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Mathbox for Saveliy Skresanov |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sigardiv | Structured version Visualization version GIF version |
Description: If signed area between vectors 𝐵 − 𝐴 and 𝐶 − 𝐴 is zero, then those vectors lie on the same line. (Contributed by Saveliy Skresanov, 22-Sep-2017.) |
Ref | Expression |
---|---|
sigar | ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) |
sigardiv.a | ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) |
sigardiv.b | ⊢ (𝜑 → ¬ 𝐶 = 𝐴) |
sigardiv.c | ⊢ (𝜑 → ((𝐵 − 𝐴)𝐺(𝐶 − 𝐴)) = 0) |
Ref | Expression |
---|---|
sigardiv | ⊢ (𝜑 → ((𝐵 − 𝐴) / (𝐶 − 𝐴)) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sigardiv.a | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) | |
2 | 1 | simp2d 1134 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
3 | 1 | simp1d 1133 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
4 | 2, 3 | subcld 10734 | . . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ) |
5 | 1 | simp3d 1135 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
6 | 5, 3 | subcld 10734 | . . . . . 6 ⊢ (𝜑 → (𝐶 − 𝐴) ∈ ℂ) |
7 | sigardiv.b | . . . . . . . 8 ⊢ (𝜑 → ¬ 𝐶 = 𝐴) | |
8 | 7 | neqned 2975 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
9 | 5, 3, 8 | subne0d 10743 | . . . . . 6 ⊢ (𝜑 → (𝐶 − 𝐴) ≠ 0) |
10 | 4, 6, 9 | cjdivd 14370 | . . . . 5 ⊢ (𝜑 → (∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴))) = ((∗‘(𝐵 − 𝐴)) / (∗‘(𝐶 − 𝐴)))) |
11 | 4 | cjcld 14343 | . . . . . . 7 ⊢ (𝜑 → (∗‘(𝐵 − 𝐴)) ∈ ℂ) |
12 | 6 | cjcld 14343 | . . . . . . 7 ⊢ (𝜑 → (∗‘(𝐶 − 𝐴)) ∈ ℂ) |
13 | 6, 9 | cjne0d 14350 | . . . . . . 7 ⊢ (𝜑 → (∗‘(𝐶 − 𝐴)) ≠ 0) |
14 | 11, 12, 6, 13, 9 | divcan5rd 11178 | . . . . . 6 ⊢ (𝜑 → (((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)) / ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴))) = ((∗‘(𝐵 − 𝐴)) / (∗‘(𝐶 − 𝐴)))) |
15 | 11, 6 | mulcld 10397 | . . . . . . . 8 ⊢ (𝜑 → ((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)) ∈ ℂ) |
16 | sigar | . . . . . . . . . . 11 ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) | |
17 | 16 | sigarval 41948 | . . . . . . . . . 10 ⊢ (((𝐵 − 𝐴) ∈ ℂ ∧ (𝐶 − 𝐴) ∈ ℂ) → ((𝐵 − 𝐴)𝐺(𝐶 − 𝐴)) = (ℑ‘((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)))) |
18 | 4, 6, 17 | syl2anc 579 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝐴)𝐺(𝐶 − 𝐴)) = (ℑ‘((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)))) |
19 | sigardiv.c | . . . . . . . . 9 ⊢ (𝜑 → ((𝐵 − 𝐴)𝐺(𝐶 − 𝐴)) = 0) | |
20 | 18, 19 | eqtr3d 2815 | . . . . . . . 8 ⊢ (𝜑 → (ℑ‘((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴))) = 0) |
21 | 15, 20 | reim0bd 14347 | . . . . . . 7 ⊢ (𝜑 → ((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)) ∈ ℝ) |
22 | 6, 12 | mulcomd 10398 | . . . . . . . 8 ⊢ (𝜑 → ((𝐶 − 𝐴) · (∗‘(𝐶 − 𝐴))) = ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴))) |
23 | 6 | cjmulrcld 14353 | . . . . . . . 8 ⊢ (𝜑 → ((𝐶 − 𝐴) · (∗‘(𝐶 − 𝐴))) ∈ ℝ) |
24 | 22, 23 | eqeltrrd 2859 | . . . . . . 7 ⊢ (𝜑 → ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴)) ∈ ℝ) |
25 | 12, 6, 13, 9 | mulne0d 11027 | . . . . . . 7 ⊢ (𝜑 → ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴)) ≠ 0) |
26 | 21, 24, 25 | redivcld 11203 | . . . . . 6 ⊢ (𝜑 → (((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)) / ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴))) ∈ ℝ) |
27 | 14, 26 | eqeltrrd 2859 | . . . . 5 ⊢ (𝜑 → ((∗‘(𝐵 − 𝐴)) / (∗‘(𝐶 − 𝐴))) ∈ ℝ) |
28 | 10, 27 | eqeltrd 2858 | . . . 4 ⊢ (𝜑 → (∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴))) ∈ ℝ) |
29 | 28 | cjred 14373 | . . 3 ⊢ (𝜑 → (∗‘(∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴)))) = (∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴)))) |
30 | 4, 6, 9 | divcld 11151 | . . . 4 ⊢ (𝜑 → ((𝐵 − 𝐴) / (𝐶 − 𝐴)) ∈ ℂ) |
31 | 30 | cjcjd 14346 | . . 3 ⊢ (𝜑 → (∗‘(∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴)))) = ((𝐵 − 𝐴) / (𝐶 − 𝐴))) |
32 | 29, 31 | eqtr3d 2815 | . 2 ⊢ (𝜑 → (∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴))) = ((𝐵 − 𝐴) / (𝐶 − 𝐴))) |
33 | 32, 28 | eqeltrrd 2859 | 1 ⊢ (𝜑 → ((𝐵 − 𝐴) / (𝐶 − 𝐴)) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ‘cfv 6135 (class class class)co 6922 ↦ cmpt2 6924 ℂcc 10270 ℝcr 10271 0cc0 10272 · cmul 10277 − cmin 10606 / cdiv 11032 ∗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: sigarcol 41962 sharhght 41963 |
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