Theorem lnhl 25510

Description: Either a point C on the line AB is on the same side as A or on the opposite side. (Contributed by Thierry Arnoux, 21-Sep-2020.)

Hypotheses

Ref	Expression
ishlg.p	|- P = ( Base ` G )
ishlg.i	|- I = (Itv ` G )
ishlg.k	|- K = (hlG ` G )
ishlg.a	|- ( ph -> A e. P )
ishlg.b	|- ( ph -> B e. P )
ishlg.c	|- ( ph -> C e. P )
hlln.1	|- ( ph -> G e. TarskiG )
hltr.d	|- ( ph -> D e. P )
lnhl.l	|- L = (LineG ` G )
lnhl.1	|- ( ph -> C e. ( A L B ) )

Assertion

Ref	Expression
lnhl	|- ( ph -> ( C ( K ` B ) A \/ B e. ( A I C ) ) )

Proof of Theorem lnhl

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 |- ( ( ph /\ C = B ) -> C = B )
2		ishlg.p	. . . . . 6 |- P = ( Base ` G )
3		eqid 2622	. . . . . 6 |- ( dist ` G ) = ( dist ` G )
4		ishlg.i	. . . . . 6 |- I = (Itv ` G )
5		hlln.1	. . . . . 6 |- ( ph -> G e. TarskiG )
6		ishlg.a	. . . . . 6 |- ( ph -> A e. P )
7		ishlg.c	. . . . . 6 |- ( ph -> C e. P )
8	2, 3, 4, 5, 6, 7	tgbtwntriv2 25382	. . . . 5 |- ( ph -> C e. ( A I C ) )
9	8	adantr 481	. . . 4 |- ( ( ph /\ C = B ) -> C e. ( A I C ) )
10	1, 9	eqeltrrd 2702	. . 3 |- ( ( ph /\ C = B ) -> B e. ( A I C ) )
11	10	olcd 408	. 2 |- ( ( ph /\ C = B ) -> ( C ( K ` B ) A \/ B e. ( A I C ) ) )
12		lnhl.1	. . . . . 6 |- ( ph -> C e. ( A L B ) )
13		lnhl.l	. . . . . . 7 |- L = (LineG ` G )
14		ishlg.b	. . . . . . 7 |- ( ph -> B e. P )
15	2, 13, 4, 5, 6, 14, 12	tglngne 25445	. . . . . . 7 |- ( ph -> A =/= B )
16	2, 13, 4, 5, 6, 14, 15, 7	tgellng 25448	. . . . . 6 |- ( ph -> ( C e. ( A L B ) <-> ( C e. ( A I B ) \/ A e. ( C I B ) \/ B e. ( A I C ) ) ) )
17	12, 16	mpbid 222	. . . . 5 |- ( ph -> ( C e. ( A I B ) \/ A e. ( C I B ) \/ B e. ( A I C ) ) )
18		df-3or 1038	. . . . 5 |- ( ( C e. ( A I B ) \/ A e. ( C I B ) \/ B e. ( A I C ) ) <-> ( ( C e. ( A I B ) \/ A e. ( C I B ) ) \/ B e. ( A I C ) ) )
19	17, 18	sylib 208	. . . 4 |- ( ph -> ( ( C e. ( A I B ) \/ A e. ( C I B ) ) \/ B e. ( A I C ) ) )
20	19	adantr 481	. . 3 |- ( ( ph /\ C =/= B ) -> ( ( C e. ( A I B ) \/ A e. ( C I B ) ) \/ B e. ( A I C ) ) )
21		ishlg.k	. . . . . . . 8 |- K = (hlG ` G )
22	2, 4, 21, 7, 6, 14, 5	ishlg 25497	. . . . . . 7 |- ( ph -> ( C ( K ` B ) A <-> ( C =/= B /\ A =/= B /\ ( C e. ( B I A ) \/ A e. ( B I C ) ) ) ) )
23	22	adantr 481	. . . . . 6 |- ( ( ph /\ C =/= B ) -> ( C ( K ` B ) A <-> ( C =/= B /\ A =/= B /\ ( C e. ( B I A ) \/ A e. ( B I C ) ) ) ) )
24		df-3an 1039	. . . . . 6 |- ( ( C =/= B /\ A =/= B /\ ( C e. ( B I A ) \/ A e. ( B I C ) ) ) <-> ( ( C =/= B /\ A =/= B ) /\ ( C e. ( B I A ) \/ A e. ( B I C ) ) ) )
25	23, 24	syl6bb 276	. . . . 5 |- ( ( ph /\ C =/= B ) -> ( C ( K ` B ) A <-> ( ( C =/= B /\ A =/= B ) /\ ( C e. ( B I A ) \/ A e. ( B I C ) ) ) ) )
26		simpr 477	. . . . . . 7 |- ( ( ph /\ C =/= B ) -> C =/= B )
27	15	adantr 481	. . . . . . 7 |- ( ( ph /\ C =/= B ) -> A =/= B )
28	26, 27	jca 554	. . . . . 6 |- ( ( ph /\ C =/= B ) -> ( C =/= B /\ A =/= B ) )
29	28	biantrurd 529	. . . . 5 |- ( ( ph /\ C =/= B ) -> ( ( C e. ( B I A ) \/ A e. ( B I C ) ) <-> ( ( C =/= B /\ A =/= B ) /\ ( C e. ( B I A ) \/ A e. ( B I C ) ) ) ) )
30	5	adantr 481	. . . . . . 7 |- ( ( ph /\ C =/= B ) -> G e. TarskiG )
31	14	adantr 481	. . . . . . 7 |- ( ( ph /\ C =/= B ) -> B e. P )
32	7	adantr 481	. . . . . . 7 |- ( ( ph /\ C =/= B ) -> C e. P )
33	6	adantr 481	. . . . . . 7 |- ( ( ph /\ C =/= B ) -> A e. P )
34	2, 3, 4, 30, 31, 32, 33	tgbtwncomb 25384	. . . . . 6 |- ( ( ph /\ C =/= B ) -> ( C e. ( B I A ) <-> C e. ( A I B ) ) )
35	2, 3, 4, 30, 31, 33, 32	tgbtwncomb 25384	. . . . . 6 |- ( ( ph /\ C =/= B ) -> ( A e. ( B I C ) <-> A e. ( C I B ) ) )
36	34, 35	orbi12d 746	. . . . 5 |- ( ( ph /\ C =/= B ) -> ( ( C e. ( B I A ) \/ A e. ( B I C ) ) <-> ( C e. ( A I B ) \/ A e. ( C I B ) ) ) )
37	25, 29, 36	3bitr2d 296	. . . 4 |- ( ( ph /\ C =/= B ) -> ( C ( K ` B ) A <-> ( C e. ( A I B ) \/ A e. ( C I B ) ) ) )
38	37	orbi1d 739	. . 3 |- ( ( ph /\ C =/= B ) -> ( ( C ( K ` B ) A \/ B e. ( A I C ) ) <-> ( ( C e. ( A I B ) \/ A e. ( C I B ) ) \/ B e. ( A I C ) ) ) )
39	20, 38	mpbird 247	. 2 |- ( ( ph /\ C =/= B ) -> ( C ( K ` B ) A \/ B e. ( A I C ) ) )
40	11, 39	pm2.61dane 2881	1 |- ( ph -> ( C ( K ` B ) A \/ B e. ( A I C ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  hlGchlg 25495

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

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-reu 2919  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-hlg 25496

This theorem is referenced by:  hlpasch  25648