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Theorem colperpexlem1 26508
Description: Lemma for colperp 26507. 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 26439 . . . 4 (𝜑 → (𝑀𝑄) ∈ 𝑃)
12 colperpexlem.c . . . . . 6 (𝜑𝐶𝑃)
131, 2, 3, 7, 8, 4, 9, 10, 12mircl 26439 . . . . 5 (𝜑 → (𝑀𝐶) ∈ 𝑃)
14 eqid 2819 . . . . . 6 (𝑆𝐵) = (𝑆𝐵)
151, 2, 3, 7, 8, 4, 6, 14, 12mircl 26439 . . . . 5 (𝜑 → ((𝑆𝐵)‘𝐶) ∈ 𝑃)
161, 2, 3, 7, 8, 4, 9, 10, 15mircl 26439 . . . . 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 26439 . . . . . . . 8 (𝜑 → (𝑁𝐶) ∈ 𝑃)
2017, 19eqeltrd 2911 . . . . . . 7 (𝜑 → (𝐾‘(𝑀𝐶)) ∈ 𝑃)
21 colperpexlem.k . . . . . . . 8 𝐾 = (𝑆𝑄)
221, 2, 3, 7, 8, 4, 5, 21, 13mirbtwn 26436 . . . . . . 7 (𝜑𝑄 ∈ ((𝐾‘(𝑀𝐶))𝐼(𝑀𝐶)))
231, 2, 3, 4, 20, 5, 13, 22tgbtwncom 26266 . . . . . 6 (𝜑𝑄 ∈ ((𝑀𝐶)𝐼(𝐾‘(𝑀𝐶))))
2418fveq1i 6664 . . . . . . . 8 (𝑁𝐶) = ((𝑆𝐵)‘𝐶)
2517, 24syl6eq 2870 . . . . . . 7 (𝜑 → (𝐾‘(𝑀𝐶)) = ((𝑆𝐵)‘𝐶))
2625oveq2d 7164 . . . . . 6 (𝜑 → ((𝑀𝐶)𝐼(𝐾‘(𝑀𝐶))) = ((𝑀𝐶)𝐼((𝑆𝐵)‘𝐶)))
2723, 26eleqtrd 2913 . . . . 5 (𝜑𝑄 ∈ ((𝑀𝐶)𝐼((𝑆𝐵)‘𝐶)))
281, 2, 3, 4, 13, 5, 15, 27tgbtwncom 26266 . . . . . . 7 (𝜑𝑄 ∈ (((𝑆𝐵)‘𝐶)𝐼(𝑀𝐶)))
291, 2, 3, 7, 8, 4, 9, 10, 15, 5, 13, 28mirbtwni 26449 . . . . . 6 (𝜑 → (𝑀𝑄) ∈ ((𝑀‘((𝑆𝐵)‘𝐶))𝐼(𝑀‘(𝑀𝐶))))
301, 2, 3, 7, 8, 4, 9, 10, 12mirmir 26440 . . . . . . 7 (𝜑 → (𝑀‘(𝑀𝐶)) = 𝐶)
3130oveq2d 7164 . . . . . 6 (𝜑 → ((𝑀‘((𝑆𝐵)‘𝐶))𝐼(𝑀‘(𝑀𝐶))) = ((𝑀‘((𝑆𝐵)‘𝐶))𝐼𝐶))
3229, 31eleqtrd 2913 . . . . 5 (𝜑 → (𝑀𝑄) ∈ ((𝑀‘((𝑆𝐵)‘𝐶))𝐼𝐶))
331, 2, 3, 4, 13, 15axtgcgrrflx 26240 . . . . . 6 (𝜑 → ((𝑀𝐶) ((𝑆𝐵)‘𝐶)) = (((𝑆𝐵)‘𝐶) (𝑀𝐶)))
341, 2, 3, 7, 8, 4, 9, 10, 15, 13miriso 26448 . . . . . 6 (𝜑 → ((𝑀‘((𝑆𝐵)‘𝐶)) (𝑀‘(𝑀𝐶))) = (((𝑆𝐵)‘𝐶) (𝑀𝐶)))
3530oveq2d 7164 . . . . . 6 (𝜑 → ((𝑀‘((𝑆𝐵)‘𝐶)) (𝑀‘(𝑀𝐶))) = ((𝑀‘((𝑆𝐵)‘𝐶)) 𝐶))
3633, 34, 353eqtr2d 2860 . . . . 5 (𝜑 → ((𝑀𝐶) ((𝑆𝐵)‘𝐶)) = ((𝑀‘((𝑆𝐵)‘𝐶)) 𝐶))
3725oveq2d 7164 . . . . . . 7 (𝜑 → (𝑄 (𝐾‘(𝑀𝐶))) = (𝑄 ((𝑆𝐵)‘𝐶)))
381, 2, 3, 7, 8, 4, 5, 21, 13mircgr 26435 . . . . . . 7 (𝜑 → (𝑄 (𝐾‘(𝑀𝐶))) = (𝑄 (𝑀𝐶)))
3937, 38eqtr3d 2856 . . . . . 6 (𝜑 → (𝑄 ((𝑆𝐵)‘𝐶)) = (𝑄 (𝑀𝐶)))
401, 2, 3, 7, 8, 4, 9, 10, 5, 13miriso 26448 . . . . . 6 (𝜑 → ((𝑀𝑄) (𝑀‘(𝑀𝐶))) = (𝑄 (𝑀𝐶)))
4130oveq2d 7164 . . . . . 6 (𝜑 → ((𝑀𝑄) (𝑀‘(𝑀𝐶))) = ((𝑀𝑄) 𝐶))
4239, 40, 413eqtr2d 2860 . . . . 5 (𝜑 → (𝑄 ((𝑆𝐵)‘𝐶)) = ((𝑀𝑄) 𝐶))
