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Theorem colperpexlem1 24230
Description: Lemma for colperp 24229. 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
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  |-  ( 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
)
111, 2, 3, 7, 8, 4, 9, 10, 5mircl 24168 . . . 4  |-  ( ph  ->  ( M `  Q
)  e.  P )
12 colperpexlem.c . . . . . 6  |-  ( ph  ->  C  e.  P )
131, 2, 3, 7, 8, 4, 9, 10, 12mircl 24168 . . . . 5  |-  ( ph  ->  ( M `  C
)  e.  P )
14 eqid 2457 . . . . . 6  |-  ( S `
 B )  =  ( S `  B
)
151, 2, 3, 7, 8, 4, 6, 14, 12mircl 24168 . . . . 5  |-  ( ph  ->  ( ( S `  B ) `  C
)  e.  P )
161, 2, 3, 7, 8, 4, 9, 10, 15mircl 24168 . . . . 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
)
191, 2, 3, 7, 8, 4, 6, 18, 12mircl 24168 . . . . . . . 8  |-  ( ph  ->  ( N `  C
)  e.  P )
2017, 19eqeltrd 2545 . . . . . . 7  |-  ( ph  ->  ( K `  ( M `  C )
)  e.  P )
21 colperpexlem.k . . . . . . . 8  |-  K  =  ( S `  Q
)
221, 2, 3, 7, 8, 4, 5, 21, 13mirbtwn 24165 . . . . . . 7  |-  ( ph  ->  Q  e.  ( ( K `  ( M `
 C ) ) I ( M `  C ) ) )
231, 2, 3, 4, 20, 5, 13, 22tgbtwncom 24005 . . . . . 6  |-  ( ph  ->  Q  e.  ( ( M `  C ) I ( K `  ( M `  C ) ) ) )
2418fveq1i 5873 . . . . . . . 8  |-  ( N `
 C )  =  ( ( S `  B ) `  C
)
2517, 24syl6eq 2514 . . . . . . 7  |-  ( ph  ->  ( K `  ( M `  C )
)  =  ( ( S `  B ) `
 C ) )
2625oveq2d 6312 . . . . . 6  |-  ( ph  ->  ( ( M `  C ) I ( K `  ( M `
 C ) ) )  =  ( ( M `  C ) I ( ( S `
 B ) `  C ) ) )
2723, 26eleqtrd 2547 . . . . 5  |-  ( ph  ->  Q  e.  ( ( M `  C ) I ( ( S `
 B ) `  C ) ) )
281, 2, 3, 4, 13, 5, 15, 27tgbtwncom 24005 . . . . . . 7  |-  ( ph  ->  Q  e.  ( ( ( S `  B
) `  C )
I ( M `  C ) ) )
291, 2, 3, 7, 8, 4, 9, 10, 15, 5, 13, 28mirbtwni 24177 . . . . . 6  |-  ( ph  ->  ( M `  Q
)  e.  ( ( M `  ( ( S `  B ) `
 C ) ) I ( M `  ( M `  C ) ) ) )
301, 2, 3, 7, 8, 4, 9, 10, 12mirmir 24169 . . . . . . 7  |-  ( ph  ->  ( M `  ( M `  C )
)  =  C )
3130oveq2d 6312 . . . . . 6  |-  ( ph  ->  ( ( M `  ( ( S `  B ) `  C
) ) I ( M `  ( M `
 C ) ) )  =  ( ( M `  ( ( S `  B ) `
 C ) ) I C ) )
3229, 31eleqtrd 2547 . . . . 5  |-  ( ph  ->  ( M `  Q
)  e.  ( ( M `  ( ( S `  B ) `
 C ) ) I C ) )
331, 2, 3, 4, 13, 15axtgcgrrflx 23985 . . . . . 6  |-  ( ph  ->  ( ( M `  C )  .-  (
( S `  B
) `  C )
)  =  ( ( ( S `  B
) `  C )  .-  ( M `  C
) ) )
341, 2, 3, 7, 8, 4, 9, 10, 15, 13miriso 24176 . . . . . 6  |-  ( ph  ->  ( ( M `  ( ( S `  B ) `  C
) )  .-  ( M `  ( M `  C ) ) )  =  ( ( ( S `  B ) `
 C )  .-  ( M `  C ) ) )
3530oveq2d 6312 . . . . . 6  |-  ( ph  ->  ( ( M `  ( ( S `  B ) `  C
) )  .-  ( M `  ( M `  C ) ) )  =  ( ( M `
 ( ( S `
 B ) `  C ) )  .-  C ) )
3633, 34, 353eqtr2d 2504 . . . . 5  |-  ( ph  ->  ( ( M `  C )  .-  (
( S `  B
) `  C )
)  =  ( ( M `  ( ( S `  B ) `
 C ) ) 
.-  C ) )
3725oveq2d 6312 . . . . . . 7  |-  ( ph  ->  ( Q  .-  ( K `  ( M `  C ) ) )  =  ( Q  .-  ( ( S `  B ) `  C
) ) )
381, 2, 3, 7, 8, 4, 5, 21, 13mircgr 24164 . . . . . . 7  |-  ( ph  ->  ( Q  .-  ( K `  ( M `  C ) ) )  =  ( Q  .-  ( M `  C ) ) )
