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 1067 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
3 | 1 | simp1d 1066 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
4 | 2, 3 | subcld 10271 | . . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ) |
5 | 1 | simp3d 1068 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
6 | 5, 3 | subcld 10271 | . . . . . 6 ⊢ (𝜑 → (𝐶 − 𝐴) ∈ ℂ) |
7 | sigardiv.b | . . . . . . . 8 ⊢ (𝜑 → ¬ 𝐶 = 𝐴) | |
8 | 7 | neqned 2789 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
9 | 5, 3, 8 | subne0d 10280 | . . . . . 6 ⊢ (𝜑 → (𝐶 − 𝐴) ≠ 0) |
10 | 4, 6, 9 | cjdivd 13811 | . . . . 5 ⊢ (𝜑 → (∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴))) = ((∗‘(𝐵 − 𝐴)) / (∗‘(𝐶 − 𝐴)))) |
11 | 4 | cjcld 13784 | . . . . . . 7 ⊢ (𝜑 → (∗‘(𝐵 − 𝐴)) ∈ ℂ) |
12 | 6 | cjcld 13784 | . . . . . . 7 ⊢ (𝜑 → (∗‘(𝐶 − 𝐴)) ∈ ℂ) |
13 | 6, 9 | cjne0d 13791 | . . . . . . 7 ⊢ (𝜑 → (∗‘(𝐶 − 𝐴)) ≠ 0) |
14 | 11, 12, 6, 13, 9 | divcan5rd 10707 | . . . . . 6 ⊢ (𝜑 → (((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)) / ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴))) = ((∗‘(𝐵 − 𝐴)) / (∗‘(𝐶 − 𝐴)))) |
15 | 11, 6 | mulcld 9939 | . . . . . . . 8 ⊢ (𝜑 → ((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)) ∈ ℂ) |
16 | sigar | . . . . . . . . . . 11 ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) | |
17 | 16 | sigarval 39688 | . . . . . . . . . 10 ⊢ (((𝐵 − 𝐴) ∈ ℂ ∧ (𝐶 − 𝐴) ∈ ℂ) → ((𝐵 − 𝐴)𝐺(𝐶 − 𝐴)) = (ℑ‘((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)))) |
18 | 4, 6, 17 | syl2anc 691 | . . . . . . . . 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 2646 | . . . . . . . 8 ⊢ (𝜑 → (ℑ‘((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴))) = 0) |
21 | 15, 20 | reim0bd 13788 | . . . . . . 7 ⊢ (𝜑 → ((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)) ∈ ℝ) |
22 | 6, 12 | mulcomd 9940 | . . . . . . . 8 ⊢ (𝜑 → ((𝐶 − 𝐴) · (∗‘(𝐶 − 𝐴))) = ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴))) |
23 | 6 | cjmulrcld 13794 | . . . . . . . 8 ⊢ (𝜑 → ((𝐶 − 𝐴) · (∗‘(𝐶 − 𝐴))) ∈ ℝ) |
24 | 22, 23 | eqeltrrd 2689 | . . . . . . 7 ⊢ (𝜑 → ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴)) ∈ ℝ) |
25 | 12, 6, 13, 9 | mulne0d 10558 | . . . . . . 7 ⊢ (𝜑 → ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴)) ≠ 0) |
26 | 21, 24, 25 | redivcld 10732 | . . . . . 6 ⊢ (𝜑 → (((∗‘(𝐵 − 𝐴)) · (𝐶 − 𝐴)) / ((∗‘(𝐶 − 𝐴)) · (𝐶 − 𝐴))) ∈ ℝ) |
27 | 14, 26 | eqeltrrd 2689 | . . . . 5 ⊢ (𝜑 → ((∗‘(𝐵 − 𝐴)) / (∗‘(𝐶 − 𝐴))) ∈ ℝ) |
28 | 10, 27 | eqeltrd 2688 | . . . 4 ⊢ (𝜑 → (∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴))) ∈ ℝ) |
29 | 28 | cjred 13814 | . . 3 ⊢ (𝜑 → (∗‘(∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴)))) = (∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴)))) |
30 | 4, 6, 9 | divcld 10680 | . . . 4 ⊢ (𝜑 → ((𝐵 − 𝐴) / (𝐶 − 𝐴)) ∈ ℂ) |
31 | 30 | cjcjd 13787 | . . 3 ⊢ (𝜑 → (∗‘(∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴)))) = ((𝐵 − 𝐴) / (𝐶 − 𝐴))) |
32 | 29, 31 | eqtr3d 2646 | . 2 ⊢ (𝜑 → (∗‘((𝐵 − 𝐴) / (𝐶 − 𝐴))) = ((𝐵 − 𝐴) / (𝐶 − 𝐴))) |
33 | 32, 28 | eqeltrrd 2689 | 1 ⊢ (𝜑 → ((𝐵 − 𝐴) / (𝐶 − 𝐴)) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ℂcc 9813 ℝcr 9814 0cc0 9815 · cmul 9820 − cmin 10145 / cdiv 10563 ∗ccj 13684 ℑcim 13686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-2 10956 df-cj 13687 df-re 13688 df-im 13689 |
This theorem is referenced by: sigarcol 39702 sharhght 39703 |
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