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Mirrors > Home > MPE Home > Th. List > cjreim | Structured version Visualization version 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 10627 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | ax-icn 10596 | . . . 4 ⊢ i ∈ ℂ | |
3 | recn 10627 | . . . 4 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
4 | mulcl 10621 | . . . 4 ⊢ ((i ∈ ℂ ∧ 𝐵 ∈ ℂ) → (i · 𝐵) ∈ ℂ) | |
5 | 2, 3, 4 | sylancr 589 | . . 3 ⊢ (𝐵 ∈ ℝ → (i · 𝐵) ∈ ℂ) |
6 | cjadd 14500 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (i · 𝐵) ∈ ℂ) → (∗‘(𝐴 + (i · 𝐵))) = ((∗‘𝐴) + (∗‘(i · 𝐵)))) | |
7 | 1, 5, 6 | syl2an 597 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (∗‘(𝐴 + (i · 𝐵))) = ((∗‘𝐴) + (∗‘(i · 𝐵)))) |
8 | cjre 14498 | . . 3 ⊢ (𝐴 ∈ ℝ → (∗‘𝐴) = 𝐴) | |
9 | cjmul 14501 | . . . . 5 ⊢ ((i ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(i · 𝐵)) = ((∗‘i) · (∗‘𝐵))) | |
10 | 2, 3, 9 | sylancr 589 | . . . 4 ⊢ (𝐵 ∈ ℝ → (∗‘(i · 𝐵)) = ((∗‘i) · (∗‘𝐵))) |
11 | cji 14518 | . . . . . 6 ⊢ (∗‘i) = -i | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (∗‘i) = -i) |
13 | cjre 14498 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (∗‘𝐵) = 𝐵) | |
14 | 12, 13 | oveq12d 7174 | . . . 4 ⊢ (𝐵 ∈ ℝ → ((∗‘i) · (∗‘𝐵)) = (-i · 𝐵)) |
15 | mulneg1 11076 | . . . . 5 ⊢ ((i ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-i · 𝐵) = -(i · 𝐵)) | |
16 | 2, 3, 15 | sylancr 589 | . . . 4 ⊢ (𝐵 ∈ ℝ → (-i · 𝐵) = -(i · 𝐵)) |
17 | 10, 14, 16 | 3eqtrd 2860 | . . 3 ⊢ (𝐵 ∈ ℝ → (∗‘(i · 𝐵)) = -(i · 𝐵)) |
18 | 8, 17 | oveqan12d 7175 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((∗‘𝐴) + (∗‘(i · 𝐵))) = (𝐴 + -(i · 𝐵))) |
19 | negsub 10934 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (i · 𝐵) ∈ ℂ) → (𝐴 + -(i · 𝐵)) = (𝐴 − (i · 𝐵))) | |
20 | 1, 5, 19 | syl2an 597 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + -(i · 𝐵)) = (𝐴 − (i · 𝐵))) |
21 | 7, 18, 20 | 3eqtrd 2860 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (∗‘(𝐴 + (i · 𝐵))) = (𝐴 − (i · 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 ici 10539 + caddc 10540 · cmul 10542 − cmin 10870 -cneg 10871 ∗ccj 14455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-2 11701 df-cj 14458 df-re 14459 df-im 14460 |
This theorem is referenced by: cjreim2 14520 dipcj 28491 lnophmlem2 29794 |
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