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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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