431, 2, 3, 7, 8, 4, 9, 10, 6mirmir 26440 . . . . . . . . . 10 (𝜑 → (𝑀‘(𝑀𝐵)) = 𝐵)
44 eqidd 2820 . . . . . . . . . 10 (𝜑 → (𝑀𝐵) = (𝑀𝐵))
45 eqidd 2820 . . . . . . . . . 10 (𝜑 → (𝑀𝐶) = (𝑀𝐶))
4643, 44, 45s3eqd 14218 . . . . . . . . 9 (𝜑 → ⟨“(𝑀‘(𝑀𝐵))(𝑀𝐵)(𝑀𝐶)”⟩ = ⟨“𝐵(𝑀𝐵)(𝑀𝐶)”⟩)
471, 2, 3, 7, 8, 4, 9, 10, 6mircl 26439 . . . . . . . . . 10 (𝜑 → (𝑀𝐵) ∈ 𝑃)
48 simpr 487 . . . . . . . . . . . . . . 15 ((𝜑𝐴 = 𝐵) → 𝐴 = 𝐵)
4948fveq2d 6667 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐵) → (𝑀𝐴) = (𝑀𝐵))
504adantr 483 . . . . . . . . . . . . . . 15 ((𝜑𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
519adantr 483 . . . . . . . . . . . . . . 15 ((𝜑𝐴 = 𝐵) → 𝐴𝑃)
521, 2, 3, 7, 8, 50, 51, 10mircinv 26446 . . . . . . . . . . . . . 14 ((𝜑𝐴 = 𝐵) → (𝑀𝐴) = 𝐴)
5349, 52eqtr3d 2856 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐵) → (𝑀𝐵) = 𝐴)
54 eqidd 2820 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐵) → 𝐵 = 𝐵)
55 eqidd 2820 . . . . . . . . . . . . 13 ((𝜑𝐴 = 𝐵) → 𝐶 = 𝐶)
5653, 54, 55s3eqd 14218 . . . . . . . . . . . 12 ((𝜑𝐴 = 𝐵) → ⟨“(𝑀𝐵)𝐵𝐶”⟩ = ⟨“𝐴𝐵𝐶”⟩)
57 colperpexlem.1 . . . . . . . . . . . . 13 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
5857adantr 483 . . . . . . . . . . . 12 ((𝜑𝐴 = 𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
5956, 58eqeltrd 2911 . . . . . . . . . . 11 ((𝜑𝐴 = 𝐵) → ⟨“(𝑀𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
604adantr 483 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐺 ∈ TarskiG)
619adantr 483 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐴𝑃)
626adantr 483 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐵𝑃)
6312adantr 483 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐶𝑃)
641, 2, 3, 7, 8, 60, 61, 10, 62mircl 26439 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → (𝑀𝐵) ∈ 𝑃)
6557adantr 483 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
66 simpr 487 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → 𝐴𝐵)
671, 2, 3, 7, 8, 60, 61, 10, 62mirbtwn 26436 . . . . . . . . . . . . . 14 ((𝜑𝐴𝐵) → 𝐴 ∈ ((𝑀𝐵)𝐼𝐵))
681, 7, 3, 60, 64, 62, 61, 67btwncolg1 26333 . . . . . . . . . . . . 13 ((𝜑𝐴𝐵) → (𝐴 ∈ ((𝑀𝐵)𝐿𝐵) ∨ (𝑀𝐵) = 𝐵))
691, 7, 3, 60, 64, 62, 61, 68colcom 26336 . . . . . . . . . . . 12 ((𝜑𝐴𝐵) → (𝐴 ∈ (𝐵𝐿(𝑀𝐵)) ∨ 𝐵 = (𝑀𝐵)))
