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Theorem colperpexlem1 25340
Description: Lemma for colperp 25339. 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 25274 . . . 4 (𝜑 → (𝑀𝑄) ∈ 𝑃)
12 colperpexlem.c . . . . . 6 (𝜑𝐶𝑃)
131, 2, 3, 7, 8, 4, 9, 10, 12mircl 25274 . . . . 5 (𝜑 → (𝑀𝐶) ∈ 𝑃)
14 eqid 2609 . . . . . 6 (𝑆𝐵) = (𝑆𝐵)
151, 2, 3, 7, 8, 4, 6, 14, 12mircl 25274 . . . . 5 (𝜑 → ((𝑆𝐵)‘𝐶) ∈ 𝑃)
161, 2, 3, 7, 8, 4, 9, 10, 15mircl 25274 . . . . 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 25274 . . . . . . . 8 (𝜑 → (𝑁𝐶) ∈ 𝑃)
2017, 19eqeltrd 2687 . . . . . . 7 (𝜑 → (𝐾‘(𝑀𝐶)) ∈ 𝑃)
21 colperpexlem.k . . . . . . . 8 𝐾 = (𝑆𝑄)
221, 2, 3, 7, 8, 4, 5, 21, 13mirbtwn 25271 . . . . . . 7 (𝜑𝑄 ∈ ((𝐾‘(𝑀𝐶))𝐼(𝑀𝐶)))
231, 2, 3, 4, 20, 5, 13, 22tgbtwncom 25100 . . . . . 6 (𝜑𝑄 ∈ ((𝑀𝐶)𝐼(𝐾‘(𝑀𝐶))))
2418fveq1i 6089 . . . . . . . 8 (𝑁𝐶) = ((𝑆𝐵)‘𝐶)
2517, 24syl6eq 2659 . . . . . . 7 (𝜑 → (𝐾‘(𝑀𝐶)) = ((𝑆𝐵)‘𝐶))
2625oveq2d 6543 . . . . . 6 (𝜑 → ((𝑀𝐶)𝐼(𝐾‘(𝑀𝐶))) = ((𝑀𝐶)𝐼((𝑆𝐵)‘𝐶)))
2723, 26eleqtrd 2689 . . . . 5 (𝜑𝑄 ∈ ((𝑀𝐶)𝐼((𝑆𝐵)‘𝐶)))
281, 2, 3, 4, 13, 5, 15, 27tgbtwncom 25100 . . . . . . 7 (𝜑𝑄 ∈ (((𝑆𝐵)‘𝐶)𝐼(𝑀𝐶)))
291, 2, 3, 7, 8, 4, 9, 10, 15, 5, 13, 28mirbtwni 25284 . . . . . 6 (𝜑 → (𝑀𝑄) ∈ ((𝑀‘((𝑆𝐵)‘𝐶))𝐼(𝑀‘(𝑀𝐶))))
301, 2, 3, 7, 8, 4, 9, 10, 12mirmir 25275 . . . . . . 7 (𝜑 → (𝑀‘(𝑀𝐶)) = 𝐶)
3130oveq2d 6543 . . . . . 6 (𝜑 → ((𝑀‘((𝑆𝐵)‘𝐶))𝐼(𝑀‘(𝑀𝐶))) = ((𝑀‘((𝑆𝐵)‘𝐶))𝐼𝐶))
3229, 31eleqtrd 2689 . . . . 5 (𝜑 → (𝑀𝑄) ∈ ((𝑀‘((𝑆𝐵)‘𝐶))𝐼𝐶))
331, 2, 3, 4, 13, 15axtgcgrrflx 25078 . . . . . 6 (𝜑 → ((𝑀𝐶) ((𝑆𝐵)‘𝐶)) = (((𝑆𝐵)‘𝐶) (𝑀𝐶)))
341, 2, 3, 7, 8, 4, 9, 10, 15, 13miriso 25283 . . . . . 6 (𝜑 → ((𝑀‘((𝑆𝐵)‘𝐶)) (𝑀‘(𝑀𝐶))) = (((𝑆𝐵)‘𝐶) (𝑀𝐶)))
3530oveq2d 6543 . . . . . 6 (𝜑 → ((𝑀‘((𝑆𝐵)‘𝐶)) (𝑀‘(𝑀𝐶))) = ((𝑀‘((𝑆𝐵)‘𝐶)) 𝐶))
3633, 34, 353eqtr2d 2649 . . . . 5 (𝜑 → ((𝑀𝐶) ((𝑆𝐵)‘𝐶)) = ((𝑀‘((𝑆𝐵)‘𝐶)) 𝐶))
3725oveq2d 6543 . . . . . . 7 (𝜑 → (𝑄 (𝐾‘(𝑀𝐶))) = (𝑄 ((𝑆𝐵)‘𝐶)))
381, 2, 3, 7, 8, 4, 5, 21, 13mircgr 25270 . . . . . . 7 (𝜑 → (𝑄 (𝐾‘(𝑀𝐶))) = (𝑄 (𝑀𝐶)))
3937, 38eqtr3d 2645 . . . . . 6 (𝜑 → (𝑄 ((𝑆𝐵)‘𝐶)) = (𝑄 (𝑀𝐶)))
401, 2, 3, 7, 8, 4, 9, 10, 5, 13miriso 25283 . . . . . 6 (𝜑 → ((𝑀𝑄) (𝑀‘(𝑀𝐶))) = (𝑄 (𝑀𝐶)))
4130oveq2d 6543 . . . . . 6 (𝜑 → ((𝑀𝑄) (𝑀‘(𝑀𝐶))) = ((𝑀𝑄) 𝐶))
4239, 40, 413eqtr2d 2649 . . . . 5 (𝜑 → (𝑄 ((𝑆𝐵)‘𝐶)) = ((𝑀𝑄) 𝐶))
