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Mirrors > Home > MPE Home > Th. List > imsub | Structured version Visualization version GIF version |
Description: Imaginary part distributes over subtraction. (Contributed by NM, 18-Mar-2005.) |
Ref | Expression |
---|---|
imsub | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negcl 11359 | . . . 4 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ) | |
2 | imadd 14973 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ -𝐵 ∈ ℂ) → (ℑ‘(𝐴 + -𝐵)) = ((ℑ‘𝐴) + (ℑ‘-𝐵))) | |
3 | 1, 2 | sylan2 593 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 + -𝐵)) = ((ℑ‘𝐴) + (ℑ‘-𝐵))) |
4 | imneg 14972 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (ℑ‘-𝐵) = -(ℑ‘𝐵)) | |
5 | 4 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘-𝐵) = -(ℑ‘𝐵)) |
6 | 5 | oveq2d 7367 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((ℑ‘𝐴) + (ℑ‘-𝐵)) = ((ℑ‘𝐴) + -(ℑ‘𝐵))) |
7 | 3, 6 | eqtrd 2776 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 + -𝐵)) = ((ℑ‘𝐴) + -(ℑ‘𝐵))) |
8 | negsub 11407 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | |
9 | 8 | fveq2d 6843 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 + -𝐵)) = (ℑ‘(𝐴 − 𝐵))) |
10 | imcl 14950 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | |
11 | 10 | recnd 11141 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℂ) |
12 | imcl 14950 | . . . 4 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) ∈ ℝ) | |
13 | 12 | recnd 11141 | . . 3 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) ∈ ℂ) |
14 | negsub 11407 | . . 3 ⊢ (((ℑ‘𝐴) ∈ ℂ ∧ (ℑ‘𝐵) ∈ ℂ) → ((ℑ‘𝐴) + -(ℑ‘𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵))) | |
15 | 11, 13, 14 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((ℑ‘𝐴) + -(ℑ‘𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵))) |
16 | 7, 9, 15 | 3eqtr3d 2784 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7351 ℂcc 11007 + caddc 11012 − cmin 11343 -cneg 11344 ℑcim 14937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-2 12174 df-cj 14938 df-re 14939 df-im 14940 |
This theorem is referenced by: imsubd 15056 imcn2 15438 caucvgr 15514 tanregt0 25841 logneg2 25916 logcnlem4 25946 atancj 26206 atanlogaddlem 26209 atanlogsublem 26211 |
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