Theorem colperpexlem1 25622

Description: Lemma for colperp 25621. 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	|- ( ph -> G e. TarskiG )
colperpexlem.s	|- S = (pInvG ` G )
colperpexlem.m	|- M = ( S ` A )
colperpexlem.n	|- N = ( S ` B )
colperpexlem.k	|- K = ( S ` Q )
colperpexlem.a	|- ( ph -> A e. P )
colperpexlem.b	|- ( ph -> B e. P )
colperpexlem.c	|- ( ph -> C e. P )
colperpexlem.q	|- ( ph -> Q e. P )
colperpexlem.1	|- ( ph -> <" A B C "> e. (∟G ` G ) )
colperpexlem.2	|- ( ph -> ( K ` ( M ` C ) ) = ( N ` C ) )

Assertion

Ref	Expression
colperpexlem1	|- ( ph -> <" B A Q "> e. (∟G ` G ) )

Proof of Theorem colperpexlem1

Step	Hyp	Ref	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 |- ( ph -> G e. TarskiG )
5		colperpexlem.q	. . . 4 |- ( ph -> Q e. P )
6		colperpexlem.b	. . . 4 |- ( ph -> B e. P )
7		colperpex.l	. . . . 5 |- L = (LineG ` G )
8		colperpexlem.s	. . . . 5 |- S = (pInvG ` G )
9		colperpexlem.a	. . . . 5 |- ( ph -> A e. P )
10		colperpexlem.m	. . . . 5 |- M = ( S ` A )
11	1, 2, 3, 7, 8, 4, 9, 10, 5	mircl 25556	. . . 4 |- ( ph -> ( M ` Q ) e. P )
12		colperpexlem.c	. . . . . 6 |- ( ph -> C e. P )
13	1, 2, 3, 7, 8, 4, 9, 10, 12	mircl 25556	. . . . 5 |- ( ph -> ( M ` C ) e. P )
14		eqid 2622	. . . . . 6 |- ( S ` B ) = ( S ` B )
15	1, 2, 3, 7, 8, 4, 6, 14, 12	mircl 25556	. . . . 5 |- ( ph -> ( ( S ` B ) ` C ) e. P )
16	1, 2, 3, 7, 8, 4, 9, 10, 15	mircl 25556	. . . . 5 |- ( ph -> ( M ` ( ( S ` B ) ` C ) ) e. P )
17		colperpexlem.2	. . . . . . . 8 |- ( ph -> ( K ` ( M ` C ) ) = ( N ` C ) )
18		colperpexlem.n	. . . . . . . . 9 |- N = ( S ` B )
19	1, 2, 3, 7, 8, 4, 6, 18, 12	mircl 25556	. . . . . . . 8 |- ( ph -> ( N ` C ) e. P )
20	17, 19	eqeltrd 2701	. . . . . . 7 |- ( ph -> ( K ` ( M ` C ) ) e. P )
21		colperpexlem.k	. . . . . . . 8 |- K = ( S ` Q )
22	1, 2, 3, 7, 8, 4, 5, 21, 13	mirbtwn 25553	. . . . . . 7 |- ( ph -> Q e. ( ( K ` ( M ` C ) ) I ( M ` C ) ) )
23	1, 2, 3, 4, 20, 5, 13, 22	tgbtwncom 25383	. . . . . 6 |- ( ph -> Q e. ( ( M ` C ) I ( K ` ( M ` C ) ) ) )
24	18	fveq1i 6192	. . . . . . . 8 |- ( N ` C ) = ( ( S ` B ) ` C )
25	17, 24	syl6eq 2672	. . . . . . 7 |- ( ph -> ( K ` ( M ` C ) ) = ( ( S ` B ) ` C ) )
26	25	oveq2d 6666	. . . . . 6 |- ( ph -> ( ( M ` C ) I ( K ` ( M ` C ) ) ) = ( ( M ` C ) I ( ( S ` B ) ` C ) ) )
27	23, 26	eleqtrd 2703	. . . . 5 |- ( ph -> Q e. ( ( M ` C ) I ( ( S ` B ) ` C ) ) )
28	1, 2, 3, 4, 13, 5, 15, 27	tgbtwncom 25383	. . . . . . 7 |- ( ph -> Q e. ( ( ( S ` B ) ` C ) I ( M ` C ) ) )
29	1, 2, 3, 7, 8, 4, 9, 10, 15, 5, 13, 28	mirbtwni 25566	. . . . . 6 |- ( ph -> ( M ` Q ) e. ( ( M ` ( ( S ` B ) ` C ) ) I ( M ` ( M ` C ) ) ) )
30	1, 2, 3, 7, 8, 4, 9, 10, 12	mirmir 25557	. . . . . . 7 |- ( ph -> ( M ` ( M ` C ) ) = C )
31	30	oveq2d 6666	. . . . . 6 |- ( ph -> ( ( M ` ( ( S ` B ) ` C ) ) I ( M ` ( M ` C ) ) ) = ( ( M ` ( ( S ` B ) ` C ) ) I C ) )
32	29, 31	eleqtrd 2703	. . . . 5 |- ( ph -> ( M ` Q ) e. ( ( M ` ( ( S ` B ) ` C ) ) I C ) )
33	1, 2, 3, 4, 13, 15	axtgcgrrflx 25361	. . . . . 6 |- ( ph -> ( ( M ` C ) .- ( ( S ` B ) ` C ) ) = ( ( ( S ` B ) ` C ) .- ( M ` C ) ) )
