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Mathbox for Saveliy Skresanov |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sigarmf | Structured version Visualization version GIF version |
Description: Signed area is additive (with respect to subtraction) by the first argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.) |
Ref | Expression |
---|---|
sigar | ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) |
Ref | Expression |
---|---|
sigarmf | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶)𝐺𝐵) = ((𝐴𝐺𝐵) − (𝐶𝐺𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cjsub 14296 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (∗‘(𝐴 − 𝐶)) = ((∗‘𝐴) − (∗‘𝐶))) | |
2 | 1 | oveq1d 6937 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((∗‘(𝐴 − 𝐶)) · 𝐵) = (((∗‘𝐴) − (∗‘𝐶)) · 𝐵)) |
3 | 2 | 3adant2 1122 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((∗‘(𝐴 − 𝐶)) · 𝐵) = (((∗‘𝐴) − (∗‘𝐶)) · 𝐵)) |
4 | simp1 1127 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐴 ∈ ℂ) | |
5 | 4 | cjcld 14343 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (∗‘𝐴) ∈ ℂ) |
6 | simp3 1129 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐶 ∈ ℂ) | |
7 | 6 | cjcld 14343 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (∗‘𝐶) ∈ ℂ) |
8 | simp2 1128 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℂ) | |
9 | 5, 7, 8 | subdird 10832 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((∗‘𝐴) − (∗‘𝐶)) · 𝐵) = (((∗‘𝐴) · 𝐵) − ((∗‘𝐶) · 𝐵))) |
10 | 3, 9 | eqtrd 2813 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((∗‘(𝐴 − 𝐶)) · 𝐵) = (((∗‘𝐴) · 𝐵) − ((∗‘𝐶) · 𝐵))) |
11 | 10 | fveq2d 6450 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (ℑ‘((∗‘(𝐴 − 𝐶)) · 𝐵)) = (ℑ‘(((∗‘𝐴) · 𝐵) − ((∗‘𝐶) · 𝐵)))) |
12 | 5, 8 | mulcld 10397 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((∗‘𝐴) · 𝐵) ∈ ℂ) |
13 | 7, 8 | mulcld 10397 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((∗‘𝐶) · 𝐵) ∈ ℂ) |
14 | 12, 13 | imsubd 14364 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (ℑ‘(((∗‘𝐴) · 𝐵) − ((∗‘𝐶) · 𝐵))) = ((ℑ‘((∗‘𝐴) · 𝐵)) − (ℑ‘((∗‘𝐶) · 𝐵)))) |
15 | 11, 14 | eqtrd 2813 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (ℑ‘((∗‘(𝐴 − 𝐶)) · 𝐵)) = ((ℑ‘((∗‘𝐴) · 𝐵)) − (ℑ‘((∗‘𝐶) · 𝐵)))) |
16 | 4, 6 | subcld 10734 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 − 𝐶) ∈ ℂ) |
17 | sigar | . . . 4 ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) | |
18 | 17 | sigarval 41948 | . . 3 ⊢ (((𝐴 − 𝐶) ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐶)𝐺𝐵) = (ℑ‘((∗‘(𝐴 − 𝐶)) · 𝐵))) |
19 | 16, 8, 18 | syl2anc 579 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶)𝐺𝐵) = (ℑ‘((∗‘(𝐴 − 𝐶)) · 𝐵))) |
20 | 17 | sigarval 41948 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐺𝐵) = (ℑ‘((∗‘𝐴) · 𝐵))) |
21 | 20 | 3adant3 1123 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴𝐺𝐵) = (ℑ‘((∗‘𝐴) · 𝐵))) |
22 | 3simpc 1143 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) | |
23 | 22 | ancomd 455 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ)) |
24 | 17 | sigarval 41948 | . . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶𝐺𝐵) = (ℑ‘((∗‘𝐶) · 𝐵))) |
25 | 23, 24 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐶𝐺𝐵) = (ℑ‘((∗‘𝐶) · 𝐵))) |
26 | 21, 25 | oveq12d 6940 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐺𝐵) − (𝐶𝐺𝐵)) = ((ℑ‘((∗‘𝐴) · 𝐵)) − (ℑ‘((∗‘𝐶) · 𝐵)))) |
27 | 15, 19, 26 | 3eqtr4d 2823 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶)𝐺𝐵) = ((𝐴𝐺𝐵) − (𝐶𝐺𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ‘cfv 6135 (class class class)co 6922 ↦ cmpt2 6924 ℂcc 10270 · cmul 10277 − cmin 10606 ∗ccj 14243 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: sigarms 41954 sigarexp 41957 sigaradd 41964 |
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