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Theorem colperpexlem1 25667
 Description: Lemma for colperp 25666. 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 25601 . . . 4 (𝜑 → (𝑀𝑄) ∈ 𝑃)
12 colperpexlem.c . . . . . 6 (𝜑𝐶𝑃)
131, 2, 3, 7, 8, 4, 9, 10, 12mircl 25601 . . . . 5 (𝜑 → (𝑀𝐶) ∈ 𝑃)
14 eqid 2651 . . . . . 6 (𝑆𝐵) = (𝑆𝐵)
151, 2, 3, 7, 8, 4, 6, 14, 12mircl 25601 . . . . 5 (𝜑 → ((𝑆𝐵)‘𝐶) ∈ 𝑃)
161, 2, 3, 7, 8, 4, 9, 10, 15mircl 25601 . . . . 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 25601 . . . . . . . 8 (𝜑 → (𝑁𝐶) ∈ 𝑃)
2017, 19eqeltrd 2730 . . . . . . 7 (𝜑 → (𝐾‘(𝑀𝐶)) ∈ 𝑃)
21 colperpexlem.k . . . . . . . 8 𝐾 = (𝑆𝑄)
221, 2, 3, 7, 8, 4, 5, 21, 13mirbtwn 25598 . . . . . . 7 (𝜑𝑄 ∈ ((𝐾‘(𝑀𝐶))𝐼(𝑀𝐶)))
231, 2, 3, 4, 20, 5, 13, 22tgbtwncom 25428 . . . . . 6 (𝜑𝑄 ∈ ((𝑀𝐶)𝐼(𝐾‘(𝑀𝐶))))
2418fveq1i 6230 . . . . . . . 8 (𝑁𝐶) = ((𝑆𝐵)‘𝐶)
2517, 24syl6eq 2701 . . . . . . 7 (𝜑 → (𝐾‘(𝑀𝐶)) = ((𝑆𝐵)‘𝐶))
2625oveq2d 6706 . . . . . 6 (𝜑 → ((𝑀𝐶)𝐼(𝐾‘(𝑀𝐶))) = ((𝑀𝐶)𝐼((𝑆𝐵)‘𝐶)))
2723, 26eleqtrd 2732 . . . . 5 (𝜑𝑄 ∈ ((𝑀𝐶)𝐼((𝑆𝐵)‘𝐶)))
281, 2, 3, 4, 13, 5, 15, 27tgbtwncom 25428 . . . . . . 7 (𝜑𝑄 ∈ (((𝑆𝐵)‘𝐶)𝐼(𝑀𝐶)))
291, 2, 3, 7, 8, 4, 9, 10, 15, 5, 13, 28mirbtwni 25611 . . . . . 6 (𝜑 → (𝑀𝑄) ∈ ((𝑀‘((𝑆𝐵)‘𝐶))𝐼(𝑀‘(𝑀𝐶))))
301, 2, 3, 7, 8, 4, 9, 10, 12mirmir 25602 . . . . . . 7 (𝜑 → (𝑀‘(𝑀𝐶)) = 𝐶)
3130oveq2d 6706 . . . . . 6 (𝜑 → ((𝑀‘((𝑆𝐵)‘𝐶))𝐼(𝑀‘(𝑀𝐶))) = ((𝑀‘((𝑆𝐵)‘𝐶))𝐼𝐶))
3229, 31eleqtrd 2732 . . . . 5 (𝜑 → (𝑀𝑄) ∈ ((𝑀‘((𝑆𝐵)‘𝐶))𝐼𝐶))
331, 2, 3, 4, 13, 15axtgcgrrflx 25406 . . . . . 6 (𝜑 → ((𝑀𝐶) ((𝑆𝐵)‘𝐶)) = (((𝑆𝐵)‘𝐶) (𝑀𝐶)))
341, 2, 3, 7, 8, 4, 9, 10, 15, 13miriso 25610 . . . . . 6 (𝜑 → ((𝑀‘((𝑆𝐵)‘𝐶)) (𝑀‘(𝑀𝐶))) = (((𝑆𝐵)‘𝐶) (𝑀𝐶)))
3530oveq2d 6706 . . . . . 6 (𝜑 → ((𝑀‘((𝑆𝐵)‘𝐶)) (𝑀‘(𝑀𝐶))) = ((𝑀‘((𝑆𝐵)‘𝐶)) 𝐶))
3633, 34, 353eqtr2d 2691 . . . . 5 (𝜑 → ((𝑀𝐶) ((𝑆𝐵)‘𝐶)) = ((𝑀‘((𝑆𝐵)‘𝐶)) 𝐶))
3725oveq2d 6706 . . . . . . 7 (𝜑 → (𝑄 (𝐾‘(𝑀𝐶))) = (𝑄 ((𝑆𝐵)‘𝐶)))
381, 2, 3, 7, 8, 4, 5, 21, 13mircgr 25597 . . . . . . 7 (𝜑 → (𝑄 (𝐾‘(𝑀𝐶))) = (𝑄 (𝑀𝐶)))
3937, 38eqtr3d 2687 . . . . . 6 (𝜑 → (𝑄 ((𝑆𝐵)‘𝐶)) = (𝑄 (𝑀𝐶)))
401, 2, 3, 7, 8, 4, 9, 10, 5, 13miriso 25610 . . . . . 6 (𝜑 → ((𝑀𝑄) (𝑀‘(𝑀𝐶))) = (𝑄 (𝑀𝐶)))
4130oveq2d 6706 . . . . . 6 (𝜑 → ((𝑀𝑄) (𝑀‘(𝑀𝐶))) = ((𝑀𝑄) 𝐶))
4239, 40, 413eqtr2d 2691 . . . . 5 (𝜑 → (𝑄 ((𝑆𝐵)‘𝐶)) = ((𝑀𝑄) 𝐶))