34	1, 2, 3, 7, 8, 4, 9, 10, 15, 13	miriso 25565	. . . . . 6 |- ( ph -> ( ( M ` ( ( S ` B ) ` C ) ) .- ( M ` ( M ` C ) ) ) = ( ( ( S ` B ) ` C ) .- ( M ` C ) ) )
35	30	oveq2d 6666	. . . . . 6 |- ( ph -> ( ( M ` ( ( S ` B ) ` C ) ) .- ( M ` ( M ` C ) ) ) = ( ( M ` ( ( S ` B ) ` C ) ) .- C ) )
36	33, 34, 35	3eqtr2d 2662	. . . . 5 |- ( ph -> ( ( M ` C ) .- ( ( S ` B ) ` C ) ) = ( ( M ` ( ( S ` B ) ` C ) ) .- C ) )
37	25	oveq2d 6666	. . . . . . 7 |- ( ph -> ( Q .- ( K ` ( M ` C ) ) ) = ( Q .- ( ( S ` B ) ` C ) ) )
38	1, 2, 3, 7, 8, 4, 5, 21, 13	mircgr 25552	. . . . . . 7 |- ( ph -> ( Q .- ( K ` ( M ` C ) ) ) = ( Q .- ( M ` C ) ) )
39	37, 38	eqtr3d 2658	. . . . . 6 |- ( ph -> ( Q .- ( ( S ` B ) ` C ) ) = ( Q .- ( M ` C ) ) )
40	1, 2, 3, 7, 8, 4, 9, 10, 5, 13	miriso 25565	. . . . . 6 |- ( ph -> ( ( M ` Q ) .- ( M ` ( M ` C ) ) ) = ( Q .- ( M ` C ) ) )
41	30	oveq2d 6666	. . . . . 6 |- ( ph -> ( ( M ` Q ) .- ( M ` ( M ` C ) ) ) = ( ( M ` Q ) .- C ) )
42	39, 40, 41	3eqtr2d 2662	. . . . 5 |- ( ph -> ( Q .- ( ( S ` B ) ` C ) ) = ( ( M ` Q ) .- C ) )
43	1, 2, 3, 7, 8, 4, 9, 10, 6	mirmir 25557	. . . . . . . . . 10 |- ( ph -> ( M ` ( M ` B ) ) = B )
44		eqidd 2623	. . . . . . . . . 10 |- ( ph -> ( M ` B ) = ( M ` B ) )
45		eqidd 2623	. . . . . . . . . 10 |- ( ph -> ( M ` C ) = ( M ` C ) )
46	43, 44, 45	s3eqd 13609	. . . . . . . . 9 |- ( ph -> <" ( M ` ( M ` B ) ) ( M ` B ) ( M ` C ) "> = <" B ( M ` B ) ( M ` C ) "> )
47	1, 2, 3, 7, 8, 4, 9, 10, 6	mircl 25556	. . . . . . . . . 10 |- ( ph -> ( M ` B ) e. P )
48		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ph /\ A = B ) -> A = B )
49	48	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( ph /\ A = B ) -> ( M ` A ) = ( M ` B ) )
50	4	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ A = B ) -> G e. TarskiG )
51	9	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ A = B ) -> A e. P )
52	1, 2, 3, 7, 8, 50, 51, 10	mircinv 25563	. . . . . . . . . . . . . 14 |- ( ( ph /\ A = B ) -> ( M ` A ) = A )
53	49, 52	eqtr3d 2658	. . . . . . . . . . . . 13 |- ( ( ph /\ A = B ) -> ( M ` B ) = A )
54		eqidd 2623	. . . . . . . . . . . . 13 |- ( ( ph /\ A = B ) -> B = B )
55		eqidd 2623	. . . . . . . . . . . . 13 |- ( ( ph /\ A = B ) -> C = C )
56	53, 54, 55	s3eqd 13609	. . . . . . . . . . . 12 |- ( ( ph /\ A = B ) -> <" ( M ` B ) B C "> = <" A B C "> )
57		colperpexlem.1	. . . . . . . . . . . . 13 |- ( ph -> <" A B C "> e. (∟G ` G ) )
58	57	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ A = B ) -> <" A B C "> e. (∟G ` G ) )
59	56, 58	eqeltrd 2701	. . . . . . . . . . 11 |- ( ( ph /\ A = B ) -> <" ( M ` B ) B C "> e. (∟G ` G ) )
60	4	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= B ) -> G e. TarskiG )
61	9	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= B ) -> A e. P )
62	6	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= B ) -> B e. P )
63	12	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= B ) -> C e. P )
