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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 10972 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | remul 14851 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵)))) | |
3 | 1, 2 | sylan 580 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵)))) |
4 | rere 14844 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (ℜ‘𝐴) = 𝐴) | |
5 | 4 | adantr 481 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘𝐴) = 𝐴) |
6 | 5 | oveq1d 7287 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → ((ℜ‘𝐴) · (ℜ‘𝐵)) = (𝐴 · (ℜ‘𝐵))) |
7 | reim0 14840 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (ℑ‘𝐴) = 0) | |
8 | 7 | oveq1d 7287 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((ℑ‘𝐴) · (ℑ‘𝐵)) = (0 · (ℑ‘𝐵))) |
9 | imcl 14833 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) ∈ ℝ) | |
10 | 9 | recnd 11014 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) ∈ ℂ) |
11 | 10 | mul02d 11184 | . . . 4 ⊢ (𝐵 ∈ ℂ → (0 · (ℑ‘𝐵)) = 0) |
12 | 8, 11 | sylan9eq 2800 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → ((ℑ‘𝐴) · (ℑ‘𝐵)) = 0) |
13 | 6, 12 | oveq12d 7290 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (((ℜ‘𝐴) · (ℜ‘𝐵)) − ((ℑ‘𝐴) · (ℑ‘𝐵))) = ((𝐴 · (ℜ‘𝐵)) − 0)) |
14 | recl 14832 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℝ) | |
15 | 14 | recnd 11014 | . . . 4 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℂ) |
16 | mulcl 10966 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐵) ∈ ℂ) → (𝐴 · (ℜ‘𝐵)) ∈ ℂ) | |
17 | 1, 15, 16 | syl2an 596 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (𝐴 · (ℜ‘𝐵)) ∈ ℂ) |
18 | 17 | subid1d 11332 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → ((𝐴 · (ℜ‘𝐵)) − 0) = (𝐴 · (ℜ‘𝐵))) |
19 | 3, 13, 18 | 3eqtrd 2784 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘(𝐴 · 𝐵)) = (𝐴 · (ℜ‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ‘cfv 6432 (class class class)co 7272 ℂcc 10880 ℝcr 10881 0cc0 10882 · cmul 10887 − cmin 11216 ℜcre 14819 ℑcim 14820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-po 5504 df-so 5505 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-er 8490 df-en 8726 df-dom 8727 df-sdom 8728 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-div 11644 df-2 12047 df-cj 14821 df-re 14822 df-im 14823 |
This theorem is referenced by: rediv 14853 remul2d 14949 abscxp 25858 asinsin 26053 |
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