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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 9905 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | ax-icn 9874 | . . . 4 ⊢ i ∈ ℂ | |
3 | recn 9905 | . . . 4 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
4 | mulcl 9899 | . . . 4 ⊢ ((i ∈ ℂ ∧ 𝐵 ∈ ℂ) → (i · 𝐵) ∈ ℂ) | |
5 | 2, 3, 4 | sylancr 694 | . . 3 ⊢ (𝐵 ∈ ℝ → (i · 𝐵) ∈ ℂ) |
6 | cjadd 13729 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (i · 𝐵) ∈ ℂ) → (∗‘(𝐴 + (i · 𝐵))) = ((∗‘𝐴) + (∗‘(i · 𝐵)))) | |
7 | 1, 5, 6 | syl2an 493 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (∗‘(𝐴 + (i · 𝐵))) = ((∗‘𝐴) + (∗‘(i · 𝐵)))) |
8 | cjre 13727 | . . 3 ⊢ (𝐴 ∈ ℝ → (∗‘𝐴) = 𝐴) | |
9 | cjmul 13730 | . . . . 5 ⊢ ((i ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(i · 𝐵)) = ((∗‘i) · (∗‘𝐵))) | |
10 | 2, 3, 9 | sylancr 694 | . . . 4 ⊢ (𝐵 ∈ ℝ → (∗‘(i · 𝐵)) = ((∗‘i) · (∗‘𝐵))) |
11 | cji 13747 | . . . . . 6 ⊢ (∗‘i) = -i | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (∗‘i) = -i) |
13 | cjre 13727 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (∗‘𝐵) = 𝐵) | |
14 | 12, 13 | oveq12d 6567 | . . . 4 ⊢ (𝐵 ∈ ℝ → ((∗‘i) · (∗‘𝐵)) = (-i · 𝐵)) |
15 | mulneg1 10345 | . . . . 5 ⊢ ((i ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-i · 𝐵) = -(i · 𝐵)) | |
16 | 2, 3, 15 | sylancr 694 | . . . 4 ⊢ (𝐵 ∈ ℝ → (-i · 𝐵) = -(i · 𝐵)) |
17 | 10, 14, 16 | 3eqtrd 2648 | . . 3 ⊢ (𝐵 ∈ ℝ → (∗‘(i · 𝐵)) = -(i · 𝐵)) |
18 | 8, 17 | oveqan12d 6568 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((∗‘𝐴) + (∗‘(i · 𝐵))) = (𝐴 + -(i · 𝐵))) |
19 | negsub 10208 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (i · 𝐵) ∈ ℂ) → (𝐴 + -(i · 𝐵)) = (𝐴 − (i · 𝐵))) | |
20 | 1, 5, 19 | syl2an 493 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + -(i · 𝐵)) = (𝐴 − (i · 𝐵))) |
21 | 7, 18, 20 | 3eqtrd 2648 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (∗‘(𝐴 + (i · 𝐵))) = (𝐴 − (i · 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 ici 9817 + caddc 9818 · cmul 9820 − cmin 10145 -cneg 10146 ∗ccj 13684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-2 10956 df-cj 13687 df-re 13688 df-im 13689 |
This theorem is referenced by: cjreim2 13749 dipcj 26953 lnophmlem2 28260 |
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