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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 2738 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐴 = 𝐴) | |
| 2 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 = 𝐷) | |
| 3 | eqidd 2738 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐷 = 𝐷) | |
| 4 | 1, 2, 3 | s3eqd 14903 | . . 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 2737 | . . . . 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 28710 | . . . 4 ⊢ (𝜑 → 〈“𝐴𝐷𝐷”〉 ∈ (∟G‘𝐺)) | 
| 14 | 13 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 〈“𝐴𝐷𝐷”〉 ∈ (∟G‘𝐺)) | 
| 15 | 4, 14 | eqeltrd 2841 | . 2 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 〈“𝐴𝐶𝐷”〉 ∈ (∟G‘𝐺)) | 
| 16 | 10 | adantr 480 | . . 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 28558 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | 
| 20 | 5, 7, 8, 10, 11, 17, 19 | tgelrnln 28638 | . . . 4 ⊢ (𝜑 → (𝐴𝐿𝐵) ∈ ran 𝐿) | 
| 21 | 20 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → (𝐴𝐿𝐵) ∈ ran 𝐿) | 
| 22 | 5, 8, 7, 10, 20, 18 | tglnpt 28557 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | 
| 23 | 22 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 𝐶 ∈ 𝑃) | 
| 24 | 12 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 𝐷 ∈ 𝑃) | 
| 25 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 𝐶 ≠ 𝐷) | |
| 26 | 5, 7, 8, 16, 23, 24, 25 | tgelrnln 28638 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → (𝐶𝐿𝐷) ∈ ran 𝐿) | 
| 27 | 18 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 𝐶 ∈ (𝐴𝐿𝐵)) | 
| 28 | 5, 7, 8, 16, 23, 24, 25 | tglinerflx1 28641 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 𝐶 ∈ (𝐶𝐿𝐷)) | 
| 29 | 27, 28 | elind 4200 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 𝐶 ∈ ((𝐴𝐿𝐵) ∩ (𝐶𝐿𝐷))) | 
| 30 | 5, 7, 8, 10, 11, 17, 19 | tglinerflx1 28641 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐴𝐿𝐵)) | 
| 31 | 30 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 𝐴 ∈ (𝐴𝐿𝐵)) | 
| 32 | 5, 7, 8, 16, 23, 24, 25 | tglinerflx2 28642 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 𝐷 ∈ (𝐶𝐿𝐷)) | 
| 33 | perprag.5 | . . . 4 ⊢ (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐶𝐿𝐷)) | |
| 34 | 33 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐶𝐿𝐷)) | 
| 35 | 5, 6, 7, 8, 16, 21, 26, 29, 31, 32, 34 | isperp2d 28724 | . 2 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝐷) → 〈“𝐴𝐶𝐷”〉 ∈ (∟G‘𝐺)) | 
| 36 | 15, 35 | pm2.61dane 3029 | 1 ⊢ (𝜑 → 〈“𝐴𝐶𝐷”〉 ∈ (∟G‘𝐺)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ran crn 5686 ‘cfv 6561 (class class class)co 7431 〈“cs3 14881 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 LineGclng 28442 pInvGcmir 28660 ∟Gcrag 28701 ⟂Gcperpg 28703 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-trkgc 28456 df-trkgb 28457 df-trkgcb 28458 df-trkg 28461 df-cgrg 28519 df-mir 28661 df-rag 28702 df-perpg 28704 | 
| This theorem is referenced by: perpdragALT 28735 perpdrag 28736 colperpexlem3 28740 mideulem2 28742 opphllem 28743 opphllem5 28759 opphllem6 28760 trgcopy 28812 | 
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