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Theorem hltr 25505
Description: The half-line relation is transitive. Theorem 6.7 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 23-Feb-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 )
hltr.1 |- ( ph -> A ( K ` D ) B )
hltr.2 |- ( ph -> B ( K ` D ) C )
Assertion
Ref Expression
hltr |- ( ph -> A ( K ` D ) C )
Proof of Theorem hltr
StepHypRef Expression
1 ishlg.p . . . 4 |- P = ( Base ` G )
2 ishlg.i . . . 4 |- I = (Itv ` G )
3 ishlg.k . . . 4 |- K = (hlG ` G )
4 ishlg.a . . . 4 |- ( ph -> A e. P )
5 ishlg.b . . . 4 |- ( ph -> B e. P )
6 hltr.d . . . 4 |- ( ph -> D e. P )
7 hlln.1 . . . 4 |- ( ph -> G e. TarskiG )
8 hltr.1 . . . 4 |- ( ph -> A ( K ` D ) B )
91, 2, 3, 4, 5, 6, 7, 8hlne1 25500 . . 3 |- ( ph -> A =/= D )
10 ishlg.c . . . 4 |- ( ph -> C e. P )
11 hltr.2 . . . 4 |- ( ph -> B ( K ` D ) C )
121, 2, 3, 5, 10, 6, 7, 11hlne2 25501 . . 3 |- ( ph -> C =/= D )
13 eqid 2622 . . . . . . 7 |- ( dist ` G ) = ( dist ` G )
147ad2antrr 762 . . . . . . 7 |- ( ( ( ph /\ A e. ( D I B ) ) /\ B e. ( D I C ) ) -> G e. TarskiG )
156ad2antrr 762 . . . . . . 7 |- ( ( ( ph /\ A e. ( D I B ) ) /\ B e. ( D I C ) ) -> D e. P )
164ad2antrr 762 . . . . . . 7 |- ( ( ( ph /\ A e. ( D I B ) ) /\ B e. ( D I C ) ) -> A e. P )
175ad2antrr 762 . . . . . . 7 |- ( ( ( ph /\ A e. ( D I B ) ) /\ B e. ( D I C ) ) -> B e. P )
1810ad2antrr 762 . . . . . . 7 |- ( ( ( ph /\ A e. ( D I B ) ) /\ B e. ( D I C ) ) -> C e. P )
19 simplr 792 . . . . . . 7 |- ( ( ( ph /\ A e. ( D I B ) ) /\ B e. ( D I C ) ) -> A e. ( D I B ) )
20 simpr 477 . . . . . . 7 |- ( ( ( ph /\ A e. ( D I B ) ) /\ B e. ( D I C ) ) -> B e. ( D I C ) )
211, 13, 2, 14, 15, 16, 17, 18, 19, 20tgbtwnexch 25393 . . . . . 6 |- ( ( ( ph /\ A e. ( D I B ) ) /\ B e. ( D I C ) ) -> A e. ( D I C ) )
2221orcd 407 . . . . 5 |- ( ( ( ph /\ A e. ( D I B ) ) /\ B e. ( D I C ) ) -> ( A e. ( D I C ) \/ C e. ( D I A ) ) )
237ad2antrr 762 . . . . . 6 |- ( ( ( ph /\ A e. ( D I B ) ) /\ C e. ( D I B ) ) -> G e. TarskiG )
246ad2antrr 762 . . . . . 6 |- ( ( ( ph /\ A e. ( D I B ) ) /\ C e. ( D I B ) ) -> D e. P )
254ad2antrr 762 . . . . . 6 |- ( ( ( ph /\ A e. ( D I B ) ) /\ C e. ( D I B ) ) -> A e. P )
2610ad2antrr 762 . . . . . 6 |- ( ( ( ph /\ A e. ( D I B ) ) /\ C e. ( D I B ) ) -> C e. P )
275ad2antrr 762 . . . . . 6 |- ( ( ( ph /\ A e. ( D I B ) ) /\ C e. ( D I B ) ) -> B e. P )
28 simplr 792 . . . . . 6 |- ( ( ( ph /\ A e. ( D I B ) ) /\ C e. ( D I B ) ) -> A e. ( D I B ) )
29 simpr 477 . . . . . 6 |- ( ( ( ph /\ A e. ( D I B ) ) /\ C e. ( D I B ) ) -> C e. ( D I B ) )
301, 2, 23, 24, 25, 26, 27, 28, 29tgbtwnconn3 25472 . . . . 5 |- ( ( ( ph /\ A e. ( D I B ) ) /\ C e. ( D I B ) ) -> ( A e. ( D I C ) \/ C e. ( D I A ) ) )
311, 2, 3, 5, 10, 6, 7ishlg 25497 . . . . . . . 8 |- ( ph -> ( B ( K ` D ) C <-> ( B =/= D /\ C =/= D /\ ( B e. ( D I C ) \/ C e. ( D I B ) ) ) ) )
3211, 31mpbid 222 . . . . . . 7 |- ( ph -> ( B =/= D /\ C =/= D /\ ( B e. ( D I C ) \/ C e. ( D I B ) ) ) )
3332simp3d 1075 . . . . . 6 |- ( ph -> ( B e. ( D I C ) \/ C e. ( D I B ) ) )
3433adantr 481 . . . . 5 |- ( ( ph /\ A e. ( D I B ) ) -> ( B e. ( D I C ) \/ C e. ( D I B ) ) )
3522, 30, 34mpjaodan 827 . . . 4 |- ( ( ph /\ A e. ( D I B ) ) -> ( A e. ( D I C ) \/ C e. ( D I A ) ) )
