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Theorem ragcgr 28730
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 2736 . . . 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 480 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐺 ∈ TarskiG)
7 israg.b . . . . . 6 (𝜑𝐵𝑃)
87adantr 480 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐵𝑃)
9 israg.c . . . . . 6 (𝜑𝐶𝑃)
109adantr 480 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐶𝑃)
11 ragcgr.e . . . . . 6 (𝜑𝐸𝑃)
1211adantr 480 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐸𝑃)
13 ragcgr.f . . . . . 6 (𝜑𝐹𝑃)
1413adantr 480 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐹𝑃)
15 ragcgr.c . . . . . 6 = (cgrG‘𝐺)
16 israg.a . . . . . . 7 (𝜑𝐴𝑃)
1716adantr 480 . . . . . 6 ((𝜑𝐵 = 𝐶) → 𝐴𝑃)
18 ragcgr.d . . . . . . 7 (𝜑𝐷𝑃)
1918adantr 480 . . . . . 6 ((𝜑𝐵 = 𝐶) → 𝐷𝑃)
20 ragcgr.2 . . . . . . 7 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
2120adantr 480 . . . . . 6 ((𝜑𝐵 = 𝐶) → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
222, 3, 4, 15, 6, 17, 8, 10, 19, 12, 14, 21cgr3simp2 28544 . . . . 5 ((𝜑𝐵 = 𝐶) → (𝐵 𝐶) = (𝐸 𝐹))
23 simpr 484 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐵 = 𝐶)
242, 3, 4, 6, 8, 10, 12, 14, 22, 23tgcgreq 28505 . . . 4 ((𝜑𝐵 = 𝐶) → 𝐸 = 𝐹)
25 eqidd 2736 . . . 4 ((𝜑𝐵 = 𝐶) → 𝐹 = 𝐹)
261, 24, 25s3eqd 14900 . . 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 28725 . . 3 ((𝜑𝐵 = 𝐶) → ⟨“𝐷𝐹𝐹”⟩ ∈ (∟G‘𝐺))
3026, 29eqeltrd 2839 . 2 ((𝜑𝐵 = 𝐶) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
31 ragcgr.1 . . . . . 6 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
3231adantr 480 . . . . 5 ((𝜑𝐵𝐶) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
335adantr 480 . . . . . 6 ((𝜑𝐵𝐶) → 𝐺 ∈ TarskiG)
3416adantr 480 . . . . . 6 ((𝜑𝐵𝐶) → 𝐴𝑃)
357adantr 480 . . . . . 6 ((𝜑𝐵𝐶) → 𝐵𝑃)
369adantr 480 . . . . . 6 ((𝜑𝐵𝐶) → 𝐶𝑃)
372, 3, 4, 27, 28, 33, 34, 35, 36israg 28720 . . . . 5 ((𝜑𝐵𝐶) → (⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝐴 𝐶) = (𝐴 ((𝑆𝐵)‘𝐶))))
3832, 37mpbid 232 . . . 4 ((𝜑𝐵𝐶) → (𝐴 𝐶) = (𝐴 ((𝑆𝐵)‘𝐶)))
3913adantr 480 . . . . 5 ((𝜑𝐵𝐶) → 𝐹𝑃)
4018adantr 480 . . . . 5 ((𝜑𝐵𝐶) → 𝐷𝑃)
4111adantr 480 . . . . . 6 ((𝜑𝐵𝐶) → 𝐸𝑃)
4220adantr 480 . . . . . 6 ((𝜑𝐵𝐶) → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
432, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42cgr3simp3 28545 . . . . 5 ((𝜑𝐵𝐶) → (𝐶 𝐴) = (𝐹 𝐷))
442, 3, 4, 33, 36, 34, 39, 40, 43tgcgrcomlr 28503 . . . 4 ((𝜑𝐵𝐶) → (𝐴 𝐶) = (𝐷 𝐹))
