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Mirrors > Home > MPE Home > Th. List > remul2 | Structured version Visualization version GIF version |
Description: Real part of a product. (Contributed by Mario Carneiro, 2-Aug-2014.) |
Ref | Expression |
---|---|
remul2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (𝐴 · (ℜ‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 11222 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | remul 15102 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵)))) | |
3 | 1, 2 | sylan 579 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵)))) |
4 | rere 15095 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (ℜ‘𝐴) = 𝐴) | |
5 | 4 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘𝐴) = 𝐴) |
6 | 5 | oveq1d 7429 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → ((ℜ‘𝐴) · (ℜ‘𝐵)) = (𝐴 · (ℜ‘𝐵))) |
7 | reim0 15091 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (ℑ‘𝐴) = 0) | |
8 | 7 | oveq1d 7429 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((ℑ‘𝐴) · (ℑ‘𝐵)) = (0 · (ℑ‘𝐵))) |
9 | imcl 15084 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) ∈ ℝ) | |
10 | 9 | recnd 11266 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) ∈ ℂ) |
11 | 10 | mul02d 11436 | . . . 4 ⊢ (𝐵 ∈ ℂ → (0 · (ℑ‘𝐵)) = 0) |
12 | 8, 11 | sylan9eq 2788 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → ((ℑ‘𝐴) · (ℑ‘𝐵)) = 0) |
13 | 6, 12 | oveq12d 7432 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵))) = ((𝐴 · (ℜ‘𝐵)) − 0)) |
14 | recl 15083 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℝ) | |
15 | 14 | recnd 11266 | . . . 4 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℂ) |
16 | mulcl 11216 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐵) ∈ ℂ) → (𝐴 · (ℜ‘𝐵)) ∈ ℂ) | |
17 | 1, 15, 16 | syl2an 595 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (𝐴 · (ℜ‘𝐵)) ∈ ℂ) |
18 | 17 | subid1d 11584 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → ((𝐴 · (ℜ‘𝐵)) − 0) = (𝐴 · (ℜ‘𝐵))) |
19 | 3, 13, 18 | 3eqtrd 2772 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (𝐴 · (ℜ‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ‘cfv 6542 (class class class)co 7414 ℂcc 11130 ℝcr 11131 0cc0 11132 · cmul 11137 − cmin 11468 ℜcre 15070 ℑcim 15071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-2 12299 df-cj 15072 df-re 15073 df-im 15074 |
This theorem is referenced by: rediv 15104 remul2d 15200 abscxp 26619 asinsin 26817 |
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