Theorem tgbtwnxfr 25425

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.

Step	Hyp	Ref	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 )
5	4	ad2antrr 762	. . . 4 |- ( ( ( ph /\ e e. P ) /\ ( e e. ( D I F ) /\ <" A B C "> .~ <" D e F "> ) ) -> G e. TarskiG )
6		simplr 792	. . . 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 )
8	7	ad2antrr 762	. . . 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 )
10	9	ad2antrr 762	. . . . 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 )
12	11	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ e e. P ) /\ ( e e. ( D I F ) /\ <" A B C "> .~ <" D e F "> ) ) -> F e. P )
13		simprl 794	. . . . 5 |- ( ( ( ph /\ e e. P ) /\ ( e e. ( D I F ) /\ <" A B C "> .~ <" D e F "> ) ) -> e e. ( D I F ) )
14		eqidd 2623	. . . . 5 |- ( ( ( ph /\ e e. P ) /\ ( e e. ( D I F ) /\ <" A B C "> .~ <" D e F "> ) ) -> ( D .- F ) = ( D .- F ) )
15		eqidd 2623	. . . . 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 )
18	17	ad2antrr 762	. . . . . . . 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 )
20	19	ad2antrr 762	. . . . . . . 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 )
22	21	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ e e. P ) /\ ( e e. ( D I F ) /\ <" A B C "> .~ <" D e F "> ) ) -> C e. P )
23		simprr 796	. . . . . . . . 9 |- ( ( ( ph /\ e e. P ) /\ ( e e. ( D I F ) /\ <" A B C "> .~ <" D e F "> ) ) -> <" A B C "> .~ <" D e F "> )
24	1, 2, 3, 16, 5, 18, 20, 22, 10, 6, 12, 23	trgcgrcom 25423	. . . . . . . 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 "> )
26	25	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ e e. P ) /\ ( e e. ( D I F ) /\ <" A B C "> .~ <" D e F "> ) ) -> <" A B C "> .~ <" D E F "> )
27	1, 2, 3, 16, 5, 10, 6, 12, 18, 20, 22, 24, 10, 8, 12, 26	cgr3tr 25424	. . . . . . 7 |- ( ( ( ph /\ e e. P ) /\ ( e e. ( D I F ) /\ <" A B C "> .~ <" D e F "> ) ) -> <" D e F "> .~ <" D E F "> )
28	1, 2, 3, 16, 5, 10, 6, 12, 10, 8, 12, 27	trgcgrcom 25423	. . . . . 6 |- ( ( ( ph /\ e e. P ) /\ ( e e. ( D I F ) /\ <" A B C "> .~ <" D e F "> ) ) -> <" D E F "> .~ <" D e F "> )
29	1, 2, 3, 16, 5, 10, 8, 12, 10, 6, 12, 28	cgr3simp1 25415	. . . . 5 |- ( ( ( ph /\ e e. P ) /\ ( e e. ( D I F ) /\ <" A B C "> .~ <" D e F "> ) ) -> ( D .- E ) = ( D .- e ) )
30	1, 2, 3, 16, 5, 10, 8, 12, 10, 6, 12, 28	cgr3simp2 25416	. . . . . 6 |- ( ( ( ph /\ e e. P ) /\ ( e e. ( D I F ) /\ <" A B C "> .~ <" D e F "> ) ) -> ( E .- F ) = ( e .- F ) )
31	1, 2, 3, 5, 8, 12, 6, 12, 30	tgcgrcomlr 25375	. . . . 5 |- ( ( ( ph /\ e e. P ) /\ ( e e. ( D I F ) /\ <" A B C "> .~ <" D e F "> ) ) -> ( F .- E ) = ( F .- e ) )
32	1, 2, 3, 5, 10, 6, 12, 8, 10, 6, 12, 6, 13, 13, 14, 15, 29, 31	tgifscgr 25403	. . . 4 |- ( ( ( ph /\ e e. P ) /\ ( e e. ( D I F ) /\ <" A B C "> .~ <" D e F "> ) ) -> ( e .- E ) = ( e .- e ) )
33	1, 2, 3, 5, 6, 8, 6, 32	axtgcgrid 25362	. . 3 |- ( ( ( ph /\ e e. P ) /\ ( e e. ( D I F ) /\ <" A B C "> .~ <" D e F "> ) ) -> e = E )
34	33, 13	eqeltrrd 2702	. 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 ) )
36	1, 2, 3, 16, 4, 17, 19, 21, 9, 7, 11, 25	cgr3simp3 25417	. . . 4 |- ( ph -> ( C .- A ) = ( F .- D ) )
37	1, 2, 3, 4, 21, 17, 11, 9, 36	tgcgrcomlr 25375	. . 3 |- ( ph -> ( A .- C ) = ( D .- F ) )
38	1, 2, 3, 16, 4, 17, 19, 21, 9, 11, 35, 37	tgcgrxfr 25413	. 2 |- ( ph -> E. e e. P ( e e. ( D I F ) /\ <" A B C "> .~ <" D e F "> ) )
39	34, 38	r19.29a 3078	1 |- ( ph -> E e. ( D I F ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  cgrGccgrg 25405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-cgrg 25406

This theorem is referenced by:  lnxfr  25461  tgfscgr  25463  legov  25480  legov2  25481  legtrd  25484  mirbtwni  25566  cgrabtwn  25717  cgrahl  25718