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Theorem colperpexlem1 27091
Description: Lemma for colperp 27090. First part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 27-Oct-2019.)
Hypotheses
Ref Expression
colperpex.p 𝑃 = (Base‘𝐺)
colperpex.d = (dist‘𝐺)
colperpex.i 𝐼 = (Itv‘𝐺)
colperpex.l 𝐿 = (LineG‘𝐺)
colperpex.g (𝜑𝐺 ∈ TarskiG)
colperpexlem.s 𝑆 = (pInvG‘𝐺)
colperpexlem.m 𝑀 = (𝑆𝐴)
colperpexlem.n 𝑁 = (𝑆𝐵)
colperpexlem.k 𝐾 = (𝑆𝑄)
colperpexlem.a (𝜑𝐴𝑃)
colperpexlem.b (𝜑𝐵𝑃)
colperpexlem.c (𝜑𝐶𝑃)
colperpexlem.q (𝜑𝑄𝑃)
colperpexlem.1 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
colperpexlem.2 (𝜑 → (𝐾‘(𝑀𝐶)) = (𝑁𝐶))
Assertion
Ref Expression
colperpexlem1 (𝜑 → ⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺))

Proof of Theorem colperpexlem1
StepHypRef Expression
1 colperpex.p . . . 4 𝑃 = (Base‘𝐺)
2 colperpex.d . . . 4 = (dist‘𝐺)
3 colperpex.i . . . 4 𝐼 = (Itv‘𝐺)
4 colperpex.g . . . 4 (𝜑𝐺 ∈ TarskiG)
5 colperpexlem.q . . . 4 (𝜑𝑄𝑃)
6 colperpexlem.b . . . 4 (𝜑𝐵𝑃)
7 colperpex.l . . . . 5 𝐿 = (LineG‘𝐺)
8 colperpexlem.s . . . . 5 𝑆 = (pInvG‘𝐺)
9 colperpexlem.a . . . . 5 (𝜑𝐴𝑃)
10 colperpexlem.m . . . . 5 𝑀 = (𝑆𝐴)
111, 2, 3, 7, 8, 4, 9, 10, 5mircl 27022 . . . 4 (𝜑 → (𝑀𝑄) ∈ 𝑃)
12 colperpexlem.c . . . . . 6 (𝜑𝐶𝑃)
131, 2, 3, 7, 8, 4, 9, 10, 12mircl 27022 . . . . 5 (𝜑 → (𝑀𝐶) ∈ 𝑃)
14 eqid 2738 . . . . . 6 (𝑆𝐵) = (𝑆𝐵)
151, 2, 3, 7, 8, 4, 6, 14, 12mircl 27022 . . . . 5 (𝜑 → ((𝑆𝐵)‘𝐶) ∈ 𝑃)
161, 2, 3, 7, 8, 4, 9, 10, 15mircl 27022 . . . . 5 (𝜑 → (𝑀‘((𝑆𝐵)‘𝐶)) ∈ 𝑃)
17 colperpexlem.2 . . . . . . . 8 (𝜑 → (𝐾‘(𝑀𝐶)) = (𝑁𝐶))
18 colperpexlem.n . . . . . . . . 9 𝑁 = (𝑆𝐵)
191, 2, 3, 7, 8, 4, 6, 18, 12mircl 27022 . . . . . . . 8 (𝜑 → (𝑁𝐶) ∈ 𝑃)
2017, 19eqeltrd 2839 . . . . . . 7 (𝜑 → (𝐾‘(𝑀𝐶)) ∈ 𝑃)
21 colperpexlem.k . . . . . . . 8 𝐾 = (𝑆𝑄)
221, 2, 3, 7, 8, 4, 5, 21, 13mirbtwn 27019 . . . . . . 7 (𝜑𝑄 ∈ ((𝐾‘(𝑀𝐶))𝐼(𝑀𝐶)))
231, 2, 3, 4, 20, 5, 13, 22tgbtwncom 26849 . . . . . 6 (𝜑𝑄 ∈ ((𝑀𝐶)𝐼(𝐾‘(𝑀𝐶))))
2418fveq1i 6775 . . . . . . . 8 (𝑁𝐶) = ((𝑆𝐵)‘𝐶)