3937, 38eqtr3d 2500 . . . . . 6  |-  ( ph  ->  ( Q  .-  (
( S `  B
) `  C )
)  =  ( Q 
.-  ( M `  C ) ) )
401, 2, 3, 7, 8, 4, 9, 10, 5, 13miriso 24176 . . . . . 6  |-  ( ph  ->  ( ( M `  Q )  .-  ( M `  ( M `  C ) ) )  =  ( Q  .-  ( M `  C ) ) )
4130oveq2d 6312 . . . . . 6  |-  ( ph  ->  ( ( M `  Q )  .-  ( M `  ( M `  C ) ) )  =  ( ( M `
 Q )  .-  C ) )
4239, 40, 413eqtr2d 2504 . . . . 5  |-  ( ph  ->  ( Q  .-  (
( S `  B
) `  C )
)  =  ( ( M `  Q ) 
.-  C ) )
431, 2, 3, 7, 8, 4, 9, 10, 6mirmir 24169 . . . . . . . . . 10  |-  ( ph  ->  ( M `  ( M `  B )
)  =  B )
44 eqidd 2458 . . . . . . . . . 10  |-  ( ph  ->  ( M `  B
)  =  ( M `
 B ) )
45 eqidd 2458 . . . . . . . . . 10  |-  ( ph  ->  ( M `  C
)  =  ( M `
 C ) )
4643, 44, 45s3eqd 12840 . . . . . . . . 9  |-  ( ph  ->  <" ( M `
 ( M `  B ) ) ( M `  B ) ( M `  C
) ">  =  <" B ( M `
 B ) ( M `  C ) "> )
471, 2, 3, 7, 8, 4, 9, 10, 6mircl 24168 . . . . . . . . . 10  |-  ( ph  ->  ( M `  B
)  e.  P )
48 simpr 461 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  A  =  B )  ->  A  =  B )
4948fveq2d 5876 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  =  B )  ->  ( M `  A )  =  ( M `  B ) )
504adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  A  =  B )  ->  G  e. TarskiG )
519adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  A  =  B )  ->  A  e.  P )
521, 2, 3, 7, 8, 50, 51, 10mircinv 24174 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  =  B )  ->  ( M `  A )  =  A )
5349, 52eqtr3d 2500 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  =  B )  ->  ( M `  B )  =  A )
54 eqidd 2458 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  =  B )  ->  B  =  B )
55 eqidd 2458 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  =  B )  ->  C  =  C )
5653, 54, 55s3eqd 12840 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =  B )  ->  <" ( M `  B ) B C ">  =  <" A B C "> )
57 colperpexlem.1 . . . . . . . . . . . . 13  |-  ( ph  ->  <" A B C ">  e.  (∟G `  G ) )
5857adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =  B )  ->  <" A B C ">  e.  (∟G `  G ) )
5956, 58eqeltrd 2545 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =  B )  ->  <" ( M `  B ) B C ">  e.  (∟G `  G ) )
604adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  B )  ->  G  e. TarskiG )
619adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  B )  ->  A  e.  P )
626adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  B )  ->  B  e.  P )
6312adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  B )  ->  C  e.  P )
641, 2, 3, 7, 8, 60, 61, 10, 62mircl 24168 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  B )  ->  ( M `  B )  e.  P
)
6557adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  B )  ->  <" A B C ">  e.  (∟G `  G ) )
66 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  B )  ->  A  =/=  B )
671, 2, 3, 7, 8, 60, 61, 10, 62mirbtwn 24165 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  =/=  B )  ->  A  e.  ( ( M `  B ) I B ) )
681, 7, 3, 60, 64, 62, 61, 67btwncolg1 24068 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  =/=  B )  ->  ( A  e.  ( ( M `  B ) L B )  \/  ( M `
