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Theorem ragcgr 25823
 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 2772 . . . 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 466 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐺 ∈ TarskiG)
7 israg.b . . . . . 6 (𝜑𝐵𝑃)
87adantr 466 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐵𝑃)
9 israg.c . . . . . 6 (𝜑𝐶𝑃)
109adantr 466 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐶𝑃)
11 ragcgr.e . . . . . 6 (𝜑𝐸𝑃)
1211adantr 466 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐸𝑃)
13 ragcgr.f . . . . . 6 (𝜑𝐹𝑃)
1413adantr 466 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐹𝑃)
15 ragcgr.c . . . . . 6 = (cgrG‘𝐺)
16 israg.a . . . . . . 7 (𝜑𝐴𝑃)
1716adantr 466 . . . . . 6 ((𝜑𝐵 = 𝐶) → 𝐴𝑃)
18 ragcgr.d . . . . . . 7 (𝜑𝐷𝑃)
1918adantr 466 . . . . . 6 ((𝜑𝐵 = 𝐶) → 𝐷𝑃)
20 ragcgr.2 . . . . . . 7 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
2120adantr 466 . . . . . 6 ((𝜑𝐵 = 𝐶) → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
222, 3, 4, 15, 6, 17, 8, 10, 19, 12, 14, 21cgr3simp2 25637 . . . . 5 ((𝜑𝐵 = 𝐶) → (𝐵 𝐶) = (𝐸 𝐹))
23 simpr 471 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐵 = 𝐶)
242, 3, 4, 6, 8, 10, 12, 14, 22, 23tgcgreq 25598 . . . 4 ((𝜑𝐵 = 𝐶) → 𝐸 = 𝐹)
25 eqidd 2772 . . . 4 ((𝜑𝐵 = 𝐶) → 𝐹 = 𝐹)
261, 24, 25s3eqd 13818 . . 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 25818 . . 3 ((𝜑𝐵 = 𝐶) → ⟨“𝐷𝐹𝐹”⟩ ∈ (∟G‘𝐺))
3026, 29eqeltrd 2850 . 2 ((𝜑𝐵 = 𝐶) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
31 ragcgr.1 . . . . . 6 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
3231adantr 466 . . . . 5 ((𝜑𝐵𝐶) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
335adantr 466 . . . . . 6 ((𝜑𝐵𝐶) → 𝐺 ∈ TarskiG)
3416adantr 466 . . . . . 6 ((𝜑𝐵𝐶) → 𝐴𝑃)
357adantr 466 . . . . . 6 ((𝜑𝐵𝐶) → 𝐵𝑃)
369adantr 466 . . . . . 6 ((𝜑𝐵𝐶) → 𝐶𝑃)
372, 3, 4, 27, 28, 33, 34, 35, 36israg 25813 . . . . 5 ((𝜑𝐵𝐶) → (⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝐴 𝐶) = (𝐴 ((𝑆𝐵)‘𝐶))))
3832, 37mpbid 222 . . . 4 ((𝜑𝐵𝐶) → (𝐴 𝐶) = (𝐴 ((𝑆𝐵)‘𝐶)))
3913adantr 466 . . . . 5 ((𝜑𝐵𝐶) → 𝐹𝑃)
4018adantr 466 . . . . 5 ((𝜑𝐵𝐶) → 𝐷𝑃)
4111adantr 466 . . . . . 6 ((𝜑𝐵𝐶) → 𝐸𝑃)
4220adantr 466 . . . . . 6 ((𝜑𝐵𝐶) → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
432, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42cgr3simp3 25638 . . . . 5 ((𝜑𝐵𝐶) → (𝐶 𝐴) = (𝐹 𝐷))
442, 3, 4, 33, 36, 34, 39, 40, 43tgcgrcomlr 25596 . . . 4 ((𝜑𝐵𝐶) → (𝐴 𝐶) = (𝐷 𝐹))
45 eqid 2771 . . . . . 6 (𝑆𝐵) = (𝑆𝐵)
462, 3, 4, 27, 28, 33, 35, 45, 36mircl 25777 . . . . 5 ((𝜑𝐵𝐶) → ((𝑆𝐵)‘𝐶) ∈ 𝑃)
