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Mirrors > Home > MPE Home > Th. List > immul2 | Structured version Visualization version GIF version |
Description: Imaginary part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.) |
Ref | Expression |
---|---|
immul2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (𝐴 · (ℑ‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 11196 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | immul 15079 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℑ‘𝐵)) + ((ℑ‘𝐴) · (ℜ‘𝐵)))) | |
3 | 1, 2 | sylan 581 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℑ‘𝐵)) + ((ℑ‘𝐴) · (ℜ‘𝐵)))) |
4 | rere 15065 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (ℜ‘𝐴) = 𝐴) | |
5 | 4 | adantr 482 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘𝐴) = 𝐴) |
6 | 5 | oveq1d 7419 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → ((ℜ‘𝐴) · (ℑ‘𝐵)) = (𝐴 · (ℑ‘𝐵))) |
7 | reim0 15061 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (ℑ‘𝐴) = 0) | |
8 | 7 | oveq1d 7419 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((ℑ‘𝐴) · (ℜ‘𝐵)) = (0 · (ℜ‘𝐵))) |
9 | recl 15053 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℝ) | |
10 | 9 | recnd 11238 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℂ) |
11 | 10 | mul02d 11408 | . . . 4 ⊢ (𝐵 ∈ ℂ → (0 · (ℜ‘𝐵)) = 0) |
12 | 8, 11 | sylan9eq 2793 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → ((ℑ‘𝐴) · (ℜ‘𝐵)) = 0) |
13 | 6, 12 | oveq12d 7422 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (((ℜ‘𝐴) · (ℑ‘𝐵)) + ((ℑ‘𝐴) · (ℜ‘𝐵))) = ((𝐴 · (ℑ‘𝐵)) + 0)) |
14 | imcl 15054 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) ∈ ℝ) | |
15 | 14 | recnd 11238 | . . . 4 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) ∈ ℂ) |
16 | mulcl 11190 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐵) ∈ ℂ) → (𝐴 · (ℑ‘𝐵)) ∈ ℂ) | |
17 | 1, 15, 16 | syl2an 597 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (𝐴 · (ℑ‘𝐵)) ∈ ℂ) |
18 | 17 | addridd 11410 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → ((𝐴 · (ℑ‘𝐵)) + 0) = (𝐴 · (ℑ‘𝐵))) |
19 | 3, 13, 18 | 3eqtrd 2777 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (𝐴 · (ℑ‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ‘cfv 6540 (class class class)co 7404 ℂcc 11104 ℝcr 11105 0cc0 11106 + caddc 11109 · cmul 11111 ℜcre 15040 ℑcim 15041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-2 12271 df-cj 15042 df-re 15043 df-im 15044 |
This theorem is referenced by: imdiv 15081 immul2d 15171 cxpsqrtlem 26192 atantan 26408 |
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