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Theorem tgbtwnxfr 23125
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 2455 . . . . 5  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  ( D  .-  F )  =  ( D  .-  F
) )
15 eqidd 2455 . . . . 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 23123 . . . . . . . 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 23124 . . . . . . 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 23123 . . . . . 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 23118 . . . . 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 23119 . . . . . 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 23078 . . . . 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 23107 . . . 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 23067 . . 3  |-  ( ( ( ph  /\  e  e.  P )  /\  (
e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )  ->  e  =  E )
3433, 13eqeltrrd 2543 . 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 23120 . . . 4  |-  ( ph  ->  ( C  .-  A
)  =  ( F 
.-  D ) )
371, 2, 3, 4, 21, 17, 11, 9, 36tgcgrcomlr 23078 . . 3  |-  ( ph  ->  ( A  .-  C
)  =  ( D 
.-  F ) )
381, 2, 3, 16, 4, 17, 19, 21, 9, 11, 35, 37tgcgrxfr 23116 . 2  |-  ( ph  ->  E. e  e.  P  ( e  e.  ( D I F )  /\  <" A B C ">  .~  <" D e F "> ) )
3934, 38r19.29a 2968 1  |-  ( ph  ->  E  e.  ( D I F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   <"cs3 12591   Basecbs 14296   distcds 14370  TarskiGcstrkg 23032  Itvcitv 23039  cgrGccgrg 23109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8224  df-cda 8452  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-fzo 11670  df-hash 12225  df-word 12351  df-concat 12353  df-s1 12354  df-s2 12597  df-s3 12598  df-trkgc 23051  df-trkgb 23052  df-trkgcb 23053  df-trkg 23057  df-cgrg 23110
This theorem is referenced by:  lnxfr  23145  tgfscgr  23147  legov  23164  legov2  23165  legtrd  23168  mirbtwni  23225
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