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 1135 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
3 | 1 | simp1d 1134 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
4 | 2, 3 | subcld 10985 | . . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ) |
5 | 1 | simp3d 1136 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
6 | 5, 3 | subcld 10985 | . . . . . 6 ⊢ (𝜑 → (𝐶 − 𝐴) ∈ ℂ) |
7 | sigardiv.b | . . . . . . . 8 ⊢ (𝜑 → ¬ 𝐶 = 𝐴) | |
8 | 7 | neqned 3020 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
9 | 5, 3, 8 | subne0d 10994 | . . . . . 6 ⊢ (𝜑 → (𝐶 − 𝐴) ≠ 0) |
10 | 4, 6, 9 | cjdivd 14570 | . . . . 5 ⊢ (𝜑 → (∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴))) = ((∗‘(𝐵 − 𝐴)) / (∗‘(𝐶 − 𝐴)))) |
11 | 4 | cjcld 14543 | . . . . . . 7 ⊢ (𝜑 → (∗‘(𝐵 − 𝐴)) ∈ ℂ) |
12 | 6 | cjcld 14543 | . . . . . . 7 ⊢ (𝜑 → (∗‘(𝐶 − 𝐴)) ∈ ℂ) |
13 | 6, 9 | cjne0d 14550 | . . . . . . 7 ⊢ (𝜑 → (∗‘(𝐶 − 𝐴)) ≠ 0) |
14 | 11, 12, 6, 13, 9 | divcan5rd 11431 | . . . . . 6 ⊢ (𝜑 → (((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)) / ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴))) = ((∗‘(𝐵 − 𝐴)) / (∗‘(𝐶 − 𝐴)))) |
15 | 11, 6 | mulcld 10649 | . . . . . . . 8 ⊢ (𝜑 → ((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)) ∈ ℂ) |
16 | sigar | . . . . . . . . . . 11 ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) | |
17 | 16 | sigarval 42984 | . . . . . . . . . 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 2855 | . . . . . . . 8 ⊢ (𝜑 → (ℑ‘((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴))) = 0) |
21 | 15, 20 | reim0bd 14547 | . . . . . . 7 ⊢ (𝜑 → ((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)) ∈ ℝ) |
22 | 6, 12 | mulcomd 10650 | . . . . . . . 8 ⊢ (𝜑 → ((𝐶 − 𝐴) · (∗‘(𝐶 − 𝐴))) = ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴))) |
23 | 6 | cjmulrcld 14553 | . . . . . . . 8 ⊢ (𝜑 → ((𝐶 − 𝐴) · (∗‘(𝐶 − 𝐴))) ∈ ℝ) |
24 | 22, 23 | eqeltrrd 2911 | . . . . . . 7 ⊢ (𝜑 → ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴)) ∈ ℝ) |
25 | 12, 6, 13, 9 | mulne0d 11280 | . . . . . . 7 ⊢ (𝜑 → ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴)) ≠ 0) |
26 | 21, 24, 25 | redivcld 11456 | . . . . . 6 ⊢ (𝜑 → (((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)) / ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴))) ∈ ℝ) |
27 | 14, 26 | eqeltrrd 2911 | . . . . 5 ⊢ (𝜑 → ((∗‘(𝐵 − 𝐴)) / (∗‘(𝐶 − 𝐴))) ∈ ℝ) |
28 | 10, 27 | eqeltrd 2910 | . . . 4 ⊢ (𝜑 → (∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴))) ∈ ℝ) |
29 | 28 | cjred 14573 | . . 3 ⊢ (𝜑 → (∗‘(∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴)))) = (∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴)))) |
30 | 4, 6, 9 | divcld 11404 | . . . 4 ⊢ (𝜑 → ((𝐵 − 𝐴) / (𝐶 − 𝐴)) ∈ ℂ) |
31 | 30 | cjcjd 14546 | . . 3 ⊢ (𝜑 → (∗‘(∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴)))) = ((𝐵 − 𝐴) / (𝐶 − 𝐴))) |
32 | 29, 31 | eqtr3d 2855 | . 2 ⊢ (𝜑 → (∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴))) = ((𝐵 − 𝐴) / (𝐶 − 𝐴))) |
33 | 32, 28 | eqeltrrd 2911 | 1 ⊢ (𝜑 → ((𝐵 − 𝐴) / (𝐶 − 𝐴)) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 ℂcc 10523 ℝcr 10524 0cc0 10525 · cmul 10530 − cmin 10858 / cdiv 11285 ∗ccj 14443 ℑcim 14445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-2 11688 df-cj 14446 df-re 14447 df-im 14448 |
This theorem is referenced by: sigarcol 42998 sharhght 42999 |
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