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Theorem colperpexlem1 28521
Description: Lemma for colperp 28520. 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 28452 . . . 4 (𝜑 → (𝑀𝑄) ∈ 𝑃)
12 colperpexlem.c . . . . . 6 (𝜑𝐶𝑃)
131, 2, 3, 7, 8, 4, 9, 10, 12mircl 28452 . . . . 5 (𝜑 → (𝑀𝐶) ∈ 𝑃)
14 eqid 2727 . . . . . 6 (𝑆𝐵) = (𝑆𝐵)
151, 2, 3, 7, 8, 4, 6, 14, 12mircl 28452 . . . . 5 (𝜑 → ((𝑆𝐵)‘𝐶) ∈ 𝑃)
161, 2, 3, 7, 8, 4, 9, 10, 15mircl 28452 . . . . 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 28452 . . . . . . . 8 (𝜑 → (𝑁𝐶) ∈ 𝑃)
2017, 19eqeltrd 2828 . . . . . . 7 (𝜑 → (𝐾‘(𝑀𝐶)) ∈ 𝑃)
21 colperpexlem.k . . . . . . . 8 𝐾 = (𝑆𝑄)
221, 2, 3, 7, 8, 4, 5, 21, 13mirbtwn 28449 . . . . . . 7 (𝜑𝑄 ∈ ((𝐾‘(𝑀𝐶))𝐼(𝑀𝐶)))
231, 2, 3, 4, 20, 5, 13, 22tgbtwncom 28279 . . . . . 6 (𝜑𝑄 ∈ ((𝑀𝐶)𝐼(𝐾‘(𝑀𝐶))))
2418fveq1i 6892 . . . . . . . 8 (𝑁𝐶) = ((𝑆𝐵)‘𝐶)
2517, 24eqtrdi 2783 . . . . . . 7 (𝜑 → (𝐾‘(𝑀𝐶)) = ((𝑆𝐵)‘𝐶))
2625oveq2d 7430 . . . . . 6 (𝜑 → ((𝑀𝐶)𝐼(𝐾‘(𝑀𝐶))) = ((𝑀𝐶)𝐼((𝑆𝐵)‘𝐶)))
2723, 26eleqtrd 2830 . . . . 5 (𝜑𝑄 ∈ ((𝑀𝐶)𝐼((𝑆𝐵)‘𝐶)))
281, 2, 3, 4, 13, 5, 15, 27tgbtwncom 28279 . . . . . . 7 (𝜑𝑄 ∈ (((𝑆𝐵)‘𝐶)𝐼(𝑀𝐶)))
291, 2, 3, 7, 8, 4, 9, 10, 15, 5, 13, 28mirbtwni 28462 . . . . . 6 (𝜑 → (𝑀𝑄) ∈ ((𝑀‘((𝑆𝐵)‘𝐶))𝐼(𝑀‘(𝑀𝐶))))
301, 2, 3, 7, 8, 4, 9, 10, 12mirmir 28453 . . . . . . 7 (𝜑 → (𝑀‘(𝑀𝐶)) = 𝐶)
3130oveq2d 7430 . . . . . 6 (𝜑 → ((𝑀‘((𝑆𝐵)‘𝐶))𝐼(𝑀‘(𝑀𝐶))) = ((𝑀‘((𝑆𝐵)‘𝐶))𝐼𝐶))
3229, 31eleqtrd 2830 . . . . 5 (𝜑 → (𝑀𝑄) ∈ ((𝑀‘((𝑆𝐵)‘𝐶))𝐼𝐶))
331, 2, 3, 4, 13, 15axtgcgrrflx 28253 . . . . . 6 (𝜑 → ((𝑀𝐶) ((𝑆𝐵)‘𝐶)) = (((𝑆𝐵)‘𝐶) (𝑀𝐶)))
341, 2, 3, 7, 8, 4, 9, 10, 15, 13miriso 28461 . . . . . 6 (𝜑 → ((𝑀‘((𝑆𝐵)‘𝐶)) (𝑀‘(𝑀𝐶))) = (((𝑆𝐵)‘𝐶) (𝑀𝐶)))
3530oveq2d 7430 . . . . . 6 (𝜑 → ((𝑀‘((𝑆𝐵)‘𝐶)) (𝑀‘(𝑀𝐶))) = ((𝑀‘((𝑆𝐵)‘𝐶)) 𝐶))
3633, 34, 353eqtr2d 2773 . . . . 5 (𝜑 → ((𝑀𝐶) ((𝑆𝐵)‘𝐶)) = ((𝑀‘((𝑆𝐵)‘𝐶)) 𝐶))
3725oveq2d 7430 . . . . . . 7 (𝜑 → (𝑄 (𝐾‘(𝑀𝐶))) = (𝑄 ((𝑆𝐵)‘𝐶)))
381, 2, 3, 7, 8, 4, 5, 21, 13mircgr 28448 . . . . . . 7 (𝜑 → (𝑄 (𝐾‘(𝑀𝐶))) = (𝑄 (𝑀𝐶)))
3937, 38eqtr3d 2769 . . . . . 6 (𝜑 → (𝑄 ((𝑆𝐵)‘𝐶)) = (𝑄 (𝑀𝐶)))
401, 2, 3, 7, 8, 4, 9, 10, 5, 13miriso 28461 . . . . . 6 (𝜑 → ((𝑀𝑄) (𝑀‘(𝑀𝐶))) = (𝑄 (𝑀𝐶)))