367ad2antrr 762 . . . . . 6 |- ( ( ( ph /\ B e. ( D I A ) ) /\ B e. ( D I C ) ) -> G e. TarskiG )
376ad2antrr 762 . . . . . 6 |- ( ( ( ph /\ B e. ( D I A ) ) /\ B e. ( D I C ) ) -> D e. P )
385ad2antrr 762 . . . . . 6 |- ( ( ( ph /\ B e. ( D I A ) ) /\ B e. ( D I C ) ) -> B e. P )
394ad2antrr 762 . . . . . 6 |- ( ( ( ph /\ B e. ( D I A ) ) /\ B e. ( D I C ) ) -> A e. P )
4010ad2antrr 762 . . . . . 6 |- ( ( ( ph /\ B e. ( D I A ) ) /\ B e. ( D I C ) ) -> C e. P )
4132simp1d 1073 . . . . . . . 8 |- ( ph -> B =/= D )
4241necomd 2849 . . . . . . 7 |- ( ph -> D =/= B )
4342ad2antrr 762 . . . . . 6 |- ( ( ( ph /\ B e. ( D I A ) ) /\ B e. ( D I C ) ) -> D =/= B )
44 simplr 792 . . . . . 6 |- ( ( ( ph /\ B e. ( D I A ) ) /\ B e. ( D I C ) ) -> B e. ( D I A ) )
45 simpr 477 . . . . . 6 |- ( ( ( ph /\ B e. ( D I A ) ) /\ B e. ( D I C ) ) -> B e. ( D I C ) )
461, 2, 36, 37, 38, 39, 40, 43, 44, 45tgbtwnconn1 25470 . . . . 5 |- ( ( ( ph /\ B e. ( D I A ) ) /\ B e. ( D I C ) ) -> ( A e. ( D I C ) \/ C e. ( D I A ) ) )
477ad2antrr 762 . . . . . . 7 |- ( ( ( ph /\ B e. ( D I A ) ) /\ C e. ( D I B ) ) -> G e. TarskiG )
486ad2antrr 762 . . . . . . 7 |- ( ( ( ph /\ B e. ( D I A ) ) /\ C e. ( D I B ) ) -> D e. P )
4910ad2antrr 762 . . . . . . 7 |- ( ( ( ph /\ B e. ( D I A ) ) /\ C e. ( D I B ) ) -> C e. P )
505ad2antrr 762 . . . . . . 7 |- ( ( ( ph /\ B e. ( D I A ) ) /\ C e. ( D I B ) ) -> B e. P )
514ad2antrr 762 . . . . . . 7 |- ( ( ( ph /\ B e. ( D I A ) ) /\ C e. ( D I B ) ) -> A e. P )
52 simpr 477 . . . . . . 7 |- ( ( ( ph /\ B e. ( D I A ) ) /\ C e. ( D I B ) ) -> C e. ( D I B ) )
53 simplr 792 . . . . . . 7 |- ( ( ( ph /\ B e. ( D I A ) ) /\ C e. ( D I B ) ) -> B e. ( D I A ) )
541, 13, 2, 47, 48, 49, 50, 51, 52, 53tgbtwnexch 25393 . . . . . 6 |- ( ( ( ph /\ B e. ( D I A ) ) /\ C e. ( D I B ) ) -> C e. ( D I A ) )
5554olcd 408 . . . . 5 |- ( ( ( ph /\ B e. ( D I A ) ) /\ C e. ( D I B ) ) -> ( A e. ( D I C ) \/ C e. ( D I A ) ) )
5633adantr 481 . . . . 5 |- ( ( ph /\ B e. ( D I A ) ) -> ( B e. ( D I C ) \/ C e. ( D I B ) ) )
5746, 55, 56mpjaodan 827 . . . 4 |- ( ( ph /\ B e. ( D I A ) ) -> ( A e. ( D I C ) \/ C e. ( D I A ) ) )
581, 2, 3, 4, 5, 6, 7ishlg 25497 . . . . . 6 |- ( ph -> ( A ( K ` D ) B <-> ( A =/= D /\ B =/= D /\ ( A e. ( D I B ) \/ B e. ( D I A ) ) ) ) )
598, 58mpbid 222 . . . . 5 |- ( ph -> ( A =/= D /\ B =/= D /\ ( A e. ( D I B ) \/ B e. ( D I A ) ) ) )
6059simp3d 1075 . . . 4 |- ( ph -> ( A e. ( D I B ) \/ B e. ( D I A ) ) )
6135, 57, 60mpjaodan 827 . . 3 |- ( ph -> ( A e. ( D I C ) \/ C e. ( D I A ) ) )
629, 12, 613jca 1242 . 2 |- ( ph -> ( A =/= D /\ C =/= D /\ ( A e. ( D I C ) \/ C e. ( D I A ) ) ) )
631, 2, 3, 4, 10, 6, 7ishlg 25497 . 2 |- ( ph -> ( A ( K ` D ) C <-> ( A =/= D /\ C =/= D /\ ( A e. ( D I C ) \/ C e. ( D I A ) ) ) ) )
6462, 63mpbird 247 1 |- ( ph -> A ( K ` D ) C )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ 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  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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-cgrg 25406  df-hlg 25496
This theorem is referenced by:  opphllem4  25642  cgrahl1  25708  cgrahl2  25709  cgrahl  25718  acopyeu  25725  inaghl  25731
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