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Theorem lnxfr 24154
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 463 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  G  e. TarskiG )
6 lnxfr.a . . . 4  |-  ( ph  ->  A  e.  P )
76adantr 463 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  A  e.  P )
8 lnxfr.c . . . 4  |-  ( ph  ->  C  e.  P )
98adantr 463 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  C  e.  P )
10 lnxfr.b . . . 4  |-  ( ph  ->  B  e.  P )
1110adantr 463 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  B  e.  P )
12 eqid 2454 . . . 4  |-  ( dist `  G )  =  (
dist `  G )
13 lnxfr.r . . . 4  |-  .~  =  (cgrG `  G )
14 tglngval.x . . . . 5  |-  ( ph  ->  X  e.  P )
1514adantr 463 . . . 4  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  X  e.  P )
16 tglngval.y . . . . 5  |-  ( ph  ->  Y  e.  P )
1716adantr 463 . . . 4  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  Y  e.  P )
18 tgcolg.z . . . . 5  |-  ( ph  ->  Z  e.  P )
1918adantr 463 . . . 4  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  Z  e.  P )
20 lnxfr.2 . . . . 5  |-  ( ph  ->  <" X Y Z ">  .~  <" A B C "> )
2120adantr 463 . . . 4  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  <" X Y Z ">  .~  <" A B C "> )
22 simpr 459 . . . 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 24119 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  B  e.  ( A I C ) )
241, 2, 3, 5, 7, 9, 11, 23btwncolg1 24143 . 2  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  ( B  e.  ( A L C )  \/  A  =  C ) )
254adantr 463 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  G  e. TarskiG )
266adantr 463 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  A  e.  P )
278adantr 463 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  C  e.  P )
2810adantr 463 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  B  e.  P )
2916adantr 463 . . . 4  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  Y  e.  P )
3014adantr 463 . . . 4  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  X  e.  P )
3118adantr 463 . . . 4  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  Z  e.  P )
3220adantr 463 . . . . 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 24115 . . . 4  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  <" Y X Z ">  .~  <" B A C "> )
34 simpr 459 . . . 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 24119 . . 3  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  A  e.  ( B I C ) )
361, 2, 3, 25, 26, 27, 28, 35btwncolg2 24144 . 2  |-  ( (
ph  /\  X  e.  ( Y I Z ) )  ->  ( B  e.  ( A L C )  \/  A  =  C ) )
374adantr 463 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  G  e. TarskiG )
386adantr 463 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  A  e.  P )
398adantr 463 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  C  e.  P )
4010adantr 463 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  B  e.  P )
4114adantr 463 . . . 4  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  X  e.  P )
4218adantr 463 . . . 4  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  Z  e.  P )
4316adantr 463 . . . 4  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  Y  e.  P )
4420adantr 463 . . . . 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 24116 . . . 4  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  <" X Z Y ">  .~  <" A C B "> )
46 simpr 459 . . . 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 24119 . . 3  |-  ( (
ph  /\  Z  e.  ( X I Y ) )  ->  C  e.  ( A I B ) )
481, 2, 3, 37, 38, 39, 40, 47btwncolg3 24145 . 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 24142 . . 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 1293 1  |-  ( ph  ->  ( B  e.  ( A L C )  \/  A  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    \/ w3o 970    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   <"cs3 12798   Basecbs 14716   distcds 14793  TarskiGcstrkg 24023  Itvcitv 24030  LineGclng 24031  cgrGccgrg 24103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-concat 12528  df-s1 12529  df-s2 12804  df-s3 12805  df-trkgc 24042  df-trkgb 24043  df-trkgcb 24044  df-trkg 24048  df-cgrg 24104
This theorem is referenced by:  symquadlem  24267  midexlem  24270
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