64	1, 2, 3, 7, 8, 60, 61, 10, 62	mircl 25556	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= B ) -> ( M ` B ) e. P )
65	57	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= B ) -> <" A B C "> e. (∟G ` G ) )
66		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= B ) -> A =/= B )
67	1, 2, 3, 7, 8, 60, 61, 10, 62	mirbtwn 25553	. . . . . . . . . . . . . 14 |- ( ( ph /\ A =/= B ) -> A e. ( ( M ` B ) I B ) )
68	1, 7, 3, 60, 64, 62, 61, 67	btwncolg1 25450	. . . . . . . . . . . . 13 |- ( ( ph /\ A =/= B ) -> ( A e. ( ( M ` B ) L B ) \/ ( M ` B ) = B ) )
69	1, 7, 3, 60, 64, 62, 61, 68	colcom 25453	. . . . . . . . . . . 12 |- ( ( ph /\ A =/= B ) -> ( A e. ( B L ( M ` B ) ) \/ B = ( M ` B ) ) )
70	1, 2, 3, 7, 8, 60, 61, 62, 63, 64, 65, 66, 69	ragcol 25594	. . . . . . . . . . 11 |- ( ( ph /\ A =/= B ) -> <" ( M ` B ) B C "> e. (∟G ` G ) )
71	59, 70	pm2.61dane 2881	. . . . . . . . . 10 |- ( ph -> <" ( M ` B ) B C "> e. (∟G ` G ) )
72	1, 2, 3, 7, 8, 4, 47, 6, 12, 71, 10, 9	mirrag 25596	. . . . . . . . 9 |- ( ph -> <" ( M ` ( M ` B ) ) ( M ` B ) ( M ` C ) "> e. (∟G ` G ) )
73	46, 72	eqeltrrd 2702	. . . . . . . 8 |- ( ph -> <" B ( M ` B ) ( M ` C ) "> e. (∟G ` G ) )
74	1, 2, 3, 7, 8, 4, 6, 47, 13	israg 25592	. . . . . . . 8 |- ( ph -> ( <" B ( M ` B ) ( M ` C ) "> e. (∟G ` G ) <-> ( B .- ( M ` C ) ) = ( B .- ( ( S ` ( M ` B ) ) ` ( M ` C ) ) ) ) )
75	73, 74	mpbid 222	. . . . . . 7 |- ( ph -> ( B .- ( M ` C ) ) = ( B .- ( ( S ` ( M ` B ) ) ` ( M ` C ) ) ) )
76	1, 2, 3, 7, 8, 4, 9, 10, 12, 6	mirmir2 25569	. . . . . . . 8 |- ( ph -> ( M ` ( ( S ` B ) ` C ) ) = ( ( S ` ( M ` B ) ) ` ( M ` C ) ) )
77	76	oveq2d 6666	. . . . . . 7 |- ( ph -> ( B .- ( M ` ( ( S ` B ) ` C ) ) ) = ( B .- ( ( S ` ( M ` B ) ) ` ( M ` C ) ) ) )
78	75, 77	eqtr4d 2659	. . . . . 6 |- ( ph -> ( B .- ( M ` C ) ) = ( B .- ( M ` ( ( S ` B ) ` C ) ) ) )
79	1, 2, 3, 4, 6, 13, 6, 16, 78	tgcgrcomlr 25375	. . . . 5 |- ( ph -> ( ( M ` C ) .- B ) = ( ( M ` ( ( S ` B ) ` C ) ) .- B ) )
80	1, 2, 3, 7, 8, 4, 6, 14, 12	mircgr 25552	. . . . . 6 |- ( ph -> ( B .- ( ( S ` B ) ` C ) ) = ( B .- C ) )
81	1, 2, 3, 4, 6, 15, 6, 12, 80	tgcgrcomlr 25375	. . . . 5 |- ( ph -> ( ( ( S ` B ) ` C ) .- B ) = ( C .- B ) )
82	1, 2, 3, 4, 13, 5, 15, 6, 16, 11, 12, 6, 27, 32, 36, 42, 79, 81	tgifscgr 25403	. . . 4 |- ( ph -> ( Q .- B ) = ( ( M ` Q ) .- B ) )
83	1, 2, 3, 4, 5, 6, 11, 6, 82	tgcgrcomlr 25375	. . 3 |- ( ph -> ( B .- Q ) = ( B .- ( M ` Q ) ) )
84	10	fveq1i 6192	. . . 4 |- ( M ` Q ) = ( ( S ` A ) ` Q )
85	84	oveq2i 6661	. . 3 |- ( B .- ( M ` Q ) ) = ( B .- ( ( S ` A ) ` Q ) )
86	83, 85	syl6eq 2672	. 2 |- ( ph -> ( B .- Q ) = ( B .- ( ( S ` A ) ` Q ) ) )
87	1, 2, 3, 7, 8, 4, 6, 9, 5	israg 25592	. 2 |- ( ph -> ( <" B A Q "> e. (∟G ` G ) <-> ( B .- Q ) = ( B .- ( ( S ` A ) ` Q ) ) ) )
88	86, 87	mpbird 247	1 |- ( ph -> <" B A Q "> e. (∟G ` G ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  pInvGcmir 25547  ∟Gcrag 25588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-cgrg 25406  df-mir 25548  df-rag 25589

This theorem is referenced by:  colperpexlem3  25624