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Mathbox for Saveliy Skresanov |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sigarid | Structured version Visualization version GIF version |
Description: Signed area of a flat parallelogram is zero. (Contributed by Saveliy Skresanov, 20-Sep-2017.) |
Ref | Expression |
---|---|
sigar | ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) |
Ref | Expression |
---|---|
sigarid | ⊢ (𝐴 ∈ ℂ → (𝐴𝐺𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sigar | . . . 4 ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) | |
2 | 1 | sigarval 46838 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐴𝐺𝐴) = (ℑ‘((∗‘𝐴) · 𝐴))) |
3 | 2 | anidms 566 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴𝐺𝐴) = (ℑ‘((∗‘𝐴) · 𝐴))) |
4 | cjcl 15140 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
5 | id 22 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
6 | 4, 5 | mulcomd 11278 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((∗‘𝐴) · 𝐴) = (𝐴 · (∗‘𝐴))) |
7 | cjmulrcl 15179 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · (∗‘𝐴)) ∈ ℝ) | |
8 | 6, 7 | eqeltrd 2840 | . . 3 ⊢ (𝐴 ∈ ℂ → ((∗‘𝐴) · 𝐴) ∈ ℝ) |
9 | 8 | reim0d 15260 | . 2 ⊢ (𝐴 ∈ ℂ → (ℑ‘((∗‘𝐴) · 𝐴)) = 0) |
10 | 3, 9 | eqtrd 2776 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴𝐺𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6559 (class class class)co 7429 ∈ cmpo 7431 ℂcc 11149 ℝcr 11150 0cc0 11151 · cmul 11156 ∗ccj 15131 ℑcim 15133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-po 5590 df-so 5591 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-er 8741 df-en 8982 df-dom 8983 df-sdom 8984 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-div 11917 df-2 12325 df-cj 15134 df-re 15135 df-im 15136 |
This theorem is referenced by: sigarexp 46847 sigarcol 46852 |
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