Theorem lnrot2 25519

Description: Rotating the points defining a line. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.)

Hypotheses

Ref	Expression
btwnlng1.p	|- P = ( Base ` G )
btwnlng1.i	|- I = (Itv ` G )
btwnlng1.l	|- L = (LineG ` G )
btwnlng1.g	|- ( ph -> G e. TarskiG )
btwnlng1.x	|- ( ph -> X e. P )
btwnlng1.y	|- ( ph -> Y e. P )
btwnlng1.z	|- ( ph -> Z e. P )
btwnlng1.d	|- ( ph -> X =/= Y )
lnrot2.1	|- ( ph -> X e. ( Y L Z ) )
lnrot2.2	|- ( ph -> Y =/= Z )

Assertion

Ref	Expression
lnrot2	|- ( ph -> Z e. ( X L Y ) )

Proof of Theorem lnrot2

Step	Hyp	Ref	Expression
1		lnrot2.1	. 2 |- ( ph -> X e. ( Y L Z ) )
2		btwnlng1.p	. . . . . 6 |- P = ( Base ` G )
3		eqid 2622	. . . . . 6 |- ( dist ` G ) = ( dist ` G )
4		btwnlng1.i	. . . . . 6 |- I = (Itv ` G )
5		btwnlng1.g	. . . . . 6 |- ( ph -> G e. TarskiG )
6		btwnlng1.y	. . . . . 6 |- ( ph -> Y e. P )
7		btwnlng1.x	. . . . . 6 |- ( ph -> X e. P )
8		btwnlng1.z	. . . . . 6 |- ( ph -> Z e. P )
9	2, 3, 4, 5, 6, 7, 8	tgbtwncomb 25384	. . . . 5 |- ( ph -> ( X e. ( Y I Z ) <-> X e. ( Z I Y ) ) )
10		biidd 252	. . . . 5 |- ( ph -> ( Y e. ( X I Z ) <-> Y e. ( X I Z ) ) )
11	2, 3, 4, 5, 6, 8, 7	tgbtwncomb 25384	. . . . 5 |- ( ph -> ( Z e. ( Y I X ) <-> Z e. ( X I Y ) ) )
12	9, 10, 11	3orbi123d 1398	. . . 4 |- ( ph -> ( ( X e. ( Y I Z ) \/ Y e. ( X I Z ) \/ Z e. ( Y I X ) ) <-> ( X e. ( Z I Y ) \/ Y e. ( X I Z ) \/ Z e. ( X I Y ) ) ) )
13		3orrot 1044	. . . 4 |- ( ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) <-> ( X e. ( Z I Y ) \/ Y e. ( X I Z ) \/ Z e. ( X I Y ) ) )
14	12, 13	syl6bbr 278	. . 3 |- ( ph -> ( ( X e. ( Y I Z ) \/ Y e. ( X I Z ) \/ Z e. ( Y I X ) ) <-> ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) ) )
15		btwnlng1.l	. . . 4 |- L = (LineG ` G )
16		lnrot2.2	. . . 4 |- ( ph -> Y =/= Z )
17	2, 15, 4, 5, 6, 8, 16, 7	tgellng 25448	. . 3 |- ( ph -> ( X e. ( Y L Z ) <-> ( X e. ( Y I Z ) \/ Y e. ( X I Z ) \/ Z e. ( Y I X ) ) ) )
18		btwnlng1.d	. . . 4 |- ( ph -> X =/= Y )
19	2, 15, 4, 5, 7, 6, 18, 8	tgellng 25448	. . 3 |- ( ph -> ( Z e. ( X L Y ) <-> ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) ) )
20	14, 17, 19	3bitr4d 300	. 2 |- ( ph -> ( X e. ( Y L Z ) <-> Z e. ( X L Y ) ) )
21	1, 20	mpbid 222	1 |- ( ph -> Z e. ( X L Y ) )

Syntax hints:   -> wi 4   \/ 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-pw 4160  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-trkgb 25348  df-trkgcb 25349  df-trkg 25352

This theorem is referenced by:  coltr  25542  mideulem2  25626