Theorem tgcolg 25449

Description: We choose the notation ( Z e. ( X L Y ) \/ X = Y ) instead of "colinear" in order to avoid defining an additional symbol for colinearity because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 25-May-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 )

Assertion

Ref	Expression
tgcolg	|- ( ph -> ( ( Z e. ( X L Y ) \/ X = Y ) <-> ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) ) )

Proof of Theorem tgcolg

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 |- ( ( ph /\ X = Y ) -> X = Y )
2	1	olcd 408	. . 3 |- ( ( ph /\ X = Y ) -> ( Z e. ( X L Y ) \/ X = Y ) )
3		tglngval.p	. . . . . 6 |- P = ( Base ` G )
4		eqid 2622	. . . . . 6 |- ( dist ` G ) = ( dist ` G )
5		tglngval.i	. . . . . 6 |- I = (Itv ` G )
6		tglngval.g	. . . . . . 7 |- ( ph -> G e. TarskiG )
7	6	adantr 481	. . . . . 6 |- ( ( ph /\ X = Y ) -> G e. TarskiG )
8		tgcolg.z	. . . . . . 7 |- ( ph -> Z e. P )
9	8	adantr 481	. . . . . 6 |- ( ( ph /\ X = Y ) -> Z e. P )
10		tglngval.x	. . . . . . 7 |- ( ph -> X e. P )
11	10	adantr 481	. . . . . 6 |- ( ( ph /\ X = Y ) -> X e. P )
12	3, 4, 5, 7, 9, 11	tgbtwntriv2 25382	. . . . 5 |- ( ( ph /\ X = Y ) -> X e. ( Z I X ) )
13	1	oveq2d 6666	. . . . 5 |- ( ( ph /\ X = Y ) -> ( Z I X ) = ( Z I Y ) )
14	12, 13	eleqtrd 2703	. . . 4 |- ( ( ph /\ X = Y ) -> X e. ( Z I Y ) )
15	14	3mix2d 1237	. . 3 |- ( ( ph /\ X = Y ) -> ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) )
16	2, 15	2thd 255	. 2 |- ( ( ph /\ X = Y ) -> ( ( Z e. ( X L Y ) \/ X = Y ) <-> ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) ) )
17		simpr 477	. . . . . 6 |- ( ( ph /\ X =/= Y ) -> X =/= Y )
18	17	neneqd 2799	. . . . 5 |- ( ( ph /\ X =/= Y ) -> -. X = Y )
19		biorf 420	. . . . 5 |- ( -. X = Y -> ( Z e. ( X L Y ) <-> ( X = Y \/ Z e. ( X L Y ) ) ) )
20	18, 19	syl 17	. . . 4 |- ( ( ph /\ X =/= Y ) -> ( Z e. ( X L Y ) <-> ( X = Y \/ Z e. ( X L Y ) ) ) )
21		orcom 402	. . . 4 |- ( ( X = Y \/ Z e. ( X L Y ) ) <-> ( Z e. ( X L Y ) \/ X = Y ) )
22	20, 21	syl6bb 276	. . 3 |- ( ( ph /\ X =/= Y ) -> ( Z e. ( X L Y ) <-> ( Z e. ( X L Y ) \/ X = Y ) ) )
23		tglngval.l	. . . 4 |- L = (LineG ` G )
24	6	adantr 481	. . . 4 |- ( ( ph /\ X =/= Y ) -> G e. TarskiG )
25	10	adantr 481	. . . 4 |- ( ( ph /\ X =/= Y ) -> X e. P )
26		tglngval.y	. . . . 5 |- ( ph -> Y e. P )
27	26	adantr 481	. . . 4 |- ( ( ph /\ X =/= Y ) -> Y e. P )
28	8	adantr 481	. . . 4 |- ( ( ph /\ X =/= Y ) -> Z e. P )
29	3, 23, 5, 24, 25, 27, 17, 28	tgellng 25448	. . 3 |- ( ( ph /\ X =/= Y ) -> ( Z e. ( X L Y ) <-> ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) ) )
30	22, 29	bitr3d 270	. 2 |- ( ( ph /\ X =/= Y ) -> ( ( Z e. ( X L Y ) \/ X = Y ) <-> ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) ) )
31	16, 30	pm2.61dane 2881	1 |- ( ph -> ( ( Z e. ( X L Y ) \/ X = Y ) <-> ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336

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-sep 4781  ax-nul 4789  ax-pr 4906

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-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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-trkgc 25347  df-trkgcb 25349  df-trkg 25352

This theorem is referenced by:  btwncolg1  25450  btwncolg2  25451  btwncolg3  25452  colcom  25453  colrot1  25454  lnxfr  25461  lnext  25462  tgfscgr  25463  tglowdim2l  25545  outpasch  25647