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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 10622 | . . . 4 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ) | |
2 | imadd 14281 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ -𝐵 ∈ ℂ) → (ℑ‘(𝐴 + -𝐵)) = ((ℑ‘𝐴) + (ℑ‘-𝐵))) | |
3 | 1, 2 | sylan2 586 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 + -𝐵)) = ((ℑ‘𝐴) + (ℑ‘-𝐵))) |
4 | imneg 14280 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (ℑ‘-𝐵) = -(ℑ‘𝐵)) | |
5 | 4 | adantl 475 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘-𝐵) = -(ℑ‘𝐵)) |
6 | 5 | oveq2d 6938 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((ℑ‘𝐴) + (ℑ‘-𝐵)) = ((ℑ‘𝐴) + -(ℑ‘𝐵))) |
7 | 3, 6 | eqtrd 2813 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 + -𝐵)) = ((ℑ‘𝐴) + -(ℑ‘𝐵))) |
8 | negsub 10671 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | |
9 | 8 | fveq2d 6450 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 + -𝐵)) = (ℑ‘(𝐴 − 𝐵))) |
10 | imcl 14258 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ) | |
11 | 10 | recnd 10405 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℂ) |
12 | imcl 14258 | . . . 4 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) ∈ ℝ) | |
13 | 12 | recnd 10405 | . . 3 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) ∈ ℂ) |
14 | negsub 10671 | . . 3 ⊢ (((ℑ‘𝐴) ∈ ℂ ∧ (ℑ‘𝐵) ∈ ℂ) → ((ℑ‘𝐴) + -(ℑ‘𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵))) | |
15 | 11, 13, 14 | syl2an 589 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((ℑ‘𝐴) + -(ℑ‘𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵))) |
16 | 7, 9, 15 | 3eqtr3d 2821 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (ℑ‘(𝐴 − 𝐵)) = ((ℑ‘𝐴) − (ℑ‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 + caddc 10275 − cmin 10606 -cneg 10607 ℑcim 14245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-2 11438 df-cj 14246 df-re 14247 df-im 14248 |
This theorem is referenced by: imsubd 14364 imcn2 14740 caucvgr 14814 tanregt0 24723 logneg2 24798 logcnlem4 24828 atancj 25088 atanlogaddlem 25091 atanlogsublem 25093 |
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