2517, 24eqtrdi 2794 . . . . . . 7 (𝜑 → (𝐾‘(𝑀𝐶)) = ((𝑆𝐵)‘𝐶))
2625oveq2d 7291 . . . . . 6 (𝜑 → ((𝑀𝐶)𝐼(𝐾‘(𝑀𝐶))) = ((𝑀𝐶)𝐼((𝑆𝐵)‘𝐶)))
2723, 26eleqtrd 2841 . . . . 5 (𝜑𝑄 ∈ ((𝑀𝐶)𝐼((𝑆𝐵)‘𝐶)))
281, 2, 3, 4, 13, 5, 15, 27tgbtwncom 26849 . . . . . . 7 (𝜑𝑄 ∈ (((𝑆𝐵)‘𝐶)𝐼(𝑀𝐶)))
291, 2, 3, 7, 8, 4, 9, 10, 15, 5, 13, 28mirbtwni 27032 . . . . . 6 (𝜑 → (𝑀𝑄) ∈ ((𝑀‘((𝑆𝐵)‘𝐶))𝐼(𝑀‘(𝑀𝐶))))
301, 2, 3, 7, 8, 4, 9, 10, 12mirmir 27023 . . . . . . 7 (𝜑 → (𝑀‘(𝑀𝐶)) = 𝐶)
3130oveq2d 7291 . . . . . 6 (𝜑 → ((𝑀‘((𝑆𝐵)‘𝐶))𝐼(𝑀‘(𝑀𝐶))) = ((𝑀‘((𝑆𝐵)‘𝐶))𝐼𝐶))
3229, 31eleqtrd 2841 . . . . 5 (𝜑 → (𝑀𝑄) ∈ ((𝑀‘((𝑆𝐵)‘𝐶))𝐼𝐶))
331, 2, 3, 4, 13, 15axtgcgrrflx 26823 . . . . . 6 (𝜑 → ((𝑀𝐶) ((𝑆𝐵)‘𝐶)) = (((𝑆𝐵)‘𝐶) (𝑀𝐶)))
341, 2, 3, 7, 8, 4, 9, 10, 15, 13miriso 27031 . . . . . 6 (𝜑 → ((𝑀‘((𝑆𝐵)‘𝐶)) (𝑀‘(𝑀𝐶))) = (((𝑆𝐵)‘𝐶) (𝑀𝐶)))
3530oveq2d 7291 . . . . . 6 (𝜑 → ((𝑀‘((𝑆𝐵)‘𝐶)) (𝑀‘(𝑀𝐶))) = ((𝑀‘((𝑆𝐵)‘𝐶)) 𝐶))
3633, 34, 353eqtr2d 2784 . . . . 5 (𝜑 → ((𝑀𝐶) ((𝑆𝐵)‘𝐶)) = ((𝑀‘((𝑆𝐵)‘𝐶)) 𝐶))
3725oveq2d 7291 . . . . . . 7 (𝜑 → (𝑄 (𝐾‘(𝑀𝐶))) = (𝑄 ((𝑆𝐵)‘𝐶)))
381, 2, 3, 7, 8, 4, 5, 21, 13mircgr 27018 . . . . . . 7 (𝜑 → (𝑄 (𝐾‘(𝑀𝐶))) = (𝑄 (𝑀𝐶)))
3937, 38eqtr3d 2780 . . . . . 6 (𝜑 → (𝑄 ((𝑆𝐵)‘𝐶)) = (𝑄 (𝑀𝐶)))
401, 2, 3, 7, 8, 4, 9, 10, 5, 13miriso 27031 . . . . . 6 (𝜑 → ((𝑀𝑄) (𝑀‘(𝑀𝐶))) = (𝑄 (𝑀𝐶)))
4130oveq2d 7291 . . . . . 6 (𝜑 → ((𝑀𝑄) (𝑀‘(𝑀𝐶))) = ((𝑀𝑄) 𝐶))
4239, 40, 413eqtr2d 2784 . . . . 5 (𝜑 → (𝑄 ((𝑆𝐵)‘𝐶)) = ((𝑀𝑄) 𝐶))
431, 2, 3, 7, 8, 4, 9, 10, 6mirmir 27023 . . . . . . . . . 10 (𝜑 → (𝑀‘(𝑀𝐵)) = 𝐵)
44 eqidd 2739 . . . . . . . . . 10 (𝜑 → (𝑀𝐵) = (𝑀𝐵))
45 eqidd 2739 . . . . . . . . . 10 (𝜑 → (𝑀𝐶) = (𝑀𝐶))
4643, 44, 45s3eqd 14577 . . . . . . . . 9 (𝜑 → ⟨“(𝑀‘(𝑀𝐵))(𝑀𝐵)(𝑀𝐶)”⟩ = ⟨“𝐵(𝑀𝐵)(𝑀𝐶)”⟩)
471, 2, 3, 7, 8, 4, 9, 10, 6mircl 27022 . . . . . . . . . 10 (𝜑 → (𝑀𝐵) ∈ 𝑃)
48 simpr 485 . . . . . . . . . . . . . . 15 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
4948fveq2d 6778 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐵) → (𝑀𝐴) = (𝑀𝐵))
504adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
519adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝐴 = 𝐵) → 𝐴𝑃)
521, 2, 3, 7, 8, 50, 51, 10mircinv 27029 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐵) → (𝑀𝐴) = 𝐴)
5349, 52eqtr3d 2780 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐵) → (𝑀𝐵) = 𝐴)
54 eqidd 2739 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐵) → 𝐵 = 𝐵)
55 eqidd 2739 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐵) → 𝐶 = 𝐶)
5653, 54, 55s3eqd 14577 . . . . . . . . . . . 12 ((𝜑𝐴 = 𝐵) → ⟨“(𝑀𝐵)𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩)
57 colperpexlem.1 . . . . . . . . . . . . 13 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
5857adantr 481 . . . . . . . . . . . 12 ((𝜑𝐴 = 𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
5956, 58eqeltrd 2839 . . . . . . . . . . 11 ((𝜑𝐴 = 𝐵) → ⟨“(𝑀𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
604adantr 481 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐺 ∈ TarskiG)
619adantr 481 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐴𝑃)
626adantr 481 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐵𝑃)
6312adantr 481 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐶𝑃)
641, 2, 3, 7, 8, 60, 61, 10, 62mircl 27022 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → (𝑀𝐵) ∈ 𝑃)
6557adantr 481 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
66 simpr 485 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐴𝐵)
671, 2, 3, 7, 8, 60, 61, 10, 62mirbtwn 27019 . . . . . . . . . . . . . 14 ((𝜑𝐴𝐵) → 𝐴 ∈ ((𝑀𝐵)𝐼𝐵))
681, 7, 3, 60, 64, 62, 61, 67btwncolg1 26916 . . . . . . . . . . . . 13 ((𝜑𝐴𝐵) → (𝐴 ∈ ((𝑀𝐵)𝐿𝐵) ∨ (𝑀𝐵) = 𝐵))
691, 7, 3, 60, 64, 62, 61, 68colcom 26919 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → (𝐴 ∈ (𝐵𝐿(𝑀𝐵)) ∨ 𝐵 = (𝑀𝐵)))
701, 2, 3, 7, 8, 60, 61, 62, 63, 64, 65, 66, 69ragcol 27060 . . . . . . . . . . 11 ((𝜑𝐴𝐵) → ⟨“(𝑀𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
7159, 70pm2.61dane 3032 . . . . . . . . . 10 (𝜑 → ⟨“(𝑀𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