45 eqid 2735 . . . . . 6 (𝑆𝐵) = (𝑆𝐵)
462, 3, 4, 27, 28, 33, 35, 45, 36mircl 28684 . . . . 5 ((𝜑𝐵𝐶) → ((𝑆𝐵)‘𝐶) ∈ 𝑃)
47 eqid 2735 . . . . . 6 (𝑆𝐸) = (𝑆𝐸)
482, 3, 4, 27, 28, 33, 41, 47, 39mircl 28684 . . . . 5 ((𝜑𝐵𝐶) → ((𝑆𝐸)‘𝐹) ∈ 𝑃)
49 simpr 484 . . . . . . 7 ((𝜑𝐵𝐶) → 𝐵𝐶)
5049necomd 2994 . . . . . 6 ((𝜑𝐵𝐶) → 𝐶𝐵)
512, 3, 4, 27, 28, 33, 35, 45, 36mirbtwn 28681 . . . . . . 7 ((𝜑𝐵𝐶) → 𝐵 ∈ (((𝑆𝐵)‘𝐶)𝐼𝐶))
522, 3, 4, 33, 46, 35, 36, 51tgbtwncom 28511 . . . . . 6 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐶𝐼((𝑆𝐵)‘𝐶)))
532, 3, 4, 27, 28, 33, 41, 47, 39mirbtwn 28681 . . . . . . 7 ((𝜑𝐵𝐶) → 𝐸 ∈ (((𝑆𝐸)‘𝐹)𝐼𝐹))
542, 3, 4, 33, 48, 41, 39, 53tgbtwncom 28511 . . . . . 6 ((𝜑𝐵𝐶) → 𝐸 ∈ (𝐹𝐼((𝑆𝐸)‘𝐹)))
552, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42cgr3simp2 28544 . . . . . . 7 ((𝜑𝐵𝐶) → (𝐵 𝐶) = (𝐸 𝐹))
562, 3, 4, 33, 35, 36, 41, 39, 55tgcgrcomlr 28503 . . . . . 6 ((𝜑𝐵𝐶) → (𝐶 𝐵) = (𝐹 𝐸))
572, 3, 4, 27, 28, 33, 35, 45, 36mircgr 28680 . . . . . . 7 ((𝜑𝐵𝐶) → (𝐵 ((𝑆𝐵)‘𝐶)) = (𝐵 𝐶))
582, 3, 4, 27, 28, 33, 41, 47, 39mircgr 28680 . . . . . . 7 ((𝜑𝐵𝐶) → (𝐸 ((𝑆𝐸)‘𝐹)) = (𝐸 𝐹))
5955, 57, 583eqtr4d 2785 . . . . . 6 ((𝜑𝐵𝐶) → (𝐵 ((𝑆𝐵)‘𝐶)) = (𝐸 ((𝑆𝐸)‘𝐹)))
602, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42cgr3simp1 28543 . . . . . . 7 ((𝜑𝐵𝐶) → (𝐴 𝐵) = (𝐷 𝐸))
612, 3, 4, 33, 34, 35, 40, 41, 60tgcgrcomlr 28503 . . . . . 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 28488 . . . . 5 ((𝜑𝐵𝐶) → (((𝑆𝐵)‘𝐶) 𝐴) = (((𝑆𝐸)‘𝐹) 𝐷))
632, 3, 4, 33, 46, 34, 48, 40, 62tgcgrcomlr 28503 . . . 4 ((𝜑𝐵𝐶) → (𝐴 ((𝑆𝐵)‘𝐶)) = (𝐷 ((𝑆𝐸)‘𝐹)))
6438, 44, 633eqtr3d 2783 . . 3 ((𝜑𝐵𝐶) → (𝐷 𝐹) = (𝐷 ((𝑆𝐸)‘𝐹)))
652, 3, 4, 27, 28, 33, 40, 41, 39israg 28720 . . 3 ((𝜑𝐵𝐶) → (⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺) ↔ (𝐷 𝐹) = (𝐷 ((𝑆𝐸)‘𝐹))))
6664, 65mpbird 257 . 2 ((𝜑𝐵𝐶) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
6730, 66pm2.61dane 3027 1 (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938   class class class wbr 5148  cfv 6563  (class class class)co 7431  ⟨“cs3 14878  Basecbs 17245  distcds 17307  TarskiGcstrkg 28450  Itvcitv 28456  LineGclng 28457  cgrGccgrg 28533  pInvGcmir 28675  ∟Gcrag 28716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-concat 14606  df-s1 14631  df-s2 14884  df-s3 14885  df-trkgc 28471  df-trkgb 28472  df-trkgcb 28473  df-trkg 28476  df-cgrg 28534  df-mir 28676  df-rag 28717
This theorem is referenced by:  motrag  28731  footexALT  28741  footexlem1  28742  footexlem2  28743
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