431, 2, 3, 7, 8, 4, 9, 10, 6mirmir 25602 . . . . . . . . . 10 (𝜑 → (𝑀‘(𝑀𝐵)) = 𝐵)
44 eqidd 2652 . . . . . . . . . 10 (𝜑 → (𝑀𝐵) = (𝑀𝐵))
45 eqidd 2652 . . . . . . . . . 10 (𝜑 → (𝑀𝐶) = (𝑀𝐶))
4643, 44, 45s3eqd 13655 . . . . . . . . 9 (𝜑 → ⟨“(𝑀‘(𝑀𝐵))(𝑀𝐵)(𝑀𝐶)”⟩ = ⟨“𝐵(𝑀𝐵)(𝑀𝐶)”⟩)
471, 2, 3, 7, 8, 4, 9, 10, 6mircl 25601 . . . . . . . . . 10 (𝜑 → (𝑀𝐵) ∈ 𝑃)
48 simpr 476 . . . . . . . . . . . . . . 15 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
4948fveq2d 6233 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐵) → (𝑀𝐴) = (𝑀𝐵))
504adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
519adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝐴 = 𝐵) → 𝐴𝑃)
521, 2, 3, 7, 8, 50, 51, 10mircinv 25608 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐵) → (𝑀𝐴) = 𝐴)
5349, 52eqtr3d 2687 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐵) → (𝑀𝐵) = 𝐴)
54 eqidd 2652 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐵) → 𝐵 = 𝐵)
55 eqidd 2652 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐵) → 𝐶 = 𝐶)
5653, 54, 55s3eqd 13655 . . . . . . . . . . . 12 ((𝜑𝐴 = 𝐵) → ⟨“(𝑀𝐵)𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩)
57 colperpexlem.1 . . . . . . . . . . . . 13 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
5857adantr 480 . . . . . . . . . . . 12 ((𝜑𝐴 = 𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
5956, 58eqeltrd 2730 . . . . . . . . . . 11 ((𝜑𝐴 = 𝐵) → ⟨“(𝑀𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
604adantr 480 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐺 ∈ TarskiG)
619adantr 480 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐴𝑃)
626adantr 480 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐵𝑃)
6312adantr 480 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐶𝑃)
641, 2, 3, 7, 8, 60, 61, 10, 62mircl 25601 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → (𝑀𝐵) ∈ 𝑃)
6557adantr 480 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
66 simpr 476 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐴𝐵)
671, 2, 3, 7, 8, 60, 61, 10, 62mirbtwn 25598 . . . . . . . . . . . . . 14 ((𝜑𝐴𝐵) → 𝐴 ∈ ((𝑀𝐵)𝐼𝐵))
681, 7, 3, 60, 64, 62, 61, 67btwncolg1 25495 . . . . . . . . . . . . 13 ((𝜑𝐴𝐵) → (𝐴 ∈ ((𝑀𝐵)𝐿𝐵) ∨ (𝑀𝐵) = 𝐵))
