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Mirrors > Home > MPE Home > Th. List > reneg | Structured version Visualization version GIF version |
Description: Real part of negative. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 14-Jul-2014.) |
Ref | Expression |
---|---|
reneg | ⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) = -(ℜ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recl 14802 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
2 | 1 | recnd 10987 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
3 | ax-icn 10914 | . . . . . 6 ⊢ i ∈ ℂ | |
4 | imcl 14803 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | |
5 | 4 | recnd 10987 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℂ) |
6 | mulcl 10939 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ) → (i · (ℑ‘𝐴)) ∈ ℂ) | |
7 | 3, 5, 6 | sylancr 586 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · (ℑ‘𝐴)) ∈ ℂ) |
8 | 2, 7 | negdid 11328 | . . . 4 ⊢ (𝐴 ∈ ℂ → -((ℜ‘𝐴) + (i · (ℑ‘𝐴))) = (-(ℜ‘𝐴) + -(i · (ℑ‘𝐴)))) |
9 | replim 14808 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 𝐴 = ((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) | |
10 | 9 | negeqd 11198 | . . . 4 ⊢ (𝐴 ∈ ℂ → -𝐴 = -((ℜ‘𝐴) + (i · (ℑ‘𝐴)))) |
11 | mulneg2 11395 | . . . . . 6 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ) → (i · -(ℑ‘𝐴)) = -(i · (ℑ‘𝐴))) | |
12 | 3, 5, 11 | sylancr 586 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (i · -(ℑ‘𝐴)) = -(i · (ℑ‘𝐴))) |
13 | 12 | oveq2d 7284 | . . . 4 ⊢ (𝐴 ∈ ℂ → (-(ℜ‘𝐴) + (i · -(ℑ‘𝐴))) = (-(ℜ‘𝐴) + -(i · (ℑ‘𝐴)))) |
14 | 8, 10, 13 | 3eqtr4d 2789 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 = (-(ℜ‘𝐴) + (i · -(ℑ‘𝐴)))) |
15 | 14 | fveq2d 6772 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) = (ℜ‘(-(ℜ‘𝐴) + (i · -(ℑ‘𝐴))))) |
16 | 1 | renegcld 11385 | . . 3 ⊢ (𝐴 ∈ ℂ → -(ℜ‘𝐴) ∈ ℝ) |
17 | 4 | renegcld 11385 | . . 3 ⊢ (𝐴 ∈ ℂ → -(ℑ‘𝐴) ∈ ℝ) |
18 | crre 14806 | . . 3 ⊢ ((-(ℜ‘𝐴) ∈ ℝ ∧ -(ℑ‘𝐴) ∈ ℝ) → (ℜ‘(-(ℜ‘𝐴) + (i · -(ℑ‘𝐴)))) = -(ℜ‘𝐴)) | |
19 | 16, 17, 18 | syl2anc 583 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘(-(ℜ‘𝐴) + (i · -(ℑ‘𝐴)))) = -(ℜ‘𝐴)) |
20 | 15, 19 | eqtrd 2779 | 1 ⊢ (𝐴 ∈ ℂ → (ℜ‘-𝐴) = -(ℜ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2109 ‘cfv 6430 (class class class)co 7268 ℂcc 10853 ℝcr 10854 ici 10857 + caddc 10858 · cmul 10860 -cneg 11189 ℜcre 14789 ℑcim 14790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-po 5502 df-so 5503 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-2 12019 df-cj 14791 df-re 14792 df-im 14793 |
This theorem is referenced by: resub 14819 cjneg 14839 sqeqd 14858 renegi 14872 renegd 14901 cnpart 14932 asinsin 26023 ftc1anclem6 35834 sqrtcvallem4 41200 |
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