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Mirrors > Home > ILE Home > Th. List > immul2 | 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 7973 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | immul 10919 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℑ‘𝐵)) + ((ℑ‘𝐴) · (ℜ‘𝐵)))) | |
3 | 1, 2 | sylan 283 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (((ℜ‘𝐴) · (ℑ‘𝐵)) + ((ℑ‘𝐴) · (ℜ‘𝐵)))) |
4 | rere 10905 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (ℜ‘𝐴) = 𝐴) | |
5 | 4 | adantr 276 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘𝐴) = 𝐴) |
6 | 5 | oveq1d 5910 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → ((ℜ‘𝐴) · (ℑ‘𝐵)) = (𝐴 · (ℑ‘𝐵))) |
7 | reim0 10901 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (ℑ‘𝐴) = 0) | |
8 | 7 | oveq1d 5910 | . . . 4 ⊢ (𝐴 ∈ ℝ → ((ℑ‘𝐴) · (ℜ‘𝐵)) = (0 · (ℜ‘𝐵))) |
9 | recl 10893 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℝ) | |
10 | 9 | recnd 8015 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℂ) |
11 | 10 | mul02d 8378 | . . . 4 ⊢ (𝐵 ∈ ℂ → (0 · (ℜ‘𝐵)) = 0) |
12 | 8, 11 | sylan9eq 2242 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → ((ℑ‘𝐴) · (ℜ‘𝐵)) = 0) |
13 | 6, 12 | oveq12d 5913 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (((ℜ‘𝐴) · (ℑ‘𝐵)) + ((ℑ‘𝐴) · (ℜ‘𝐵))) = ((𝐴 · (ℑ‘𝐵)) + 0)) |
14 | imcl 10894 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) ∈ ℝ) | |
15 | 14 | recnd 8015 | . . . 4 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) ∈ ℂ) |
16 | mulcl 7967 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (ℑ‘𝐵) ∈ ℂ) → (𝐴 · (ℑ‘𝐵)) ∈ ℂ) | |
17 | 1, 15, 16 | syl2an 289 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (𝐴 · (ℑ‘𝐵)) ∈ ℂ) |
18 | 17 | addridd 8135 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → ((𝐴 · (ℑ‘𝐵)) + 0) = (𝐴 · (ℑ‘𝐵))) |
19 | 3, 13, 18 | 3eqtrd 2226 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 · 𝐵)) = (𝐴 · (ℑ‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 ‘cfv 5235 (class class class)co 5895 ℂcc 7838 ℝcr 7839 0cc0 7840 + caddc 7843 · cmul 7845 ℜcre 10880 ℑcim 10881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulrcl 7939 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-precex 7950 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-apti 7955 ax-pre-ltadd 7956 ax-pre-mulgt0 7957 ax-pre-mulext 7958 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-reap 8561 df-ap 8568 df-div 8659 df-2 9007 df-cj 10882 df-re 10883 df-im 10884 |
This theorem is referenced by: imdivap 10921 immul2d 11013 |
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