691, 7, 3, 60, 64, 62, 61, 68colcom 25498 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → (𝐴 ∈ (𝐵𝐿(𝑀𝐵)) ∨ 𝐵 = (𝑀𝐵)))
701, 2, 3, 7, 8, 60, 61, 62, 63, 64, 65, 66, 69ragcol 25639 . . . . . . . . . . 11 ((𝜑𝐴𝐵) → ⟨“(𝑀𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
7159, 70pm2.61dane 2910 . . . . . . . . . 10 (𝜑 → ⟨“(𝑀𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
721, 2, 3, 7, 8, 4, 47, 6, 12, 71, 10, 9mirrag 25641 . . . . . . . . 9 (𝜑 → ⟨“(𝑀‘(𝑀𝐵))(𝑀𝐵)(𝑀𝐶)”⟩ ∈ (∟G‘𝐺))
7346, 72eqeltrrd 2731 . . . . . . . 8 (𝜑 → ⟨“𝐵(𝑀𝐵)(𝑀𝐶)”⟩ ∈ (∟G‘𝐺))
741, 2, 3, 7, 8, 4, 6, 47, 13israg 25637 . . . . . . . 8 (𝜑 → (⟨“𝐵(𝑀𝐵)(𝑀𝐶)”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 (𝑀𝐶)) = (𝐵 ((𝑆‘(𝑀𝐵))‘(𝑀𝐶)))))
7573, 74mpbid 222 . . . . . . 7 (𝜑 → (𝐵 (𝑀𝐶)) = (𝐵 ((𝑆‘(𝑀𝐵))‘(𝑀𝐶))))
761, 2, 3, 7, 8, 4, 9, 10, 12, 6mirmir2 25614 . . . . . . . 8 (𝜑 → (𝑀‘((𝑆𝐵)‘𝐶)) = ((𝑆‘(𝑀𝐵))‘(𝑀𝐶)))
7776oveq2d 6706 . . . . . . 7 (𝜑 → (𝐵 (𝑀‘((𝑆𝐵)‘𝐶))) = (𝐵 ((𝑆‘(𝑀𝐵))‘(𝑀𝐶))))
7875, 77eqtr4d 2688 . . . . . 6 (𝜑 → (𝐵 (𝑀𝐶)) = (𝐵 (𝑀‘((𝑆𝐵)‘𝐶))))
791, 2, 3, 4, 6, 13, 6, 16, 78tgcgrcomlr 25420 . . . . 5 (𝜑 → ((𝑀𝐶) 𝐵) = ((𝑀‘((𝑆𝐵)‘𝐶)) 𝐵))
801, 2, 3, 7, 8, 4, 6, 14, 12mircgr 25597 . . . . . 6 (𝜑 → (𝐵 ((𝑆𝐵)‘𝐶)) = (𝐵 𝐶))
811, 2, 3, 4, 6, 15, 6, 12, 80tgcgrcomlr 25420 . . . . 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 25448 . . . 4 (𝜑 → (𝑄 𝐵) = ((𝑀𝑄) 𝐵))
831, 2, 3, 4, 5, 6, 11, 6, 82tgcgrcomlr 25420 . . 3 (𝜑 → (𝐵 𝑄) = (𝐵 (𝑀𝑄)))
8410fveq1i 6230 . . . 4 (𝑀𝑄) = ((𝑆𝐴)‘𝑄)
8584oveq2i 6701 . . 3 (𝐵 (𝑀𝑄)) = (𝐵 ((𝑆𝐴)‘𝑄))
8683, 85syl6eq 2701 . 2 (𝜑 → (𝐵 𝑄) = (𝐵 ((𝑆𝐴)‘𝑄)))
871, 2, 3, 7, 8, 4, 6, 9, 5israg 25637 . 2 (𝜑 → (⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 𝑄) = (𝐵 ((𝑆𝐴)‘𝑄))))
8886, 87mpbird 247 1 (𝜑 → ⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ‘cfv 5926  (class class class)co 6690  ⟨“cs3 13633  Basecbs 15904  distcds 15997  TarskiGcstrkg 25374  Itvcitv 25380  LineGclng 25381  pInvGcmir 25592  ∟Gcrag 25633 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-s2 13639  df-s3 13640  df-trkgc 25392  df-trkgb 25393  df-trkgcb 25394  df-trkg 25397  df-cgrg 25451  df-mir 25593  df-rag 25634 This theorem is referenced by:  colperpexlem3  25669
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