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Theorem ragcgr 25502
Description: Right angle and colinearity. Theorem 8.10 of [Schwabhauser] p. 58. (Contributed by Thierry Arnoux, 4-Sep-2019.)
Hypotheses
Ref Expression
israg.p 𝑃 = (Base‘𝐺)
israg.d = (dist‘𝐺)
israg.i 𝐼 = (Itv‘𝐺)
israg.l 𝐿 = (LineG‘𝐺)
israg.s 𝑆 = (pInvG‘𝐺)
israg.g (𝜑𝐺 ∈ TarskiG)
israg.a (𝜑𝐴𝑃)
israg.b (𝜑𝐵𝑃)
israg.c (𝜑𝐶𝑃)
ragcgr.c = (cgrG‘𝐺)
ragcgr.d (𝜑𝐷𝑃)
ragcgr.e (𝜑𝐸𝑃)
ragcgr.f (𝜑𝐹𝑃)
ragcgr.1 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
ragcgr.2 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
Assertion
Ref Expression
ragcgr (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))

Proof of Theorem ragcgr
StepHypRef Expression
1 eqidd 2622 . . . 4 ((𝜑𝐵 = 𝐶) → 𝐷 = 𝐷)
2 israg.p . . . . 5 𝑃 = (Base‘𝐺)
3 israg.d . . . . 5 = (dist‘𝐺)
4 israg.i . . . . 5 𝐼 = (Itv‘𝐺)
5 israg.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
65adantr 481 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐺 ∈ TarskiG)
7 israg.b . . . . . 6 (𝜑𝐵𝑃)
87adantr 481 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐵𝑃)
9 israg.c . . . . . 6 (𝜑𝐶𝑃)
109adantr 481 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐶𝑃)
11 ragcgr.e . . . . . 6 (𝜑𝐸𝑃)
1211adantr 481 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐸𝑃)
13 ragcgr.f . . . . . 6 (𝜑𝐹𝑃)
1413adantr 481 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐹𝑃)
15 ragcgr.c . . . . . 6 = (cgrG‘𝐺)
16 israg.a . . . . . . 7 (𝜑𝐴𝑃)
1716adantr 481 . . . . . 6 ((𝜑𝐵 = 𝐶) → 𝐴𝑃)
18 ragcgr.d . . . . . . 7 (𝜑𝐷𝑃)
1918adantr 481 . . . . . 6 ((𝜑𝐵 = 𝐶) → 𝐷𝑃)
20 ragcgr.2 . . . . . . 7 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
2120adantr 481 . . . . . 6 ((𝜑𝐵 = 𝐶) → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
222, 3, 4, 15, 6, 17, 8, 10, 19, 12, 14, 21cgr3simp2 25316 . . . . 5 ((𝜑𝐵 = 𝐶) → (𝐵 𝐶) = (𝐸 𝐹))
23 simpr 477 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐵 = 𝐶)
242, 3, 4, 6, 8, 10, 12, 14, 22, 23tgcgreq 25277 . . . 4 ((𝜑𝐵 = 𝐶) → 𝐸 = 𝐹)
25 eqidd 2622 . . . 4 ((𝜑𝐵 = 𝐶) → 𝐹 = 𝐹)
261, 24, 25s3eqd 13546 . . 3 ((𝜑𝐵 = 𝐶) → ⟨“𝐷𝐸𝐹”⟩ = ⟨“𝐷𝐹𝐹”⟩)
27 israg.l . . . 4 𝐿 = (LineG‘𝐺)
28 israg.s . . . 4 𝑆 = (pInvG‘𝐺)
292, 3, 4, 27, 28, 6, 19, 14, 12ragtrivb 25497 . . 3 ((𝜑𝐵 = 𝐶) → ⟨“𝐷𝐹𝐹”⟩ ∈ (∟G‘𝐺))
3026, 29eqeltrd 2698 . 2 ((𝜑𝐵 = 𝐶) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
31 ragcgr.1 . . . . . 6 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
3231adantr 481 . . . . 5 ((𝜑𝐵𝐶) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
335adantr 481 . . . . . 6 ((𝜑𝐵𝐶) → 𝐺 ∈ TarskiG)
3416adantr 481 . . . . . 6 ((𝜑𝐵𝐶) → 𝐴𝑃)
357adantr 481 . . . . . 6 ((𝜑𝐵𝐶) → 𝐵𝑃)
369adantr 481 . . . . . 6 ((𝜑𝐵𝐶) → 𝐶𝑃)
372, 3, 4, 27, 28, 33, 34, 35, 36israg 25492 . . . . 5 ((𝜑𝐵𝐶) → (⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝐴 𝐶) = (𝐴 ((𝑆𝐵)‘𝐶))))
3832, 37mpbid 222 . . . 4 ((𝜑𝐵𝐶) → (𝐴 𝐶) = (𝐴 ((𝑆𝐵)‘𝐶)))
