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Mirrors > Home > MPE Home > Th. List > perpdragALT | Structured version Visualization version GIF version |
Description: Deduce a right angle from perpendicular lines. (Contributed by Thierry Arnoux, 12-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
colperpex.p | ⊢ 𝑃 = (Base‘𝐺) |
colperpex.d | ⊢ − = (dist‘𝐺) |
colperpex.i | ⊢ 𝐼 = (Itv‘𝐺) |
colperpex.l | ⊢ 𝐿 = (LineG‘𝐺) |
colperpex.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
perpdrag.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
perpdrag.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
perpdrag.3 | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
perpdrag.4 | ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐶)) |
Ref | Expression |
---|---|
perpdragALT | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2735 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐴) | |
2 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
3 | eqidd 2735 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐶) | |
4 | 1, 2, 3 | s3eqd 14412 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 〈“𝐴𝐴𝐶”〉 = 〈“𝐴𝐵𝐶”〉) |
5 | colperpex.p | . . . . 5 ⊢ 𝑃 = (Base‘𝐺) | |
6 | colperpex.d | . . . . 5 ⊢ − = (dist‘𝐺) | |
7 | colperpex.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
8 | colperpex.l | . . . . 5 ⊢ 𝐿 = (LineG‘𝐺) | |
9 | eqid 2734 | . . . . 5 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺) | |
10 | colperpex.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
11 | perpdrag.3 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
12 | perpdrag.4 | . . . . . . 7 ⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐶)) | |
13 | 8, 10, 12 | perpln1 26773 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) |
14 | perpdrag.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
15 | 5, 8, 7, 10, 13, 14 | tglnpt 26612 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
16 | 5, 6, 7, 8, 9, 10, 11, 15, 11 | ragtrivb 26765 | . . . . 5 ⊢ (𝜑 → 〈“𝐶𝐴𝐴”〉 ∈ (∟G‘𝐺)) |
17 | 5, 6, 7, 8, 9, 10, 11, 15, 15, 16 | ragcom 26761 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐴𝐶”〉 ∈ (∟G‘𝐺)) |
18 | 17 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 〈“𝐴𝐴𝐶”〉 ∈ (∟G‘𝐺)) |
19 | 4, 18 | eqeltrrd 2835 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) |
20 | 10 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐺 ∈ TarskiG) |
21 | 15 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑃) |
22 | perpdrag.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
23 | 5, 8, 7, 10, 13, 22 | tglnpt 26612 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
24 | 23 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑃) |
25 | 22 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝐷) |
26 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵) | |
27 | 13 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐷 ∈ ran 𝐿) |
28 | 14 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝐷) |
29 | 5, 7, 8, 20, 21, 24, 26, 26, 27, 28, 25 | tglinethru 26699 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐷 = (𝐴𝐿𝐵)) |
30 | 25, 29 | eleqtrd 2836 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ (𝐴𝐿𝐵)) |
31 | 11 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐶 ∈ 𝑃) |
32 | 12 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐶)) |
33 | 29, 32 | eqbrtrrd 5067 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐵𝐿𝐶)) |
34 | 5, 6, 7, 8, 20, 21, 24, 30, 31, 33 | perprag 26789 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) |
35 | 19, 34 | pm2.61dane 3022 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 class class class wbr 5043 ran crn 5541 ‘cfv 6369 (class class class)co 7202 〈“cs3 14390 Basecbs 16684 distcds 16776 TarskiGcstrkg 26493 Itvcitv 26499 LineGclng 26500 pInvGcmir 26715 ∟Gcrag 26756 ⟂Gcperpg 26758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-oadd 8195 df-er 8380 df-map 8499 df-pm 8500 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-dju 9500 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-xnn0 12146 df-z 12160 df-uz 12422 df-fz 13079 df-fzo 13222 df-hash 13880 df-word 14053 df-concat 14109 df-s1 14136 df-s2 14396 df-s3 14397 df-trkgc 26511 df-trkgb 26512 df-trkgcb 26513 df-trkg 26516 df-cgrg 26574 df-mir 26716 df-rag 26757 df-perpg 26759 |
This theorem is referenced by: (None) |
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