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Mirrors > Home > MPE Home > Th. List > perprag | Structured version Visualization version GIF version |
Description: Deduce a right angle from perpendicular lines. (Contributed by Thierry Arnoux, 10-Nov-2019.) |
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) |
perprag.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
perprag.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
perprag.3 | ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐿𝐵)) |
perprag.4 | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
perprag.5 | ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐶𝐿𝐷)) |
Ref | Expression |
---|---|
perprag | ⊢ (𝜑 → 〈“𝐴𝐶𝐷”〉 ∈ (∟G‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2825 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐴 = 𝐴) | |
2 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 = 𝐷) | |
3 | eqidd 2825 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐷 = 𝐷) | |
4 | 1, 2, 3 | s3eqd 13984 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 〈“𝐴𝐶𝐷”〉 = 〈“𝐴𝐷𝐷”〉) |
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 2824 | . . . . 5 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺) | |
10 | colperpex.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
11 | perprag.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
12 | perprag.4 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
13 | 5, 6, 7, 8, 9, 10, 11, 12, 12 | ragtrivb 26013 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐷𝐷”〉 ∈ (∟G‘𝐺)) |
14 | 13 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 〈“𝐴𝐷𝐷”〉 ∈ (∟G‘𝐺)) |
15 | 4, 14 | eqeltrd 2905 | . 2 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 〈“𝐴𝐶𝐷”〉 ∈ (∟G‘𝐺)) |
16 | 10 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 𝐺 ∈ TarskiG) |
17 | perprag.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
18 | perprag.3 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐿𝐵)) | |
19 | 5, 8, 7, 10, 11, 17, 18 | tglngne 25861 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
20 | 5, 7, 8, 10, 11, 17, 19 | tgelrnln 25941 | . . . 4 ⊢ (𝜑 → (𝐴𝐿𝐵) ∈ ran 𝐿) |
21 | 20 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → (𝐴𝐿𝐵) ∈ ran 𝐿) |
22 | 5, 8, 7, 10, 20, 18 | tglnpt 25860 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
23 | 22 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 𝐶 ∈ 𝑃) |
24 | 12 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 𝐷 ∈ 𝑃) |
25 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 𝐶 ≠ 𝐷) | |
26 | 5, 7, 8, 16, 23, 24, 25 | tgelrnln 25941 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → (𝐶𝐿𝐷) ∈ ran 𝐿) |
27 | 18 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 𝐶 ∈ (𝐴𝐿𝐵)) |
28 | 5, 7, 8, 16, 23, 24, 25 | tglinerflx1 25944 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 𝐶 ∈ (𝐶𝐿𝐷)) |
29 | 27, 28 | elind 4024 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 𝐶 ∈ ((𝐴𝐿𝐵) ∩ (𝐶𝐿𝐷))) |
30 | 5, 7, 8, 10, 11, 17, 19 | tglinerflx1 25944 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐴𝐿𝐵)) |
31 | 30 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 𝐴 ∈ (𝐴𝐿𝐵)) |
32 | 5, 7, 8, 16, 23, 24, 25 | tglinerflx2 25945 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 𝐷 ∈ (𝐶𝐿𝐷)) |
33 | perprag.5 | . . . 4 ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐶𝐿𝐷)) | |
34 | 33 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐶𝐿𝐷)) |
35 | 5, 6, 7, 8, 16, 21, 26, 29, 31, 32, 34 | isperp2d 26027 | . 2 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 〈“𝐴𝐶𝐷”〉 ∈ (∟G‘𝐺)) |
36 | 15, 35 | pm2.61dane 3085 | 1 ⊢ (𝜑 → 〈“𝐴𝐶𝐷”〉 ∈ (∟G‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 2998 class class class wbr 4872 ran crn 5342 ‘cfv 6122 (class class class)co 6904 〈“cs3 13962 Basecbs 16221 distcds 16313 TarskiGcstrkg 25741 Itvcitv 25747 LineGclng 25748 pInvGcmir 25963 ∟Gcrag 26004 ⟂Gcperpg 26006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-oadd 7829 df-er 8008 df-map 8123 df-pm 8124 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-card 9077 df-cda 9304 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-nn 11350 df-2 11413 df-3 11414 df-n0 11618 df-xnn0 11690 df-z 11704 df-uz 11968 df-fz 12619 df-fzo 12760 df-hash 13410 df-word 13574 df-concat 13630 df-s1 13655 df-s2 13968 df-s3 13969 df-trkgc 25759 df-trkgb 25760 df-trkgcb 25761 df-trkg 25764 df-cgrg 25822 df-mir 25964 df-rag 26005 df-perpg 26007 |
This theorem is referenced by: perpdragALT 26035 perpdrag 26036 colperpexlem3 26040 mideulem2 26042 opphllem 26043 opphllem5 26059 opphllem6 26060 trgcopy 26112 |
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