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Mirrors > Home > MPE Home > Th. List > reim | Structured version Visualization version GIF version |
Description: The real part of a complex number in terms of the imaginary part function. (Contributed by Mario Carneiro, 31-Mar-2015.) |
Ref | Expression |
---|---|
reim | ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = (ℑ‘(i · 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 10442 | . . . 4 ⊢ i ∈ ℂ | |
2 | mulcl 10467 | . . . 4 ⊢ ((i ∈ ℂ ∧ 𝐴 ∈ ℂ) → (i · 𝐴) ∈ ℂ) | |
3 | 1, 2 | mpan 686 | . . 3 ⊢ (𝐴 ∈ ℂ → (i · 𝐴) ∈ ℂ) |
4 | imval 14300 | . . 3 ⊢ ((i · 𝐴) ∈ ℂ → (ℑ‘(i · 𝐴)) = (ℜ‘((i · 𝐴) / i))) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝐴 ∈ ℂ → (ℑ‘(i · 𝐴)) = (ℜ‘((i · 𝐴) / i))) |
6 | ine0 10923 | . . . 4 ⊢ i ≠ 0 | |
7 | divcan3 11172 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ i ∈ ℂ ∧ i ≠ 0) → ((i · 𝐴) / i) = 𝐴) | |
8 | 1, 6, 7 | mp3an23 1445 | . . 3 ⊢ (𝐴 ∈ ℂ → ((i · 𝐴) / i) = 𝐴) |
9 | 8 | fveq2d 6542 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘((i · 𝐴) / i)) = (ℜ‘𝐴)) |
10 | 5, 9 | eqtr2d 2832 | 1 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) = (ℑ‘(i · 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ‘cfv 6225 (class class class)co 7016 ℂcc 10381 0cc0 10383 ici 10385 · cmul 10388 / cdiv 11145 ℜcre 14290 ℑcim 14291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-po 5362 df-so 5363 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-im 14294 |
This theorem is referenced by: eqsqrt2d 14562 logimul 24878 logneg2 24879 atancj 25169 atanlogaddlem 25172 atanlogsublem 25174 atantan 25182 logi 32574 |
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