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Theorem ragcgr 26605
 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 2759 . . . 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 484 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐺 ∈ TarskiG)
7 israg.b . . . . . 6 (𝜑𝐵𝑃)
87adantr 484 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐵𝑃)
9 israg.c . . . . . 6 (𝜑𝐶𝑃)
109adantr 484 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐶𝑃)
11 ragcgr.e . . . . . 6 (𝜑𝐸𝑃)
1211adantr 484 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐸𝑃)
13 ragcgr.f . . . . . 6 (𝜑𝐹𝑃)
1413adantr 484 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐹𝑃)
15 ragcgr.c . . . . . 6 = (cgrG‘𝐺)
16 israg.a . . . . . . 7 (𝜑𝐴𝑃)
1716adantr 484 . . . . . 6 ((𝜑𝐵 = 𝐶) → 𝐴𝑃)
18 ragcgr.d . . . . . . 7 (𝜑𝐷𝑃)
1918adantr 484 . . . . . 6 ((𝜑𝐵 = 𝐶) → 𝐷𝑃)
20 ragcgr.2 . . . . . . 7 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
2120adantr 484 . . . . . 6 ((𝜑𝐵 = 𝐶) → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
222, 3, 4, 15, 6, 17, 8, 10, 19, 12, 14, 21cgr3simp2 26419 . . . . 5 ((𝜑𝐵 = 𝐶) → (𝐵 𝐶) = (𝐸 𝐹))
23 simpr 488 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐵 = 𝐶)
242, 3, 4, 6, 8, 10, 12, 14, 22, 23tgcgreq 26380 . . . 4 ((𝜑𝐵 = 𝐶) → 𝐸 = 𝐹)
25 eqidd 2759 . . . 4 ((𝜑𝐵 = 𝐶) → 𝐹 = 𝐹)
261, 24, 25s3eqd 14278 . . 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 26600 . . 3 ((𝜑𝐵 = 𝐶) → ⟨“𝐷𝐹𝐹”⟩ ∈ (∟G‘𝐺))
3026, 29eqeltrd 2852 . 2 ((𝜑𝐵 = 𝐶) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
31 ragcgr.1 . . . . . 6 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
3231adantr 484 . . . . 5 ((𝜑𝐵𝐶) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
335adantr 484 . . . . . 6 ((𝜑𝐵𝐶) → 𝐺 ∈ TarskiG)
3416adantr 484 . . . . . 6 ((𝜑𝐵𝐶) → 𝐴𝑃)
357adantr 484 . . . . . 6 ((𝜑𝐵𝐶) → 𝐵𝑃)
369adantr 484 . . . . . 6 ((𝜑𝐵𝐶) → 𝐶𝑃)
372, 3, 4, 27, 28, 33, 34, 35, 36israg 26595 . . . . 5 ((𝜑𝐵𝐶) → (⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺) ↔ (𝐴 𝐶) = (𝐴 ((𝑆𝐵)‘𝐶))))
3832, 37mpbid 235 . . . 4 ((𝜑𝐵𝐶) → (𝐴 𝐶) = (𝐴 ((𝑆𝐵)‘𝐶)))
3913adantr 484 . . . . 5 ((𝜑𝐵𝐶) → 𝐹𝑃)
4018adantr 484 . . . . 5 ((𝜑𝐵𝐶) → 𝐷𝑃)
4111adantr 484 . . . . . 6 ((𝜑𝐵𝐶) → 𝐸𝑃)
4220adantr 484 . . . . . 6 ((𝜑𝐵𝐶) → ⟨“𝐴𝐵𝐶”⟩ ⟨“𝐷𝐸𝐹”⟩)
432, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42cgr3simp3 26420 . . . . 5 ((𝜑𝐵𝐶) → (𝐶 𝐴) = (𝐹 𝐷))
442, 3, 4, 33, 36, 34, 39, 40, 43tgcgrcomlr 26378 . . . 4 ((𝜑𝐵𝐶) → (𝐴 𝐶) = (𝐷 𝐹))
45 eqid 2758 . . . . . 6 (𝑆𝐵) = (𝑆𝐵)
462, 3, 4, 27, 28, 33, 35, 45, 36mircl 26559 . . . . 5 ((𝜑𝐵𝐶) → ((𝑆𝐵)‘𝐶) ∈ 𝑃)
47 eqid 2758 . . . . . 6 (𝑆𝐸) = (𝑆𝐸)
482, 3, 4, 27, 28, 33, 41, 47, 39mircl 26559 . . . . 5 ((𝜑𝐵𝐶) → ((𝑆𝐸)‘𝐹) ∈ 𝑃)