701, 2, 3, 7, 8, 60, 61, 62, 63, 64, 65, 66, 69ragcol 26477 . . . . . . . . . . 11 ((𝜑𝐴𝐵) → ⟨“(𝑀𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
7159, 70pm2.61dane 3102 . . . . . . . . . 10 (𝜑 → ⟨“(𝑀𝐵)𝐵𝐶”⟩ ∈ (∟G‘𝐺))
721, 2, 3, 7, 8, 4, 47, 6, 12, 71, 10, 9mirrag 26479 . . . . . . . . 9 (𝜑 → ⟨“(𝑀‘(𝑀𝐵))(𝑀𝐵)(𝑀𝐶)”⟩ ∈ (∟G‘𝐺))
7346, 72eqeltrrd 2912 . . . . . . . 8 (𝜑 → ⟨“𝐵(𝑀𝐵)(𝑀𝐶)”⟩ ∈ (∟G‘𝐺))
741, 2, 3, 7, 8, 4, 6, 47, 13israg 26475 . . . . . . . 8 (𝜑 → (⟨“𝐵(𝑀𝐵)(𝑀𝐶)”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 (𝑀𝐶)) = (𝐵 ((𝑆‘(𝑀𝐵))‘(𝑀𝐶)))))
7573, 74mpbid 234 . . . . . . 7 (𝜑 → (𝐵 (𝑀𝐶)) = (𝐵 ((𝑆‘(𝑀𝐵))‘(𝑀𝐶))))
761, 2, 3, 7, 8, 4, 9, 10, 12, 6mirmir2 26452 . . . . . . . 8 (𝜑 → (𝑀‘((𝑆𝐵)‘𝐶)) = ((𝑆‘(𝑀𝐵))‘(𝑀𝐶)))
7776oveq2d 7164 . . . . . . 7 (𝜑 → (𝐵 (𝑀‘((𝑆𝐵)‘𝐶))) = (𝐵 ((𝑆‘(𝑀𝐵))‘(𝑀𝐶))))
7875, 77eqtr4d 2857 . . . . . 6 (𝜑 → (𝐵 (𝑀𝐶)) = (𝐵 (𝑀‘((𝑆𝐵)‘𝐶))))
791, 2, 3, 4, 6, 13, 6, 16, 78tgcgrcomlr 26258 . . . . 5 (𝜑 → ((𝑀𝐶) 𝐵) = ((𝑀‘((𝑆𝐵)‘𝐶)) 𝐵))
801, 2, 3, 7, 8, 4, 6, 14, 12mircgr 26435 . . . . . 6 (𝜑 → (𝐵 ((𝑆𝐵)‘𝐶)) = (𝐵 𝐶))
811, 2, 3, 4, 6, 15, 6, 12, 80tgcgrcomlr 26258 . . . . 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 26286 . . . 4 (𝜑 → (𝑄 𝐵) = ((𝑀𝑄) 𝐵))
831, 2, 3, 4, 5, 6, 11, 6, 82tgcgrcomlr 26258 . . 3 (𝜑 → (𝐵 𝑄) = (𝐵 (𝑀𝑄)))
8410fveq1i 6664 . . . 4 (𝑀𝑄) = ((𝑆𝐴)‘𝑄)
8584oveq2i 7159 . . 3 (𝐵 (𝑀𝑄)) = (𝐵 ((𝑆𝐴)‘𝑄))
8683, 85syl6eq 2870 . 2 (𝜑 → (𝐵 𝑄) = (𝐵 ((𝑆𝐴)‘𝑄)))
871, 2, 3, 7, 8, 4, 6, 9, 5israg 26475 . 2 (𝜑 → (⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 𝑄) = (𝐵 ((𝑆𝐴)‘𝑄))))
8886, 87mpbird 259 1 (𝜑 → ⟨“𝐵𝐴𝑄”⟩ ∈ (∟G‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  wcel 2108  wne 3014  cfv 6348  (class class class)co 7148  ⟨“cs3 14196  Basecbs 16475  distcds 16566  TarskiGcstrkg 26208  Itvcitv 26214  LineGclng 26215  pInvGcmir 26430  ∟Gcrag 26471
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12885  df-fzo 13026  df-hash 13683  df-word 13854  df-concat 13915  df-s1 13942  df-s2 14202  df-s3 14203  df-trkgc 26226  df-trkgb 26227  df-trkgcb 26228  df-trkg 26231  df-cgrg 26289  df-mir 26431  df-rag 26472
This theorem is referenced by:  colperpexlem3  26510
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