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Theorem tgbtwnxfr 23784
Description: A condition for extending betweenness to a new set of points based on congruence with another set of points. Theorem 4.6 of [Schwabhauser] p. 36. (Contributed by Thierry Arnoux, 27-Apr-2019.)
Hypotheses
Ref Expression
tgcgrxfr.p  |-  P  =  ( Base `  G
)
tgcgrxfr.m  |-  .-  =  ( dist `  G )
tgcgrxfr.i  |-  I  =  (Itv `  G )
tgcgrxfr.r  |-  .~  =  (cgrG `  G )
tgcgrxfr.g  |-  ( ph  ->  G  e. TarskiG )
tgbtwnxfr.a  |-  ( ph  ->  A  e.  P )
tgbtwnxfr.b  |-  ( ph  ->  B  e.  P )
tgbtwnxfr.c  |-  ( ph  ->  C  e.  P )
tgbtwnxfr.d  |-  ( ph  ->  D  e.  P )
tgbtwnxfr.e  |-  ( ph  ->  E  e.  P )
tgbtwnxfr.f  |-  ( ph  ->  F  e.  P )
tgbtwnxfr.2  |-  ( ph  ->  <" A B C ">  .~  <" D E F "> )
tgbtwnxfr.1  |-  ( ph  ->  B  e.  ( A I C ) )
Assertion
Ref Expression
tgbtwnxfr  |-  ( ph  ->  E  e.  ( D I F ) )

Proof of Theorem tgbtwnxfr
Dummy variable  e is distinct from all other variables.
StepHypRef Expression
1 tgcgrxfr.p . . . 4  |-  P  =  ( Base `  G
)
2 tgcgrxfr.m . . . 4  |-  .-  =  ( dist `  G )
3 tgcgrxfr.i . . . 4  |-  I  =  (Itv `  G )
4 tgcgrxfr.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
54ad2antrr 725 . . . 4  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  G  e. TarskiG )
6 simplr 754 . . . 4  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  e  e.  P )
7 tgbtwnxfr.e . . . . 5  |-  ( ph  ->  E  e.  P )
87ad2antrr 725 . . . 4  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  E  e.  P )
9 tgbtwnxfr.d . . . . . 6  |-  ( ph  ->  D  e.  P )
109ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  D  e.  P )
11 tgbtwnxfr.f . . . . . 6  |-  ( ph  ->  F  e.  P )
1211ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  F  e.  P )
13 simprl 755 . . . . 5  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  e  e.  ( D I F ) )
14 eqidd 2468 . . . . 5  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  ( D  .-  F )  =  ( D  .-  F
) )
15 eqidd 2468 . . . . 5  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  (
e  .-  F )  =  ( e  .-  F ) )
16 tgcgrxfr.r . . . . . 6  |-  .~  =  (cgrG `  G )
17 tgbtwnxfr.a . . . . . . . . 9  |-  ( ph  ->  A  e.  P )
1817ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  A  e.  P )
19 tgbtwnxfr.b . . . . . . . . 9  |-  ( ph  ->  B  e.  P )
2019ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  B  e.  P )
21 tgbtwnxfr.c . . . . . . . . 9  |-  ( ph  ->  C  e.  P )
2221ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  C  e.  P )
23 simprr 756 . . . . . . . . 9  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  <" A B C ">  .~  <" D e F "> )
241, 2, 3, 16, 5, 18, 20, 22, 10, 6, 12, 23trgcgrcom 23782 . . . . . . . 8  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  <" D
e F ">  .~ 
<" A B C "> )
25 tgbtwnxfr.2 . . . . . . . . 9  |-  ( ph  ->  <" A B C ">  .~  <" D E F "> )
2625ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  <" A B C ">  .~  <" D E F "> )
271, 2, 3, 16, 5, 10, 6, 12, 18, 20, 22, 24, 10, 8, 12, 26cgr3tr 23783 . . . . . . 7  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  <" D
e F ">  .~ 
<" D E F "> )
281, 2, 3, 16, 5, 10, 6, 12, 10, 8, 12, 27trgcgrcom 23782 . . . . . 6  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  <" D E F ">  .~  <" D e F "> )
291, 2, 3, 16, 5, 10, 8, 12, 10, 6, 12, 28cgr3simp1 23777 . . . . 5  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  ( D  .-  E )  =  ( D  .-  e
) )
301, 2, 3, 16, 5, 10, 8, 12, 10, 6, 12, 28cgr3simp2 23778 . . . . . 6  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  ( E  .-  F )  =  ( e  .-  F
) )
311, 2, 3, 5, 8, 12, 6, 12, 30tgcgrcomlr 23737 . . . . 5  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  ( F  .-  E )  =  ( F  .-  e
) )
321, 2, 3, 5, 10, 6, 12, 8, 10, 6, 12, 6, 13, 13, 14, 15, 29, 31tgifscgr 23766 . . . 4  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  (
e  .-  E )  =  ( e  .-  e ) )
331, 2, 3, 5, 6, 8, 6, 32axtgcgrid 23726 . . 3  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  e  =  E )
3433, 13eqeltrrd 2556 . 2  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  E  e.  ( D I F ) )
35 tgbtwnxfr.1 . . 3  |-  ( ph  ->  B  e.  ( A I C ) )
361, 2, 3, 16, 4, 17, 19, 21, 9, 7, 11, 25cgr3simp3 23779 . . . 4  |-  ( ph  ->  ( C  .-  A
)  =  ( F 
.-  D ) )
371, 2, 3, 4, 21, 17, 11, 9, 36tgcgrcomlr 23737 . . 3  |-  ( ph  ->  ( A  .-  C
)  =  ( D 
.-  F ) )
381, 2, 3, 16, 4, 17, 19, 21, 9, 11, 35, 37tgcgrxfr 23775 . 2  |-  ( ph  ->  E. e  e.  P  ( e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )
3934, 38r19.29a 3008 1  |-  ( ph  ->  E  e.  ( D I F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   <"cs3 12787   Basecbs 14507   distcds 14581  TarskiGcstrkg 23691  Itvcitv 23698  cgrGccgrg 23768
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-concat 12525  df-s1 12526  df-s2 12793  df-s3 12794  df-trkgc 23710  df-trkgb 23711  df-trkgcb 23712  df-trkg 23716  df-cgrg 23769
This theorem is referenced by:  lnxfr  23818  tgfscgr  23820  legov  23837  legov2  23838  legtrd  23841  mirbtwni  23903
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