p. 132. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
crred.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
crred.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
crimd | ⊢ (𝜑 → (ℑ‘(𝐴 + (i · 𝐵))) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | crred.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | crim 14046 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (ℑ‘(𝐴 + (i · 𝐵))) = 𝐵) | |
4 | 1, 2, 3 | syl2anc 696 | 1 ⊢ (𝜑 → (ℑ‘(𝐴 + (i · 𝐵))) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1624 ∈ wcel 2131 ‘cfv 6041 (class class class)co 6805 ℝcr 10119 ici 10122 + caddc 10123 · cmul 10125 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: resin4p 15059 4sqlem12 15854 4sqlem17 15859 itgim 23759 tanregt0 24476 eff1olem 24485 logf1o2 24587 basellem3 25000 2sqlem3 25336 bhmafibid1 29945 |
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