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Mathbox for Saveliy Skresanov |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sigariz | Structured version Visualization version GIF version |
Description: If signed area is zero, the signed area with swapped arguments is also zero. Deduction version. (Contributed by Saveliy Skresanov, 23-Sep-2017.) |
Ref | Expression |
---|---|
sigarimcd.sigar | ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) |
sigarimcd.a | ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) |
sigariz.a | ⊢ (𝜑 → (𝐴𝐺𝐵) = 0) |
Ref | Expression |
---|---|
sigariz | ⊢ (𝜑 → (𝐵𝐺𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sigariz.a | . . . 4 ⊢ (𝜑 → (𝐴𝐺𝐵) = 0) | |
2 | sigarimcd.a | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)) | |
3 | sigarimcd.sigar | . . . . . 6 ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) | |
4 | 3 | sigarac 42538 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐺𝐵) = -(𝐵𝐺𝐴)) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐴𝐺𝐵) = -(𝐵𝐺𝐴)) |
6 | 1, 5 | eqtr3d 2817 | . . 3 ⊢ (𝜑 → 0 = -(𝐵𝐺𝐴)) |
7 | 6 | negeqd 10680 | . 2 ⊢ (𝜑 → -0 = --(𝐵𝐺𝐴)) |
8 | neg0 10733 | . . 3 ⊢ -0 = 0 | |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → -0 = 0) |
10 | 2 | ancomd 454 | . . . 4 ⊢ (𝜑 → (𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ)) |
11 | 3, 10 | sigarimcd 42548 | . . 3 ⊢ (𝜑 → (𝐵𝐺𝐴) ∈ ℂ) |
12 | 11 | negnegd 10789 | . 2 ⊢ (𝜑 → --(𝐵𝐺𝐴) = (𝐵𝐺𝐴)) |
13 | 7, 9, 12 | 3eqtr3rd 2824 | 1 ⊢ (𝜑 → (𝐵𝐺𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ‘cfv 6188 (class class class)co 6976 ∈ cmpo 6978 ℂcc 10333 0cc0 10335 · cmul 10340 -cneg 10671 ∗ccj 14316 ℑcim 14318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-po 5326 df-so 5327 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-2 11503 df-cj 14319 df-re 14320 df-im 14321 |
This theorem is referenced by: cevathlem2 42554 |
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