 B )  =  B ) )
691, 7, 3, 60, 64, 62, 61, 68colcom 24071 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =/=  B )  ->  ( A  e.  ( B L ( M `  B ) )  \/  B  =  ( M `  B
) ) )
701, 2, 3, 7, 8, 60, 61, 62, 63, 64, 65, 66, 69ragcol 24202 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  B )  ->  <" ( M `  B ) B C ">  e.  (∟G `  G ) )
7159, 70pm2.61dane 2775 . . . . . . . . . 10  |-  ( ph  ->  <" ( M `
 B ) B C ">  e.  (∟G `  G ) )
721, 2, 3, 7, 8, 4, 47, 6, 12, 71, 10, 9mirrag 24204 . . . . . . . . 9  |-  ( ph  ->  <" ( M `
 ( M `  B ) ) ( M `  B ) ( M `  C
) ">  e.  (∟G `  G ) )
7346, 72eqeltrrd 2546 . . . . . . . 8  |-  ( ph  ->  <" B ( M `  B ) ( M `  C
) ">  e.  (∟G `  G ) )
741, 2, 3, 7, 8, 4, 6, 47, 13israg 24200 . . . . . . . 8  |-  ( ph  ->  ( <" B
( M `  B
) ( M `  C ) ">  e.  (∟G `  G )  <->  ( B  .-  ( M `
 C ) )  =  ( B  .-  ( ( S `  ( M `  B ) ) `  ( M `
 C ) ) ) ) )
7573, 74mpbid 210 . . . . . . 7  |-  ( ph  ->  ( B  .-  ( M `  C )
)  =  ( B 
.-  ( ( S `
 ( M `  B ) ) `  ( M `  C ) ) ) )
761, 2, 3, 7, 8, 4, 9, 10, 12, 6mirmir2 24180 . . . . . . . 8  |-  ( ph  ->  ( M `  (
( S `  B
) `  C )
)  =  ( ( S `  ( M `
 B ) ) `
 ( M `  C ) ) )
7776oveq2d 6312 . . . . . . 7  |-  ( ph  ->  ( B  .-  ( M `  ( ( S `  B ) `  C ) ) )  =  ( B  .-  ( ( S `  ( M `  B ) ) `  ( M `
 C ) ) ) )
7875, 77eqtr4d 2501 . . . . . 6  |-  ( ph  ->  ( B  .-  ( M `  C )
)  =  ( B 
.-  ( M `  ( ( S `  B ) `  C
) ) ) )
791, 2, 3, 4, 6, 13, 6, 16, 78tgcgrcomlr 23997 . . . . 5  |-  ( ph  ->  ( ( M `  C )  .-  B
)  =  ( ( M `  ( ( S `  B ) `
 C ) ) 
.-  B ) )
801, 2, 3, 7, 8, 4, 6, 14, 12mircgr 24164 . . . . . 6  |-  ( ph  ->  ( B  .-  (
( S `  B
) `  C )
)  =  ( B 
.-  C ) )
811, 2, 3, 4, 6, 15, 6, 12, 80tgcgrcomlr 23997 . . . . 5  |-  ( ph  ->  ( ( ( S `
 B ) `  C )  .-  B
)  =  ( C 
.-  B ) )
821, 2, 3, 4, 13, 5, 15, 6, 16, 11, 12, 6, 27, 32, 36, 42, 79, 81tgifscgr 24026 . . . 4  |-  ( ph  ->  ( Q  .-  B
)  =  ( ( M `  Q ) 
.-  B ) )
831, 2, 3, 4, 5, 6, 11, 6, 82tgcgrcomlr 23997 . . 3  |-  ( ph  ->  ( B  .-  Q
)  =  ( B 
.-  ( M `  Q ) ) )
8410fveq1i 5873 . . . 4  |-  ( M `
 Q )  =  ( ( S `  A ) `  Q
)
8584oveq2i 6307 . . 3  |-  ( B 
.-  ( M `  Q ) )  =  ( B  .-  (
( S `  A
) `  Q )
)
8683, 85syl6eq 2514 . 2  |-  ( ph  ->  ( B  .-  Q
)  =  ( B 
.-  ( ( S `
 A ) `  Q ) ) )
871, 2, 3, 7, 8, 4, 6, 9, 5israg 24200 . 2  |-  ( ph  ->  ( <" B A Q ">  e.  (∟G `  G )  <->  ( B  .-  Q )  =  ( B  .-  ( ( S `  A ) `
 Q ) ) ) )
8886, 87mpbird 232 1  |-  ( ph  ->  <" B A Q ">  e.  (∟G `  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   ` cfv 5594  (class class class)co 6296   <"cs3 12819   Basecbs 14644   distcds 14721  TarskiGcstrkg 23951  Itvcitv 23958  LineGclng 23959  pInvGcmir 24159  ∟Gcrag 24196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-concat 12548  df-s1 12549  df-s2 12825  df-s3 12826  df-trkgc 23970  df-trkgb 23971  df-trkgcb 23972  df-trkg 23976  df-cgrg 24029  df-mir 24160  df-rag 24197
This theorem is referenced by:  colperpexlem3  24232
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