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Theorem lnxfr 23777
Description: Transfer law for colinearity. Theorem 4.13 of [Schwabhauser] p. 37. (Contributed by Thierry Arnoux, 27-Apr-2019.)
Hypotheses
Ref Expression
tglngval.p  |-  P  =  ( Base `  G
)
tglngval.l  |-  L  =  (LineG `  G )
tglngval.i  |-  I  =  (Itv `  G )
tglngval.g  |-  ( ph  ->  G  e. TarskiG )
tglngval.x  |-  ( ph  ->  X  e.  P )
tglngval.y  |-  ( ph  ->  Y  e.  P )
tgcolg.z  |-  ( ph  ->  Z  e.  P )
lnxfr.r  |-  .~  =  (cgrG `  G )
lnxfr.a  |-  ( ph  ->  A  e.  P )
lnxfr.b  |-  ( ph  ->  B  e.  P )
lnxfr.c  |-  ( ph  ->  C  e.  P )
lnxfr.1  |-  ( ph  ->  ( Y  e.  ( X L Z )  \/  X  =  Z ) )
lnxfr.2  |-  ( ph  ->  <" X Y Z ">  .~  <" A B C "> )
Assertion
Ref Expression
lnxfr  |-  ( ph  ->  ( B  e.  ( A L C )  \/  A  =  C ) )

Proof of Theorem lnxfr
StepHypRef Expression
1 tglngval.p . . 3  |-  P  =  ( Base `  G
)
2 tglngval.l . . 3  |-  L  =  (LineG `  G )
3 tglngval.i . . 3  |-  I  =  (Itv `  G )
4 tglngval.g . . . 4  |-  ( ph  ->  G  e. TarskiG )
54adantr 465 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  G  e. TarskiG )
6 lnxfr.a . . . 4  |-  ( ph  ->  A  e.  P )
76adantr 465 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  A  e.  P )
8 lnxfr.c . . . 4  |-  ( ph  ->  C  e.  P )
98adantr 465 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  C  e.  P )
10 lnxfr.b . . . 4  |-  ( ph  ->  B  e.  P )
1110adantr 465 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  B  e.  P )
12 eqid 2467 . . . 4  |-  ( dist `  G )  =  (
dist `  G )
13 lnxfr.r . . . 4  |-  .~  =  (cgrG `  G )
14 tglngval.x . . . . 5  |-  ( ph  ->  X  e.  P )
1514adantr 465 . . . 4  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  X  e.  P )
16 tglngval.y . . . . 5  |-  ( ph  ->  Y  e.  P )
1716adantr 465 . . . 4  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  Y  e.  P )
18 tgcolg.z . . . . 5  |-  ( ph  ->  Z  e.  P )
1918adantr 465 . . . 4  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  Z  e.  P )
20 lnxfr.2 . . . . 5  |-  ( ph  ->  <" X Y Z ">  .~  <" A B C "> )
2120adantr 465 . . . 4  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  <" X Y Z ">  .~  <" A B C "> )
22 simpr 461 . . . 4  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  Y  e.  ( X I Z ) )
231, 12, 3, 13, 5, 15, 17, 19, 7, 11, 9, 21, 22tgbtwnxfr 23743 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  B  e.  ( A I C ) )
241, 2, 3, 5, 7, 9, 11, 23btwncolg1 23767 . 2  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  ( B  e.  ( A L C )  \/  A  =  C ) )
254adantr 465 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  G  e. TarskiG )
266adantr 465 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  A  e.  P )
278adantr 465 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  C  e.  P )
2810adantr 465 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  B  e.  P )
2916adantr 465 . . . 4  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  Y  e.  P )
3014adantr 465 . . . 4  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  X  e.  P )
3118adantr 465 . . . 4  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  Z  e.  P )
3220adantr 465 . . . . 5  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  <" X Y Z ">  .~  <" A B C "> )
331, 12, 3, 13, 25, 30, 29, 31, 26, 28, 27, 32cgr3swap12 23739 . . . 4  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  <" Y X Z ">  .~  <" B A C "> )
34 simpr 461 . . . 4  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  X  e.  ( Y I Z ) )
351, 12, 3, 13, 25, 29, 30, 31, 28, 26, 27, 33, 34tgbtwnxfr 23743 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  A  e.  ( B I C ) )
361, 2, 3, 25, 26, 27, 28, 35btwncolg2 23768 . 2  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  ( B  e.  ( A L C )  \/  A  =  C ) )
374adantr 465 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  G  e. TarskiG )
386adantr 465 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  A  e.  P )
398adantr 465 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  C  e.  P )
4010adantr 465 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  B  e.  P )
4114adantr 465 . . . 4  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  X  e.  P )
4218adantr 465 . . . 4  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  Z  e.  P )
4316adantr 465 . . . 4  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  Y  e.  P )
4420adantr 465 . . . . 5  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  <" X Y Z ">  .~  <" A B C "> )
451, 12, 3, 13, 37, 41, 43, 42, 38, 40, 39, 44cgr3swap23 23740 . . . 4  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  <" X Z Y ">  .~  <" A C B "> )
46 simpr 461 . . . 4  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  Z  e.  ( X I Y ) )
471, 12, 3, 13, 37, 41, 42, 43, 38, 39, 40, 45, 46tgbtwnxfr 23743 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  C  e.  ( A I B ) )
481, 2, 3, 37, 38, 39, 40, 47btwncolg3 23769 . 2  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  ( B  e.  ( A L C )  \/  A  =  C ) )
49 lnxfr.1 . . 3  |-  ( ph  ->  ( Y  e.  ( X L Z )  \/  X  =  Z ) )
501, 2, 3, 4, 14, 18, 16tgcolg 23766 . . 3  |-  ( ph  ->  ( ( Y  e.  ( X L Z )  \/  X  =  Z )  <->  ( Y  e.  ( X I Z )  \/  X  e.  ( Y I Z )  \/  Z  e.  ( X I Y ) ) ) )
5149, 50mpbid 210 . 2  |-  ( ph  ->  ( Y  e.  ( X I Z )  \/  X  e.  ( Y I Z )  \/  Z  e.  ( X I Y ) ) )
5224, 36, 48, 51mpjao3dan 1295 1  |-  ( ph  ->  ( B  e.  ( A L C )  \/  A  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 972    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   <"cs3 12773   Basecbs 14493   distcds 14567  TarskiGcstrkg 23650  Itvcitv 23657  LineGclng 23658  cgrGccgrg 23727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-fzo 11794  df-hash 12375  df-word 12509  df-concat 12511  df-s1 12512  df-s2 12779  df-s3 12780  df-trkgc 23669  df-trkgb 23670  df-trkgcb 23671  df-trkg 23675  df-cgrg 23728
This theorem is referenced by:  symquadlem  23871  midexlem  23874
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