3913adantr 481 . . . . 5 ((𝜑𝐵𝐶) → 𝐹𝑃)
4018adantr 481 . . . . 5 ((𝜑𝐵𝐶) → 𝐷𝑃)
4111adantr 481 . . . . . 6 ((𝜑𝐵𝐶) → 𝐸𝑃)
4220adantr 481 . . . . . 6 ((𝜑𝐵𝐶) → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
432, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42cgr3simp3 25317 . . . . 5 ((𝜑𝐵𝐶) → (𝐶 𝐴) = (𝐹 𝐷))
442, 3, 4, 33, 36, 34, 39, 40, 43tgcgrcomlr 25275 . . . 4 ((𝜑𝐵𝐶) → (𝐴 𝐶) = (𝐷 𝐹))
45 eqid 2621 . . . . . 6 (𝑆𝐵) = (𝑆𝐵)
462, 3, 4, 27, 28, 33, 35, 45, 36mircl 25456 . . . . 5 ((𝜑𝐵𝐶) → ((𝑆𝐵)‘𝐶) ∈ 𝑃)
47 eqid 2621 . . . . . 6 (𝑆𝐸) = (𝑆𝐸)
482, 3, 4, 27, 28, 33, 41, 47, 39mircl 25456 . . . . 5 ((𝜑𝐵𝐶) → ((𝑆𝐸)‘𝐹) ∈ 𝑃)
49 simpr 477 . . . . . . 7 ((𝜑𝐵𝐶) → 𝐵𝐶)
5049necomd 2845 . . . . . 6 ((𝜑𝐵𝐶) → 𝐶𝐵)
512, 3, 4, 27, 28, 33, 35, 45, 36mirbtwn 25453 . . . . . . 7 ((𝜑𝐵𝐶) → 𝐵 ∈ (((𝑆𝐵)‘𝐶)𝐼𝐶))
522, 3, 4, 33, 46, 35, 36, 51tgbtwncom 25283 . . . . . 6 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐶𝐼((𝑆𝐵)‘𝐶)))
532, 3, 4, 27, 28, 33, 41, 47, 39mirbtwn 25453 . . . . . . 7 ((𝜑𝐵𝐶) → 𝐸 ∈ (((𝑆𝐸)‘𝐹)𝐼𝐹))
542, 3, 4, 33, 48, 41, 39, 53tgbtwncom 25283 . . . . . 6 ((𝜑𝐵𝐶) → 𝐸 ∈ (𝐹𝐼((𝑆𝐸)‘𝐹)))
552, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42cgr3simp2 25316 . . . . . . 7 ((𝜑𝐵𝐶) → (𝐵 𝐶) = (𝐸 𝐹))
562, 3, 4, 33, 35, 36, 41, 39, 55tgcgrcomlr 25275 . . . . . 6 ((𝜑𝐵𝐶) → (𝐶 𝐵) = (𝐹 𝐸))
572, 3, 4, 27, 28, 33, 35, 45, 36mircgr 25452 . . . . . . 7 ((𝜑𝐵𝐶) → (𝐵 ((𝑆𝐵)‘𝐶)) = (𝐵 𝐶))
582, 3, 4, 27, 28, 33, 41, 47, 39mircgr 25452 . . . . . . 7 ((𝜑𝐵𝐶) → (𝐸 ((𝑆𝐸)‘𝐹)) = (𝐸 𝐹))
5955, 57, 583eqtr4d 2665 . . . . . 6 ((𝜑𝐵𝐶) → (𝐵 ((𝑆𝐵)‘𝐶)) = (𝐸 ((𝑆𝐸)‘𝐹)))
602, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42cgr3simp1 25315 . . . . . . 7 ((𝜑𝐵𝐶) → (𝐴 𝐵) = (𝐷 𝐸))
612, 3, 4, 33, 34, 35, 40, 41, 60tgcgrcomlr 25275 . . . . . 6 ((𝜑𝐵𝐶) → (𝐵 𝐴) = (𝐸 𝐷))
622, 3, 4, 33, 36, 35, 46, 39, 41, 48, 34, 40, 50, 52, 54, 56, 59, 43, 61axtg5seg 25264 . . . . 5 ((𝜑𝐵𝐶) → (((𝑆𝐵)‘𝐶) 𝐴) = (((𝑆𝐸)‘𝐹) 𝐷))
632, 3, 4, 33, 46, 34, 48, 40, 62tgcgrcomlr 25275 . . . 4 ((𝜑𝐵𝐶) → (𝐴 ((𝑆𝐵)‘𝐶)) = (𝐷 ((𝑆𝐸)‘𝐹)))
6438, 44, 633eqtr3d 2663 . . 3 ((𝜑𝐵𝐶) → (𝐷 𝐹) = (𝐷 ((𝑆𝐸)‘𝐹)))
652, 3, 4, 27, 28, 33, 40, 41, 39israg 25492 . . 3 ((𝜑𝐵𝐶) → (⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺) ↔ (𝐷 𝐹) = (𝐷 ((𝑆𝐸)‘𝐹))))
6664, 65mpbird 247 . 2 ((𝜑𝐵𝐶) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
6730, 66pm2.61dane 2877 1 (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4613  cfv 5847  (class class class)co 6604  ⟨“cs3 13524  Basecbs 15781  distcds 15871  TarskiGcstrkg 25229  Itvcitv 25235  LineGclng 25236  cgrGccgrg 25305  pInvGcmir 25447  ∟Gcrag 25488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-concat 13240  df-s1 13241  df-s2 13530  df-s3 13531  df-trkgc 25247  df-trkgb 25248  df-trkgcb 25249  df-trkg 25252  df-cgrg 25306  df-mir 25448  df-rag 25489
This theorem is referenced by:  motrag  25503  footex  25513
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