431, 2, 3, 7, 8, 4, 9, 10, 6mirmir 25275 . . . . . . . . . 10 (𝜑 → (𝑀‘(𝑀𝐵)) = 𝐵)
44 eqidd 2610 . . . . . . . . . 10 (𝜑 → (𝑀𝐵) = (𝑀𝐵))
45 eqidd 2610 . . . . . . . . . 10 (𝜑 → (𝑀𝐶) = (𝑀𝐶))
4643, 44, 45s3eqd 13406 . . . . . . . . 9 (𝜑 → ⟨“(𝑀‘(𝑀𝐵))(𝑀𝐵)(𝑀𝐶)”⟩ = ⟨“𝐵(𝑀𝐵)(𝑀𝐶)”⟩)
471, 2, 3, 7, 8, 4, 9, 10, 6mircl 25274 . . . . . . . . . 10 (𝜑 → (𝑀𝐵) ∈ 𝑃)
48 simpr 475 . . . . . . . . . . . . . . 15 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
4948fveq2d 6092 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐵) → (𝑀𝐴) = (𝑀𝐵))
504adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
519adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝐴 = 𝐵) → 𝐴𝑃)
521, 2, 3, 7, 8, 50, 51, 10mircinv 25281 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐵) → (𝑀𝐴) = 𝐴)
5349, 52eqtr3d 2645 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐵) → (𝑀𝐵) = 𝐴)
54 eqidd 2610 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐵) → 𝐵 = 𝐵)
55 eqidd 2610 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐵) → 𝐶 = 𝐶)
5653, 54, 55s3eqd 13406 . . . . . . . . . . . 12 ((𝜑𝐴 = 𝐵) → ⟨“(𝑀𝐵)𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩)
57 colperpexlem.1 . . . . . . . . . . . . 13 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
5857adantr 479 . . . . . . . . . . . 12 ((𝜑𝐴 = 𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
5956, 58eqeltrd 2687 . . . . . . . . . . 11 ((𝜑𝐴 = 𝐵) → ⟨“(𝑀𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
604adantr 479 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐺 ∈ TarskiG)
619adantr 479 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐴𝑃)
626adantr 479 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐵𝑃)
6312adantr 479 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐶𝑃)
641, 2, 3, 7, 8, 60, 61, 10, 62mircl 25274 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → (𝑀𝐵) ∈ 𝑃)
6557adantr 479 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
66 simpr 475 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐴𝐵)
671, 2, 3, 7, 8, 60, 61, 10, 62mirbtwn 25271 . . . . . . . . . . . . . 14 ((𝜑𝐴𝐵) → 𝐴 ∈ ((𝑀𝐵)𝐼𝐵))
681, 7, 3, 60, 64, 62, 61, 67btwncolg1 25168 . . . . . . . . . . . . 13 ((𝜑𝐴𝐵) → (𝐴 ∈ ((𝑀𝐵)𝐿𝐵) ∨ (𝑀𝐵) = 𝐵))
