Theorem tgellng 25448

Description: Property of lying on the line going through points X and Y. Definition 4.10 of [Schwabhauser] p. 36. We choose the notation Z e. ( X (LineG ` G ) Y ) instead of "colinear" because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 28-Mar-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 )
tglngval.z	|- ( ph -> X =/= Y )
tgellng.z	|- ( ph -> Z e. P )

Assertion

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

Proof of Theorem tgellng

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tglngval.p	. . . . 5 |- P = ( Base ` G )
2		tglngval.l	. . . . 5 |- L = (LineG ` G )
3		tglngval.i	. . . . 5 |- I = (Itv ` G )
4		tglngval.g	. . . . 5 |- ( ph -> G e. TarskiG )
5		tglngval.x	. . . . 5 |- ( ph -> X e. P )
6		tglngval.y	. . . . 5 |- ( ph -> Y e. P )
7		tglngval.z	. . . . 5 |- ( ph -> X =/= Y )
8	1, 2, 3, 4, 5, 6, 7	tglngval 25446	. . . 4 |- ( ph -> ( X L Y ) = { z e. P | ( z e. ( X I Y ) \/ X e. ( z I Y ) \/ Y e. ( X I z ) ) } )
9	8	eleq2d 2687	. . 3 |- ( ph -> ( Z e. ( X L Y ) <-> Z e. { z e. P | ( z e. ( X I Y ) \/ X e. ( z I Y ) \/ Y e. ( X I z ) ) } ) )
10		eleq1 2689	. . . . 5 |- ( z = Z -> ( z e. ( X I Y ) <-> Z e. ( X I Y ) ) )
11		oveq1 6657	. . . . . 6 |- ( z = Z -> ( z I Y ) = ( Z I Y ) )
12	11	eleq2d 2687	. . . . 5 |- ( z = Z -> ( X e. ( z I Y ) <-> X e. ( Z I Y ) ) )
13		oveq2 6658	. . . . . 6 |- ( z = Z -> ( X I z ) = ( X I Z ) )
14	13	eleq2d 2687	. . . . 5 |- ( z = Z -> ( Y e. ( X I z ) <-> Y e. ( X I Z ) ) )
15	10, 12, 14	3orbi123d 1398	. . . 4 |- ( z = Z -> ( ( z e. ( X I Y ) \/ X e. ( z I Y ) \/ Y e. ( X I z ) ) <-> ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) ) )
16	15	elrab 3363	. . 3 |- ( Z e. { z e. P | ( z e. ( X I Y ) \/ X e. ( z I Y ) \/ Y e. ( X I z ) ) } <-> ( Z e. P /\ ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) ) )
17	9, 16	syl6bb 276	. 2 |- ( ph -> ( Z e. ( X L Y ) <-> ( Z e. P /\ ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) ) ) )
18		tgellng.z	. . 3 |- ( ph -> Z e. P )
19	18	biantrurd 529	. 2 |- ( ph -> ( ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) <-> ( Z e. P /\ ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) ) ) )
20	17, 19	bitr4d 271	1 |- ( ph -> ( Z e. ( X L Y ) <-> ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   ` cfv 5888  (class class class)co 6650   Basecbs 15857  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-trkg 25352

This theorem is referenced by:  tgcolg  25449  hlln  25502  lnhl  25510  btwnlng1  25514  btwnlng2  25515  btwnlng3  25516  lncom  25517  lnrot1  25518  lnrot2  25519  tglineeltr  25526  colmid  25583  cgracol  25719