721, 2, 3, 7, 8, 4, 47, 6, 12, 71, 10, 9mirrag 27062 . . . . . . . . 9 (𝜑 → ⟨“(𝑀‘(𝑀𝐵))(𝑀𝐵)(𝑀𝐶)”⟩ ∈ (∟G‘𝐺))
7346, 72eqeltrrd 2840 . . . . . . . 8 (𝜑 → ⟨“𝐵(𝑀𝐵)(𝑀𝐶)”⟩ ∈ (∟G‘𝐺))
741, 2, 3, 7, 8, 4, 6, 47, 13israg 27058 . . . . . . . 8 (𝜑 → (⟨“𝐵(𝑀𝐵)(𝑀𝐶)”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 (𝑀𝐶)) = (𝐵 ((𝑆‘(𝑀𝐵))‘(𝑀𝐶)))))
7573, 74mpbid 231 . . . . . . 7 (𝜑 → (𝐵 (𝑀𝐶)) = (𝐵 ((𝑆‘(𝑀𝐵))‘(𝑀𝐶))))
761, 2, 3, 7, 8, 4, 9, 10, 12, 6mirmir2 27035 . . . . . . . 8 (𝜑 → (𝑀‘((𝑆𝐵)‘𝐶)) = ((𝑆‘(𝑀𝐵))‘(𝑀𝐶)))
7776oveq2d 7291 . . . . . . 7 (𝜑 → (𝐵 (𝑀‘((𝑆𝐵)‘𝐶))) = (𝐵 ((𝑆‘(𝑀𝐵))‘(𝑀𝐶))))
7875, 77eqtr4d 2781 . . . . . 6 (𝜑 → (𝐵 (𝑀𝐶)) = (𝐵 (𝑀‘((𝑆𝐵)‘𝐶))))
791, 2, 3, 4, 6, 13, 6, 16, 78tgcgrcomlr 26841 . . . . 5 (𝜑 → ((𝑀𝐶) 𝐵) = ((𝑀‘((𝑆𝐵)‘𝐶)) 𝐵))
801, 2, 3, 7, 8, 4, 6, 14, 12mircgr 27018 . . . . . 6 (𝜑 → (𝐵 ((𝑆𝐵)‘𝐶)) = (𝐵 𝐶))
811, 2, 3, 4, 6, 15, 6, 12, 80tgcgrcomlr 26841 . . . . 5 (𝜑 → (((𝑆𝐵)‘𝐶) 𝐵) = (𝐶 𝐵))
821, 2, 3, 4, 13, 5, 15, 6, 16, 11, 12, 6, 27, 32, 36, 42, 79, 81tgifscgr 26869 . . . 4 (𝜑 → (𝑄 𝐵) = ((𝑀𝑄) 𝐵))
831, 2, 3, 4, 5, 6, 11, 6, 82tgcgrcomlr 26841 . . 3 (𝜑 → (𝐵 𝑄) = (𝐵 (𝑀𝑄)))
8410fveq1i 6775 . . . 4 (𝑀𝑄) = ((𝑆𝐴)‘𝑄)
8584oveq2i 7286 . . 3 (𝐵 (𝑀𝑄)) = (𝐵 ((𝑆𝐴)‘𝑄))
8683, 85eqtrdi 2794 . 2 (𝜑 → (𝐵 𝑄) = (𝐵 ((𝑆𝐴)‘𝑄)))
871, 2, 3, 7, 8, 4, 6, 9, 5israg 27058 . 2 (𝜑 → (⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 𝑄) = (𝐵 ((𝑆𝐴)‘𝑄))))
8886, 87mpbird 256 1 (𝜑 → ⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  wne 2943  cfv 6433  (class class class)co 7275  ⟨“cs3 14555  Basecbs 16912  distcds 16971  TarskiGcstrkg 26788  Itvcitv 26794  LineGclng 26795  pInvGcmir 27013  ∟Gcrag 27054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-hash 14045  df-word 14218  df-concat 14274  df-s1 14301  df-s2 14561  df-s3 14562  df-trkgc 26809  df-trkgb 26810  df-trkgcb 26811  df-trkg 26814  df-cgrg 26872  df-mir 27014  df-rag 27055
This theorem is referenced by:  colperpexlem3  27093
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