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Mirrors > Home > ILE Home > Th. List > cjreim | GIF version |
Description: The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.) |
Ref | Expression |
---|---|
cjreim | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (∗‘(𝐴 + (i · 𝐵))) = (𝐴 − (i · 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 7204 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | ax-icn 7169 | . . . 4 ⊢ i ∈ ℂ | |
3 | recn 7204 | . . . 4 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
4 | mulcl 7198 | . . . 4 ⊢ ((i ∈ ℂ ∧ 𝐵 ∈ ℂ) → (i · 𝐵) ∈ ℂ) | |
5 | 2, 3, 4 | sylancr 405 | . . 3 ⊢ (𝐵 ∈ ℝ → (i · 𝐵) ∈ ℂ) |
6 | cjadd 9956 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (i · 𝐵) ∈ ℂ) → (∗‘(𝐴 + (i · 𝐵))) = ((∗‘𝐴) + (∗‘(i · 𝐵)))) | |
7 | 1, 5, 6 | syl2an 283 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (∗‘(𝐴 + (i · 𝐵))) = ((∗‘𝐴) + (∗‘(i · 𝐵)))) |
8 | cjre 9954 | . . 3 ⊢ (𝐴 ∈ ℝ → (∗‘𝐴) = 𝐴) | |
9 | cjmul 9957 | . . . . 5 ⊢ ((i ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(i · 𝐵)) = ((∗‘i) · (∗‘𝐵))) | |
10 | 2, 3, 9 | sylancr 405 | . . . 4 ⊢ (𝐵 ∈ ℝ → (∗‘(i · 𝐵)) = ((∗‘i) · (∗‘𝐵))) |
11 | cji 9974 | . . . . . 6 ⊢ (∗‘i) = -i | |
12 | 11 | a1i 9 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (∗‘i) = -i) |
13 | cjre 9954 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (∗‘𝐵) = 𝐵) | |
14 | 12, 13 | oveq12d 5582 | . . . 4 ⊢ (𝐵 ∈ ℝ → ((∗‘i) · (∗‘𝐵)) = (-i · 𝐵)) |
15 | mulneg1 7602 | . . . . 5 ⊢ ((i ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-i · 𝐵) = -(i · 𝐵)) | |
16 | 2, 3, 15 | sylancr 405 | . . . 4 ⊢ (𝐵 ∈ ℝ → (-i · 𝐵) = -(i · 𝐵)) |
17 | 10, 14, 16 | 3eqtrd 2119 | . . 3 ⊢ (𝐵 ∈ ℝ → (∗‘(i · 𝐵)) = -(i · 𝐵)) |
18 | 8, 17 | oveqan12d 5583 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((∗‘𝐴) + (∗‘(i · 𝐵))) = (𝐴 + -(i · 𝐵))) |
19 | negsub 7459 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (i · 𝐵) ∈ ℂ) → (𝐴 + -(i · 𝐵)) = (𝐴 − (i · 𝐵))) | |
20 | 1, 5, 19 | syl2an 283 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + -(i · 𝐵)) = (𝐴 − (i · 𝐵))) |
21 | 7, 18, 20 | 3eqtrd 2119 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (∗‘(𝐴 + (i · 𝐵))) = (𝐴 − (i · 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 ‘cfv 4953 (class class class)co 5564 ℂcc 7077 ℝcr 7078 ici 7081 + caddc 7082 · cmul 7084 − cmin 7382 -cneg 7383 ∗ccj 9911 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3917 ax-pow 3969 ax-pr 3993 ax-un 4217 ax-setind 4309 ax-cnex 7165 ax-resscn 7166 ax-1cn 7167 ax-1re 7168 ax-icn 7169 ax-addcl 7170 ax-addrcl 7171 ax-mulcl 7172 ax-mulrcl 7173 ax-addcom 7174 ax-mulcom 7175 ax-addass 7176 ax-mulass 7177 ax-distr 7178 ax-i2m1 7179 ax-0lt1 7180 ax-1rid 7181 ax-0id 7182 ax-rnegex 7183 ax-precex 7184 ax-cnre 7185 ax-pre-ltirr 7186 ax-pre-ltwlin 7187 ax-pre-lttrn 7188 ax-pre-apti 7189 ax-pre-ltadd 7190 ax-pre-mulgt0 7191 ax-pre-mulext 7192 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2612 df-sbc 2826 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-br 3807 df-opab 3861 df-mpt 3862 df-id 4077 df-po 4080 df-iso 4081 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-rn 4403 df-res 4404 df-ima 4405 df-iota 4918 df-fun 4955 df-fn 4956 df-f 4957 df-fv 4961 df-riota 5520 df-ov 5567 df-oprab 5568 df-mpt2 5569 df-pnf 7253 df-mnf 7254 df-xr 7255 df-ltxr 7256 df-le 7257 df-sub 7384 df-neg 7385 df-reap 7778 df-ap 7785 df-div 7864 df-2 8201 df-cj 9914 df-re 9915 df-im 9916 |
This theorem is referenced by: cjreim2 9976 cjap 9978 |
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