4130oveq2d 7430 . . . . . 6 (𝜑 → ((𝑀𝑄) (𝑀‘(𝑀𝐶))) = ((𝑀𝑄) 𝐶))
4239, 40, 413eqtr2d 2773 . . . . 5 (𝜑 → (𝑄 ((𝑆𝐵)‘𝐶)) = ((𝑀𝑄) 𝐶))
431, 2, 3, 7, 8, 4, 9, 10, 6mirmir 28453 . . . . . . . . . 10 (𝜑 → (𝑀‘(𝑀𝐵)) = 𝐵)
44 eqidd 2728 . . . . . . . . . 10 (𝜑 → (𝑀𝐵) = (𝑀𝐵))
45 eqidd 2728 . . . . . . . . . 10 (𝜑 → (𝑀𝐶) = (𝑀𝐶))
4643, 44, 45s3eqd 14839 . . . . . . . . 9 (𝜑 → ⟨“(𝑀‘(𝑀𝐵))(𝑀𝐵)(𝑀𝐶)”⟩ = ⟨“𝐵(𝑀𝐵)(𝑀𝐶)”⟩)
471, 2, 3, 7, 8, 4, 9, 10, 6mircl 28452 . . . . . . . . . 10 (𝜑 → (𝑀𝐵) ∈ 𝑃)
48 simpr 484 . . . . . . . . . . . . . . 15 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
4948fveq2d 6895 . . . . . . . . . . . . . 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 28459 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐵) → (𝑀𝐴) = 𝐴)
5349, 52eqtr3d 2769 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐵) → (𝑀𝐵) = 𝐴)
54 eqidd 2728 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐵) → 𝐵 = 𝐵)
55 eqidd 2728 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐵) → 𝐶 = 𝐶)
5653, 54, 55s3eqd 14839 . . . . . . . . . . . 12 ((𝜑𝐴 = 𝐵) → ⟨“(𝑀𝐵)𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩)
57 colperpexlem.1 . . . . . . . . . . . . 13 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
5857adantr 480 . . . . . . . . . . . 12 ((𝜑𝐴 = 𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
5956, 58eqeltrd 2828 . . . . . . . . . . 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 28452 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → (𝑀𝐵) ∈ 𝑃)
6557adantr 480 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
66 simpr 484 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐴𝐵)
671, 2, 3, 7, 8, 60, 61, 10, 62mirbtwn 28449 . . . . . . . . . . . . . 14 ((𝜑𝐴𝐵) → 𝐴 ∈ ((𝑀𝐵)𝐼𝐵))
681, 7, 3, 60, 64, 62, 61, 67btwncolg1 28346 . . . . . . . . . . . . 13 ((𝜑𝐴𝐵) → (𝐴 ∈ ((𝑀𝐵)𝐿𝐵) ∨ (𝑀𝐵) = 𝐵))
691, 7, 3, 60, 64, 62, 61, 68colcom 28349 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → (𝐴 ∈ (𝐵𝐿(𝑀𝐵)) ∨ 𝐵 = (𝑀𝐵)))