691, 7, 3, 60, 64, 62, 61, 68colcom 25171 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → (𝐴 ∈ (𝐵𝐿(𝑀𝐵)) ∨ 𝐵 = (𝑀𝐵)))
701, 2, 3, 7, 8, 60, 61, 62, 63, 64, 65, 66, 69ragcol 25312 . . . . . . . . . . 11 ((𝜑𝐴𝐵) → ⟨“(𝑀𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
7159, 70pm2.61dane 2868 . . . . . . . . . 10 (𝜑 → ⟨“(𝑀𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
721, 2, 3, 7, 8, 4, 47, 6, 12, 71, 10, 9mirrag 25314 . . . . . . . . 9 (𝜑 → ⟨“(𝑀‘(𝑀𝐵))(𝑀𝐵)(𝑀𝐶)”⟩ ∈ (∟G‘𝐺))
7346, 72eqeltrrd 2688 . . . . . . . 8 (𝜑 → ⟨“𝐵(𝑀𝐵)(𝑀𝐶)”⟩ ∈ (∟G‘𝐺))
741, 2, 3, 7, 8, 4, 6, 47, 13israg 25310 . . . . . . . 8 (𝜑 → (⟨“𝐵(𝑀𝐵)(𝑀𝐶)”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 (𝑀𝐶)) = (𝐵 ((𝑆‘(𝑀𝐵))‘(𝑀𝐶)))))
7573, 74mpbid 220 . . . . . . 7 (𝜑 → (𝐵 (𝑀𝐶)) = (𝐵 ((𝑆‘(𝑀𝐵))‘(𝑀𝐶))))
761, 2, 3, 7, 8, 4, 9, 10, 12, 6mirmir2 25287 . . . . . . . 8 (𝜑 → (𝑀‘((𝑆𝐵)‘𝐶)) = ((𝑆‘(𝑀𝐵))‘(𝑀𝐶)))
7776oveq2d 6543 . . . . . . 7 (𝜑 → (𝐵 (𝑀‘((𝑆𝐵)‘𝐶))) = (𝐵 ((𝑆‘(𝑀𝐵))‘(𝑀𝐶))))
7875, 77eqtr4d 2646 . . . . . 6 (𝜑 → (𝐵 (𝑀𝐶)) = (𝐵 (𝑀‘((𝑆𝐵)‘𝐶))))
791, 2, 3, 4, 6, 13, 6, 16, 78tgcgrcomlr 25092 . . . . 5 (𝜑 → ((𝑀𝐶) 𝐵) = ((𝑀‘((𝑆𝐵)‘𝐶)) 𝐵))
801, 2, 3, 7, 8, 4, 6, 14, 12mircgr 25270 . . . . . 6 (𝜑 → (𝐵 ((𝑆𝐵)‘𝐶)) = (𝐵 𝐶))
811, 2, 3, 4, 6, 15, 6, 12, 80tgcgrcomlr 25092 . . . . 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 25121 . . . 4 (𝜑 → (𝑄 𝐵) = ((𝑀𝑄) 𝐵))
831, 2, 3, 4, 5, 6, 11, 6, 82tgcgrcomlr 25092 . . 3 (𝜑 → (𝐵 𝑄) = (𝐵 (𝑀𝑄)))
8410fveq1i 6089 . . . 4 (𝑀𝑄) = ((𝑆𝐴)‘𝑄)
8584oveq2i 6538 . . 3 (𝐵 (𝑀𝑄)) = (𝐵 ((𝑆𝐴)‘𝑄))
8683, 85syl6eq 2659 . 2 (𝜑 → (𝐵 𝑄) = (𝐵 ((𝑆𝐴)‘𝑄)))
871, 2, 3, 7, 8, 4, 6, 9, 5israg 25310 . 2 (𝜑 → (⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 𝑄) = (𝐵 ((𝑆𝐴)‘𝑄))))
8886, 87mpbird 245 1 (𝜑 → ⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wne 2779  cfv 5790  (class class class)co 6527  ⟨“cs3 13384  Basecbs 15641  distcds 15723  TarskiGcstrkg 25046  Itvcitv 25052  LineGclng 25053  pInvGcmir 25265  ∟Gcrag 25306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-hash 12935  df-word 13100  df-concat 13102  df-s1 13103  df-s2 13390  df-s3 13391  df-trkgc 25064  df-trkgb 25065  df-trkgcb 25066  df-trkg 25069  df-cgrg 25124  df-mir 25266  df-rag 25307
This theorem is referenced by:  colperpexlem3  25342
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