49 simpr 488 . . . . . . 7 ((𝜑𝐵𝐶) → 𝐵𝐶)
5049necomd 3006 . . . . . 6 ((𝜑𝐵𝐶) → 𝐶𝐵)
512, 3, 4, 27, 28, 33, 35, 45, 36mirbtwn 26556 . . . . . . 7 ((𝜑𝐵𝐶) → 𝐵 ∈ (((𝑆𝐵)‘𝐶)𝐼𝐶))
522, 3, 4, 33, 46, 35, 36, 51tgbtwncom 26386 . . . . . 6 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐶𝐼((𝑆𝐵)‘𝐶)))
532, 3, 4, 27, 28, 33, 41, 47, 39mirbtwn 26556 . . . . . . 7 ((𝜑𝐵𝐶) → 𝐸 ∈ (((𝑆𝐸)‘𝐹)𝐼𝐹))
542, 3, 4, 33, 48, 41, 39, 53tgbtwncom 26386 . . . . . 6 ((𝜑𝐵𝐶) → 𝐸 ∈ (𝐹𝐼((𝑆𝐸)‘𝐹)))
552, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42cgr3simp2 26419 . . . . . . 7 ((𝜑𝐵𝐶) → (𝐵 𝐶) = (𝐸 𝐹))
562, 3, 4, 33, 35, 36, 41, 39, 55tgcgrcomlr 26378 . . . . . 6 ((𝜑𝐵𝐶) → (𝐶 𝐵) = (𝐹 𝐸))
572, 3, 4, 27, 28, 33, 35, 45, 36mircgr 26555 . . . . . . 7 ((𝜑𝐵𝐶) → (𝐵 ((𝑆𝐵)‘𝐶)) = (𝐵 𝐶))
582, 3, 4, 27, 28, 33, 41, 47, 39mircgr 26555 . . . . . . 7 ((𝜑𝐵𝐶) → (𝐸 ((𝑆𝐸)‘𝐹)) = (𝐸 𝐹))
5955, 57, 583eqtr4d 2803 . . . . . 6 ((𝜑𝐵𝐶) → (𝐵 ((𝑆𝐵)‘𝐶)) = (𝐸 ((𝑆𝐸)‘𝐹)))
602, 3, 4, 15, 33, 34, 35, 36, 40, 41, 39, 42cgr3simp1 26418 . . . . . . 7 ((𝜑𝐵𝐶) → (𝐴 𝐵) = (𝐷 𝐸))
612, 3, 4, 33, 34, 35, 40, 41, 60tgcgrcomlr 26378 . . . . . 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 26363 . . . . 5 ((𝜑𝐵𝐶) → (((𝑆𝐵)‘𝐶) 𝐴) = (((𝑆𝐸)‘𝐹) 𝐷))
632, 3, 4, 33, 46, 34, 48, 40, 62tgcgrcomlr 26378 . . . 4 ((𝜑𝐵𝐶) → (𝐴 ((𝑆𝐵)‘𝐶)) = (𝐷 ((𝑆𝐸)‘𝐹)))
6438, 44, 633eqtr3d 2801 . . 3 ((𝜑𝐵𝐶) → (𝐷 𝐹) = (𝐷 ((𝑆𝐸)‘𝐹)))
652, 3, 4, 27, 28, 33, 40, 41, 39israg 26595 . . 3 ((𝜑𝐵𝐶) → (⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺) ↔ (𝐷 𝐹) = (𝐷 ((𝑆𝐸)‘𝐹))))
6664, 65mpbird 260 . 2 ((𝜑𝐵𝐶) → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
6730, 66pm2.61dane 3038 1 (𝜑 → ⟨“𝐷𝐸𝐹”⟩ ∈ (∟G‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951   class class class wbr 5035  ‘cfv 6339  (class class class)co 7155  ⟨“cs3 14256  Basecbs 16546  distcds 16637  TarskiGcstrkg 26328  Itvcitv 26334  LineGclng 26335  cgrGccgrg 26408  pInvGcmir 26550  ∟Gcrag 26591 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-1st 7698  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-1o 8117  df-er 8304  df-map 8423  df-pm 8424  df-en 8533  df-dom 8534  df-sdom 8535  df-fin 8536  df-card 9406  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-nn 11680  df-2 11742  df-3 11743  df-n0 11940  df-z 12026  df-uz 12288  df-fz 12945  df-fzo 13088  df-hash 13746  df-word 13919  df-concat 13975  df-s1 14002  df-s2 14262  df-s3 14263  df-trkgc 26346  df-trkgb 26347  df-trkgcb 26348  df-trkg 26351  df-cgrg 26409  df-mir 26551  df-rag 26592 This theorem is referenced by:  motrag  26606  footexALT  26616  footexlem1  26617  footexlem2  26618
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