Theorem tgbtwnintr 25388

Description: Inner transitivity law for betweenness. Left-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.)

Hypotheses

Ref	Expression
tkgeom.p	|- P = ( Base ` G )
tkgeom.d	|- .- = ( dist ` G )
tkgeom.i	|- I = (Itv ` G )
tkgeom.g	|- ( ph -> G e. TarskiG )
tgbtwnintr.1	|- ( ph -> A e. P )
tgbtwnintr.2	|- ( ph -> B e. P )
tgbtwnintr.3	|- ( ph -> C e. P )
tgbtwnintr.4	|- ( ph -> D e. P )
tgbtwnintr.5	|- ( ph -> A e. ( B I D ) )
tgbtwnintr.6	|- ( ph -> B e. ( C I D ) )

Assertion

Ref	Expression
tgbtwnintr	|- ( ph -> B e. ( A I C ) )

Proof of Theorem tgbtwnintr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tkgeom.p	. . . 4 |- P = ( Base ` G )
2		tkgeom.d	. . . 4 |- .- = ( dist ` G )
3		tkgeom.i	. . . 4 |- I = (Itv ` G )
4		tkgeom.g	. . . . 5 |- ( ph -> G e. TarskiG )
5	4	ad2antrr 762	. . . 4 |- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I C ) /\ x e. ( B I B ) ) ) -> G e. TarskiG )
6		tgbtwnintr.2	. . . . 5 |- ( ph -> B e. P )
7	6	ad2antrr 762	. . . 4 |- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I C ) /\ x e. ( B I B ) ) ) -> B e. P )
8		simplr 792	. . . 4 |- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I C ) /\ x e. ( B I B ) ) ) -> x e. P )
9		simprr 796	. . . 4 |- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I C ) /\ x e. ( B I B ) ) ) -> x e. ( B I B ) )
10	1, 2, 3, 5, 7, 8, 9	axtgbtwnid 25365	. . 3 |- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I C ) /\ x e. ( B I B ) ) ) -> B = x )
11		simprl 794	. . 3 |- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I C ) /\ x e. ( B I B ) ) ) -> x e. ( A I C ) )
12	10, 11	eqeltrd 2701	. 2 |- ( ( ( ph /\ x e. P ) /\ ( x e. ( A I C ) /\ x e. ( B I B ) ) ) -> B e. ( A I C ) )
13		tgbtwnintr.3	. . 3 |- ( ph -> C e. P )
14		tgbtwnintr.4	. . 3 |- ( ph -> D e. P )
15		tgbtwnintr.1	. . 3 |- ( ph -> A e. P )
16		tgbtwnintr.5	. . 3 |- ( ph -> A e. ( B I D ) )
17		tgbtwnintr.6	. . 3 |- ( ph -> B e. ( C I D ) )
18	1, 2, 3, 4, 6, 13, 14, 15, 6, 16, 17	axtgpasch 25366	. 2 |- ( ph -> E. x e. P ( x e. ( A I C ) /\ x e. ( B I B ) ) )
19	12, 18	r19.29a 3078	1 |- ( ph -> B e. ( A I C ) )

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

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-trkgb 25348  df-trkg 25352

This theorem is referenced by:  tgbtwnexch3  25389  tgbtwnexch2  25391  tgbtwnconn1lem3  25469  tgbtwnconn3  25472  tgbtwnconn22  25474  tglineeltr  25526  mirconn  25573