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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 1142 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
3 | 1 | simp1d 1141 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
4 | 2, 3 | subcld 11618 | . . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ) |
5 | 1 | simp3d 1143 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
6 | 5, 3 | subcld 11618 | . . . . . 6 ⊢ (𝜑 → (𝐶 − 𝐴) ∈ ℂ) |
7 | sigardiv.b | . . . . . . . 8 ⊢ (𝜑 → ¬ 𝐶 = 𝐴) | |
8 | 7 | neqned 2945 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
9 | 5, 3, 8 | subne0d 11627 | . . . . . 6 ⊢ (𝜑 → (𝐶 − 𝐴) ≠ 0) |
10 | 4, 6, 9 | cjdivd 15259 | . . . . 5 ⊢ (𝜑 → (∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴))) = ((∗‘(𝐵 − 𝐴)) / (∗‘(𝐶 − 𝐴)))) |
11 | 4 | cjcld 15232 | . . . . . . 7 ⊢ (𝜑 → (∗‘(𝐵 − 𝐴)) ∈ ℂ) |
12 | 6 | cjcld 15232 | . . . . . . 7 ⊢ (𝜑 → (∗‘(𝐶 − 𝐴)) ∈ ℂ) |
13 | 6, 9 | cjne0d 15239 | . . . . . . 7 ⊢ (𝜑 → (∗‘(𝐶 − 𝐴)) ≠ 0) |
14 | 11, 12, 6, 13, 9 | divcan5rd 12068 | . . . . . 6 ⊢ (𝜑 → (((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)) / ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴))) = ((∗‘(𝐵 − 𝐴)) / (∗‘(𝐶 − 𝐴)))) |
15 | 11, 6 | mulcld 11279 | . . . . . . . 8 ⊢ (𝜑 → ((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)) ∈ ℂ) |
16 | sigar | . . . . . . . . . . 11 ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) | |
17 | 16 | sigarval 46806 | . . . . . . . . . 10 ⊢ (((𝐵 − 𝐴) ∈ ℂ ∧ (𝐶 − 𝐴) ∈ ℂ) → ((𝐵 − 𝐴)𝐺(𝐶 − 𝐴)) = (ℑ‘((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)))) |
18 | 4, 6, 17 | syl2anc 584 | . . . . . . . . 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 2777 | . . . . . . . 8 ⊢ (𝜑 → (ℑ‘((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴))) = 0) |
21 | 15, 20 | reim0bd 15236 | . . . . . . 7 ⊢ (𝜑 → ((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)) ∈ ℝ) |
22 | 6, 12 | mulcomd 11280 | . . . . . . . 8 ⊢ (𝜑 → ((𝐶 − 𝐴) · (∗‘(𝐶 − 𝐴))) = ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴))) |
23 | 6 | cjmulrcld 15242 | . . . . . . . 8 ⊢ (𝜑 → ((𝐶 − 𝐴) · (∗‘(𝐶 − 𝐴))) ∈ ℝ) |
24 | 22, 23 | eqeltrrd 2840 | . . . . . . 7 ⊢ (𝜑 → ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴)) ∈ ℝ) |
25 | 12, 6, 13, 9 | mulne0d 11913 | . . . . . . 7 ⊢ (𝜑 → ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴)) ≠ 0) |
26 | 21, 24, 25 | redivcld 12093 | . . . . . 6 ⊢ (𝜑 → (((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)) / ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴))) ∈ ℝ) |
27 | 14, 26 | eqeltrrd 2840 | . . . . 5 ⊢ (𝜑 → ((∗‘(𝐵 − 𝐴)) / (∗‘(𝐶 − 𝐴))) ∈ ℝ) |
28 | 10, 27 | eqeltrd 2839 | . . . 4 ⊢ (𝜑 → (∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴))) ∈ ℝ) |
29 | 28 | cjred 15262 | . . 3 ⊢ (𝜑 → (∗‘(∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴)))) = (∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴)))) |
30 | 4, 6, 9 | divcld 12041 | . . . 4 ⊢ (𝜑 → ((𝐵 − 𝐴) / (𝐶 − 𝐴)) ∈ ℂ) |
31 | 30 | cjcjd 15235 | . . 3 ⊢ (𝜑 → (∗‘(∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴)))) = ((𝐵 − 𝐴) / (𝐶 − 𝐴))) |
32 | 29, 31 | eqtr3d 2777 | . 2 ⊢ (𝜑 → (∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴))) = ((𝐵 − 𝐴) / (𝐶 − 𝐴))) |
33 | 32, 28 | eqeltrrd 2840 | 1 ⊢ (𝜑 → ((𝐵 − 𝐴) / (𝐶 − 𝐴)) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ℂcc 11151 ℝcr 11152 0cc0 11153 · cmul 11158 − cmin 11490 / cdiv 11918 ∗ccj 15132 ℑcim 15134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-2 12327 df-cj 15135 df-re 15136 df-im 15137 |
This theorem is referenced by: sigarcol 46820 sharhght 46821 |
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