47 eqid 2771 . . . . . 6 (𝑆𝐸) = (𝑆𝐸)
482, 3, 4, 27, 28, 33, 41, 47, 39mircl 25777 . . . . 5 ((𝜑𝐵𝐶) → ((𝑆𝐸)‘𝐹) ∈ 𝑃)
49 simpr 471 . . . . . . 7 ((𝜑𝐵𝐶) → 𝐵𝐶)
5049necomd 2998 . . . . . 6 ((𝜑𝐵𝐶) → 𝐶𝐵)
512, 3, 4, 27, 28, 33, 35, 45, 36mirbtwn 25774 . . . . . . 7 ((𝜑𝐵𝐶) → 𝐵 ∈ (((𝑆𝐵)‘𝐶)𝐼𝐶))
522, 3, 4, 33, 46, 35, 36, 51tgbtwncom 25604 . . . . . 6 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐶𝐼((𝑆𝐵)‘𝐶)))
532, 3, 4, 27, 28, 33, 41, 47, 39mirbtwn 25774 . . . . . . 7 ((𝜑𝐵𝐶) → 𝐸 ∈ (((𝑆𝐸)‘𝐹)𝐼𝐹))
542, 3, 4, 33, 48, 41, 39, 53tgbtwncom 25604 . . . . . 6 ((𝜑𝐵𝐶) → 𝐸 ∈ (𝐹𝐼((𝑆𝐸)‘𝐹)))
552, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42cgr3simp2 25637 . . . . . . 7 ((𝜑𝐵𝐶) → (𝐵 𝐶) = (𝐸 𝐹))
562, 3, 4, 33, 35, 36, 41, 39, 55tgcgrcomlr 25596 . . . . . 6 ((𝜑𝐵𝐶) → (𝐶 𝐵) = (𝐹 𝐸))
572, 3, 4, 27, 28, 33, 35, 45, 36mircgr 25773 . . . . . . 7 ((𝜑𝐵𝐶) → (𝐵 ((𝑆𝐵)‘𝐶)) = (𝐵 𝐶))
582, 3, 4, 27, 28, 33, 41, 47, 39mircgr 25773 . . . . . . 7 ((𝜑𝐵𝐶) → (𝐸 ((𝑆𝐸)‘𝐹)) = (𝐸 𝐹))
5955, 57, 583eqtr4d 2815 . . . . . 6 ((𝜑𝐵𝐶) → (𝐵 ((𝑆𝐵)‘𝐶)) = (𝐸 ((𝑆𝐸)‘𝐹)))
602, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42cgr3simp1 25636 . . . . . . 7 ((𝜑𝐵𝐶) → (𝐴 𝐵) = (𝐷 𝐸))
612, 3, 4, 33, 34, 35, 40, 41, 60tgcgrcomlr 25596 . . . . . 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 25585 . . . . 5 ((𝜑𝐵𝐶) → (((𝑆𝐵)‘𝐶) 𝐴) = (((𝑆𝐸)‘𝐹) 𝐷))
632, 3, 4, 33, 46, 34, 48, 40, 62tgcgrcomlr 25596 . . . 4 ((𝜑𝐵𝐶) → (𝐴 ((𝑆𝐵)‘𝐶)) = (𝐷 ((𝑆𝐸)‘𝐹)))
6438, 44, 633eqtr3d 2813 . . 3 ((𝜑𝐵𝐶) → (𝐷 𝐹) = (𝐷 ((𝑆𝐸)‘𝐹)))
652, 3, 4, 27, 28, 33, 40, 41, 39israg 25813 . . 3 ((𝜑𝐵𝐶) → (⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺) ↔ (𝐷 𝐹) = (𝐷 ((𝑆𝐸)‘𝐹))))
6664, 65mpbird 247 . 2 ((𝜑𝐵𝐶) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
6730, 66pm2.61dane 3030 1 (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145   ≠ wne 2943   class class class wbr 4787  ‘cfv 6030  (class class class)co 6796  ⟨“cs3 13796  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556  LineGclng 25557  cgrGccgrg 25626  pInvGcmir 25768  ∟Gcrag 25809 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7900  df-map 8015  df-pm 8016  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8969  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-3 11286  df-n0 11500  df-z 11585  df-uz 11894  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-concat 13497  df-s1 13498  df-s2 13802  df-s3 13803  df-trkgc 25568  df-trkgb 25569  df-trkgcb 25570  df-trkg 25573  df-cgrg 25627  df-mir 25769  df-rag 25810 This theorem is referenced by:  motrag  25824  footex  25834
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