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Theorem tgbtwnconn3 24084
Description: Inner connectivity law for betweenness. Theorem 5.3 of [Schwabhauser] p. 41. (Contributed by Thierry Arnoux, 17-May-2019.)
Hypotheses
Ref Expression
tgbtwnconn.p  |-  P  =  ( Base `  G
)
tgbtwnconn.i  |-  I  =  (Itv `  G )
tgbtwnconn.g  |-  ( ph  ->  G  e. TarskiG )
tgbtwnconn.a  |-  ( ph  ->  A  e.  P )
tgbtwnconn.b  |-  ( ph  ->  B  e.  P )
tgbtwnconn.c  |-  ( ph  ->  C  e.  P )
tgbtwnconn.d  |-  ( ph  ->  D  e.  P )
tgbtwnconn3.1  |-  ( ph  ->  B  e.  ( A I D ) )
tgbtwnconn3.2  |-  ( ph  ->  C  e.  ( A I D ) )
Assertion
Ref Expression
tgbtwnconn3  |-  ( ph  ->  ( B  e.  ( A I C )  \/  C  e.  ( A I B ) ) )

Proof of Theorem tgbtwnconn3
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 tgbtwnconn.p . . . 4  |-  P  =  ( Base `  G
)
2 eqid 2382 . . . 4  |-  ( dist `  G )  =  (
dist `  G )
3 tgbtwnconn.i . . . 4  |-  I  =  (Itv `  G )
4 tgbtwnconn.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
54adantr 463 . . . 4  |-  ( (
ph  /\  ( # `  P
)  =  1 )  ->  G  e. TarskiG )
6 tgbtwnconn.b . . . . 5  |-  ( ph  ->  B  e.  P )
76adantr 463 . . . 4  |-  ( (
ph  /\  ( # `  P
)  =  1 )  ->  B  e.  P
)
8 tgbtwnconn.a . . . . 5  |-  ( ph  ->  A  e.  P )
98adantr 463 . . . 4  |-  ( (
ph  /\  ( # `  P
)  =  1 )  ->  A  e.  P
)
10 tgbtwnconn.c . . . . 5  |-  ( ph  ->  C  e.  P )
1110adantr 463 . . . 4  |-  ( (
ph  /\  ( # `  P
)  =  1 )  ->  C  e.  P
)
12 simpr 459 . . . 4  |-  ( (
ph  /\  ( # `  P
)  =  1 )  ->  ( # `  P
)  =  1 )
131, 2, 3, 5, 7, 9, 11, 12tgldim0itv 24015 . . 3  |-  ( (
ph  /\  ( # `  P
)  =  1 )  ->  B  e.  ( A I C ) )
1413orcd 390 . 2  |-  ( (
ph  /\  ( # `  P
)  =  1 )  ->  ( B  e.  ( A I C )  \/  C  e.  ( A I B ) ) )
154ad3antrrr 727 . . . 4  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  G  e. TarskiG )
16 simplr 753 . . . 4  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  p  e.  P
)
178ad3antrrr 727 . . . 4  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  A  e.  P
)
186ad3antrrr 727 . . . 4  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  B  e.  P
)
1910ad3antrrr 727 . . . 4  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  C  e.  P
)
20 simprr 755 . . . . 5  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  A  =/=  p
)
2120necomd 2653 . . . 4  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  p  =/=  A
)
22 tgbtwnconn.d . . . . . . 7  |-  ( ph  ->  D  e.  P )
2322ad3antrrr 727 . . . . . 6  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  D  e.  P
)
24 tgbtwnconn3.1 . . . . . . 7  |-  ( ph  ->  B  e.  ( A I D ) )
2524ad3antrrr 727 . . . . . 6  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  B  e.  ( A I D ) )
26 simprl 754 . . . . . . 7  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  A  e.  ( D I p ) )
271, 2, 3, 15, 23, 17, 16, 26tgbtwncom 23999 . . . . . 6  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  A  e.  ( p I D ) )
281, 2, 3, 15, 18, 17, 16, 23, 25, 27tgbtwnintr 24004 . . . . 5  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  A  e.  ( B I p ) )
291, 2, 3, 15, 18, 17, 16, 28tgbtwncom 23999 . . . 4  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  A  e.  ( p I B ) )
30 tgbtwnconn3.2 . . . . . . . 8  |-  ( ph  ->  C  e.  ( A I D ) )
3130ad3antrrr 727 . . . . . . 7  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  C  e.  ( A I D ) )
321, 2, 3, 15, 17, 19, 23, 31tgbtwncom 23999 . . . . . 6  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  C  e.  ( D I A ) )
331, 2, 3, 15, 23, 19, 17, 16, 32, 26tgbtwnexch3 24005 . . . . 5  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  A  e.  ( C I p ) )
341, 2, 3, 15, 19, 17, 16, 33tgbtwncom 23999 . . . 4  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  A  e.  ( p I C ) )
351, 3, 15, 16, 17, 18, 19, 21, 29, 34tgbtwnconn2 24083 . . 3  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  ( B  e.  ( A I C )  \/  C  e.  ( A I B ) ) )
364adantr 463 . . . 4  |-  ( (
ph  /\  2  <_  (
# `  P )
)  ->  G  e. TarskiG )
3722adantr 463 . . . 4  |-  ( (
ph  /\  2  <_  (
# `  P )
)  ->  D  e.  P )
388adantr 463 . . . 4  |-  ( (
ph  /\  2  <_  (
# `  P )
)  ->  A  e.  P )
39 simpr 459 . . . 4  |-  ( (
ph  /\  2  <_  (
# `  P )
)  ->  2  <_  (
# `  P )
)
401, 2, 3, 36, 37, 38, 39tgbtwndiff 24017 . . 3  |-  ( (
ph  /\  2  <_  (
# `  P )
)  ->  E. p  e.  P  ( A  e.  ( D I p )  /\  A  =/=  p ) )
4135, 40r19.29a 2924 . 2  |-  ( (
ph  /\  2  <_  (
# `  P )
)  ->  ( B  e.  ( A I C )  \/  C  e.  ( A I B ) ) )
421, 8tgldimor 24013 . 2  |-  ( ph  ->  ( ( # `  P
)  =  1  \/  2  <_  ( # `  P
) ) )
4314, 41, 42mpjaodan 784 1  |-  ( ph  ->  ( B  e.  ( A I C )  \/  C  e.  ( A I B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   1c1 9404    <_ cle 9540   2c2 10502   #chash 12307   Basecbs 14634   distcds 14711  TarskiGcstrkg 23942  Itvcitv 23949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-hash 12308  df-word 12446  df-concat 12448  df-s1 12449  df-s2 12724  df-s3 12725  df-trkgc 23961  df-trkgb 23962  df-trkgcb 23963  df-trkg 23967  df-cgrg 24023
This theorem is referenced by:  tgbtwnconnln3  24085  hltr  24114  hlbtwn  24115
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