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Theorem tgbtwnconn3 23131
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 2451 . . . 4  |-  ( dist `  G )  =  (
dist `  G )
3 tgbtwnconn.i . . . 4  |-  I  =  (Itv `  G )
4 tgbtwnconn.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
54adantr 465 . . . 4  |-  ( (
ph  /\  ( # `  P
)  =  1 )  ->  G  e. TarskiG )
6 tgbtwnconn.b . . . . 5  |-  ( ph  ->  B  e.  P )
76adantr 465 . . . 4  |-  ( (
ph  /\  ( # `  P
)  =  1 )  ->  B  e.  P
)
8 tgbtwnconn.a . . . . 5  |-  ( ph  ->  A  e.  P )
98adantr 465 . . . 4  |-  ( (
ph  /\  ( # `  P
)  =  1 )  ->  A  e.  P
)
10 tgbtwnconn.c . . . . 5  |-  ( ph  ->  C  e.  P )
1110adantr 465 . . . 4  |-  ( (
ph  /\  ( # `  P
)  =  1 )  ->  C  e.  P
)
12 simpr 461 . . . 4  |-  ( (
ph  /\  ( # `  P
)  =  1 )  ->  ( # `  P
)  =  1 )
131, 2, 3, 5, 7, 9, 11, 12tgldim0itv 23077 . . 3  |-  ( (
ph  /\  ( # `  P
)  =  1 )  ->  B  e.  ( A I C ) )
1413orcd 392 . 2  |-  ( (
ph  /\  ( # `  P
)  =  1 )  ->  ( B  e.  ( A I C )  \/  C  e.  ( A I B ) ) )
154ad3antrrr 729 . . . 4  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  G  e. TarskiG )
16 simplr 754 . . . 4  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  p  e.  P
)
178ad3antrrr 729 . . . 4  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  A  e.  P
)
186ad3antrrr 729 . . . 4  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  B  e.  P
)
1910ad3antrrr 729 . . . 4  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  C  e.  P
)
20 simprr 756 . . . . 5  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  A  =/=  p
)
2120necomd 2719 . . . 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 729 . . . . . 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 729 . . . . . 6  |-  ( ( ( ( ph  /\  2  <_  ( # `  P
) )  /\  p  e.  P )  /\  ( A  e.  ( D I p )  /\  A  =/=  p ) )  ->  B  e.  ( A I D ) )
26 simprl 755 . . . . . . 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 23061 . . . . . 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 23066 . . . . 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 23061 . . . 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 729 . . . . . . 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 23061 . . . . . 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 23067 . . . . 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 23061 . . . 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 23130 . . 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 465 . . . 4  |-  ( (
ph  /\  2  <_  (
# `  P )
)  ->  G  e. TarskiG )
3722adantr 465 . . . 4  |-  ( (
ph  /\  2  <_  (
# `  P )
)  ->  D  e.  P )
388adantr 465 . . . 4  |-  ( (
ph  /\  2  <_  (
# `  P )
)  ->  A  e.  P )
39 simpr 461 . . . 4  |-  ( (
ph  /\  2  <_  (
# `  P )
)  ->  2  <_  (
# `  P )
)
401, 2, 3, 36, 37, 38, 39tgbtwndiff 23079 . . 3  |-  ( (
ph  /\  2  <_  (
# `  P )
)  ->  E. p  e.  P  ( A  e.  ( D I p )  /\  A  =/=  p ) )
4135, 40r19.29a 2960 . 2  |-  ( (
ph  /\  2  <_  (
# `  P )
)  ->  ( B  e.  ( A I C )  \/  C  e.  ( A I B ) ) )
421, 8tgldimor 23075 . 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 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4392   ` cfv 5518  (class class class)co 6192   1c1 9386    <_ cle 9522   2c2 10474   #chash 12206   Basecbs 14278   distcds 14351  TarskiGcstrkg 23007  Itvcitv 23014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-pm 7319  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-card 8212  df-cda 8440  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-fzo 11652  df-hash 12207  df-word 12333  df-concat 12335  df-s1 12336  df-s2 12579  df-s3 12580  df-trkgc 23026  df-trkgb 23027  df-trkgcb 23028  df-trkg 23032  df-cgrg 23085
This theorem is referenced by:  tgbtwnconnln3  23132
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