701, 2, 3, 7, 8, 60, 61, 62, 63, 64, 65, 66, 69ragcol 28490 . . . . . . . . . . 11 ((𝜑𝐴𝐵) → ⟨“(𝑀𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
7159, 70pm2.61dane 3024 . . . . . . . . . 10 (𝜑 → ⟨“(𝑀𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
721, 2, 3, 7, 8, 4, 47, 6, 12, 71, 10, 9mirrag 28492 . . . . . . . . 9 (𝜑 → ⟨“(𝑀‘(𝑀𝐵))(𝑀𝐵)(𝑀𝐶)”⟩ ∈ (∟G‘𝐺))
7346, 72eqeltrrd 2829 . . . . . . . 8 (𝜑 → ⟨“𝐵(𝑀𝐵)(𝑀𝐶)”⟩ ∈ (∟G‘𝐺))
741, 2, 3, 7, 8, 4, 6, 47, 13israg 28488 . . . . . . . 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 28465 . . . . . . . 8 (𝜑 → (𝑀‘((𝑆𝐵)‘𝐶)) = ((𝑆‘(𝑀𝐵))‘(𝑀𝐶)))
7776oveq2d 7430 . . . . . . 7 (𝜑 → (𝐵 (𝑀‘((𝑆𝐵)‘𝐶))) = (𝐵 ((𝑆‘(𝑀𝐵))‘(𝑀𝐶))))
7875, 77eqtr4d 2770 . . . . . 6 (𝜑 → (𝐵 (𝑀𝐶)) = (𝐵 (𝑀‘((𝑆𝐵)‘𝐶))))
791, 2, 3, 4, 6, 13, 6, 16, 78tgcgrcomlr 28271 . . . . 5 (𝜑 → ((𝑀𝐶) 𝐵) = ((𝑀‘((𝑆𝐵)‘𝐶)) 𝐵))
801, 2, 3, 7, 8, 4, 6, 14, 12mircgr 28448 . . . . . 6 (𝜑 → (𝐵 ((𝑆𝐵)‘𝐶)) = (𝐵 𝐶))
811, 2, 3, 4, 6, 15, 6, 12, 80tgcgrcomlr 28271 . . . . 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 28299 . . . 4 (𝜑 → (𝑄 𝐵) = ((𝑀𝑄) 𝐵))
831, 2, 3, 4, 5, 6, 11, 6, 82tgcgrcomlr 28271 . . 3 (𝜑 → (𝐵 𝑄) = (𝐵 (𝑀𝑄)))
8410fveq1i 6892 . . . 4 (𝑀𝑄) = ((𝑆𝐴)‘𝑄)
8584oveq2i 7425 . . 3 (𝐵 (𝑀𝑄)) = (𝐵 ((𝑆𝐴)‘𝑄))
8683, 85eqtrdi 2783 . 2 (𝜑 → (𝐵 𝑄) = (𝐵 ((𝑆𝐴)‘𝑄)))
871, 2, 3, 7, 8, 4, 6, 9, 5israg 28488 . 2 (𝜑 → (⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 𝑄) = (𝐵 ((𝑆𝐴)‘𝑄))))
8886, 87mpbird 257 1 (𝜑 → ⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wne 2935  cfv 6542  (class class class)co 7414  ⟨“cs3 14817  Basecbs 17171  distcds 17233  TarskiGcstrkg 28218  Itvcitv 28224  LineGclng 28225  pInvGcmir 28443  ∟Gcrag 28484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8718  df-map 8838  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-fz 13509  df-fzo 13652  df-hash 14314  df-word 14489  df-concat 14545  df-s1 14570  df-s2 14823  df-s3 14824  df-trkgc 28239  df-trkgb 28240  df-trkgcb 28241  df-trkg 28244  df-cgrg 28302  df-mir 28444  df-rag 28485
This theorem is referenced by:  colperpexlem3  28523
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