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Mathbox for Saveliy Skresanov |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sigarexp | Structured version Visualization version GIF version |
Description: Expand the signed area formula by linearity. (Contributed by Saveliy Skresanov, 20-Sep-2017.) |
Ref | Expression |
---|---|
sigar | ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) |
Ref | Expression |
---|---|
sigarexp | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶)𝐺(𝐵 − 𝐶)) = (((𝐴𝐺𝐵) − (𝐴𝐺𝐶)) − (𝐶𝐺𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1128 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℂ) | |
2 | simp3 1129 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → 𝐶 ∈ ℂ) | |
3 | 1, 2 | subcld 10736 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 − 𝐶) ∈ ℂ) |
4 | sigar | . . . 4 ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) | |
5 | 4 | sigarmf 41984 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 − 𝐶) ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶)𝐺(𝐵 − 𝐶)) = ((𝐴𝐺(𝐵 − 𝐶)) − (𝐶𝐺(𝐵 − 𝐶)))) |
6 | 3, 5 | syld3an2 1480 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶)𝐺(𝐵 − 𝐶)) = ((𝐴𝐺(𝐵 − 𝐶)) − (𝐶𝐺(𝐵 − 𝐶)))) |
7 | 4 | sigarms 41986 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴𝐺(𝐵 − 𝐶)) = ((𝐴𝐺𝐵) − (𝐴𝐺𝐶))) |
8 | 7 | oveq1d 6939 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴𝐺(𝐵 − 𝐶)) − (𝐶𝐺(𝐵 − 𝐶))) = (((𝐴𝐺𝐵) − (𝐴𝐺𝐶)) − (𝐶𝐺(𝐵 − 𝐶)))) |
9 | 4 | sigarms 41986 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐶𝐺(𝐵 − 𝐶)) = ((𝐶𝐺𝐵) − (𝐶𝐺𝐶))) |
10 | 2, 9 | syld3an1 1478 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐶𝐺(𝐵 − 𝐶)) = ((𝐶𝐺𝐵) − (𝐶𝐺𝐶))) |
11 | 4 | sigarid 41988 | . . . . . 6 ⊢ (𝐶 ∈ ℂ → (𝐶𝐺𝐶) = 0) |
12 | 2, 11 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐶𝐺𝐶) = 0) |
13 | 12 | oveq2d 6940 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶𝐺𝐵) − (𝐶𝐺𝐶)) = ((𝐶𝐺𝐵) − 0)) |
14 | 4 | sigarim 41981 | . . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶𝐺𝐵) ∈ ℝ) |
15 | 14 | recnd 10407 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐶𝐺𝐵) ∈ ℂ) |
16 | 2, 1, 15 | syl2anc 579 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐶𝐺𝐵) ∈ ℂ) |
17 | 16 | subid1d 10725 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶𝐺𝐵) − 0) = (𝐶𝐺𝐵)) |
18 | 10, 13, 17 | 3eqtrd 2818 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐶𝐺(𝐵 − 𝐶)) = (𝐶𝐺𝐵)) |
19 | 18 | oveq2d 6940 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐴𝐺𝐵) − (𝐴𝐺𝐶)) − (𝐶𝐺(𝐵 − 𝐶))) = (((𝐴𝐺𝐵) − (𝐴𝐺𝐶)) − (𝐶𝐺𝐵))) |
20 | 6, 8, 19 | 3eqtrd 2818 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐶)𝐺(𝐵 − 𝐶)) = (((𝐴𝐺𝐵) − (𝐴𝐺𝐶)) − (𝐶𝐺𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ‘cfv 6137 (class class class)co 6924 ↦ cmpt2 6926 ℂcc 10272 0cc0 10274 · cmul 10279 − cmin 10608 ∗ccj 14249 ℑcim 14251 |
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 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
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 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-po 5276 df-so 5277 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-2 11443 df-cj 14252 df-re 14253 df-im 14254 |
This theorem is referenced by: sigarperm 41990 |
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