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Theorem btwnconn3 24087
Description: Inner connectivity law for betweenness. Theorem 5.3 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 9-Oct-2013.)
Assertion
Ref Expression
btwnconn3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( B 
Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. )  ->  ( B  Btwn  <. A ,  C >.  \/  C  Btwn  <. A ,  B >. ) ) )

Proof of Theorem btwnconn3
StepHypRef Expression
1 simp1 960 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  N  e.  NN )
2 simp3r 989 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N ) )
3 simp2l 986 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N ) )
4 btwndiff 24011 . . 3  |-  ( ( N  e.  NN  /\  D  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) )  ->  E. p  e.  ( EE `  N
) ( A  Btwn  <. D ,  p >.  /\  A  =/=  p ) )
51, 2, 3, 4syl3anc 1187 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  E. p  e.  ( EE `  N ) ( A  Btwn  <. D ,  p >.  /\  A  =/=  p ) )
6 simprlr 742 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  /\  ( ( A  Btwn  <. D ,  p >.  /\  A  =/=  p )  /\  ( B  Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. ) ) )  ->  A  =/=  p
)
76necomd 2502 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  /\  ( ( A  Btwn  <. D ,  p >.  /\  A  =/=  p )  /\  ( B  Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. ) ) )  ->  p  =/=  A
)
8 simpl1 963 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  ->  N  e.  NN )
9 simpl2l 1013 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  ->  A  e.  ( EE `  N ) )
10 simpl2r 1014 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  ->  B  e.  ( EE `  N ) )
11 simpr 449 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  ->  p  e.  ( EE `  N ) )
12 simpl3r 1016 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  ->  D  e.  ( EE `  N ) )
13 simprrl 743 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  /\  ( ( A  Btwn  <. D ,  p >.  /\  A  =/=  p )  /\  ( B  Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. ) ) )  ->  B  Btwn  <. A ,  D >. )
148, 10, 9, 12, 13btwncomand 23999 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  /\  ( ( A  Btwn  <. D ,  p >.  /\  A  =/=  p )  /\  ( B  Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. ) ) )  ->  B  Btwn  <. D ,  A >. )
15 simprll 741 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  /\  ( ( A  Btwn  <. D ,  p >.  /\  A  =/=  p )  /\  ( B  Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. ) ) )  ->  A  Btwn  <. D ,  p >. )
168, 12, 10, 9, 11, 14, 15btwnexch3and 24005 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  /\  ( ( A  Btwn  <. D ,  p >.  /\  A  =/=  p )  /\  ( B  Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. ) ) )  ->  A  Btwn  <. B ,  p >. )
178, 9, 10, 11, 16btwncomand 23999 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  /\  ( ( A  Btwn  <. D ,  p >.  /\  A  =/=  p )  /\  ( B  Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. ) ) )  ->  A  Btwn  <. p ,  B >. )
18 simpl3l 1015 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  ->  C  e.  ( EE `  N ) )
19 simprrr 744 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  /\  ( ( A  Btwn  <. D ,  p >.  /\  A  =/=  p )  /\  ( B  Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. ) ) )  ->  C  Btwn  <. A ,  D >. )
208, 18, 9, 12, 19btwncomand 23999 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  /\  ( ( A  Btwn  <. D ,  p >.  /\  A  =/=  p )  /\  ( B  Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. ) ) )  ->  C  Btwn  <. D ,  A >. )
218, 12, 18, 9, 11, 20, 15btwnexch3and 24005 . . . . . . . 8  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  /\  ( ( A  Btwn  <. D ,  p >.  /\  A  =/=  p )  /\  ( B  Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. ) ) )  ->  A  Btwn  <. C ,  p >. )
228, 9, 18, 11, 21btwncomand 23999 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  /\  ( ( A  Btwn  <. D ,  p >.  /\  A  =/=  p )  /\  ( B  Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. ) ) )  ->  A  Btwn  <. p ,  C >. )
237, 17, 223jca 1137 . . . . . 6  |-  ( ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  /\  ( ( A  Btwn  <. D ,  p >.  /\  A  =/=  p )  /\  ( B  Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. ) ) )  ->  ( p  =/= 
A  /\  A  Btwn  <.
p ,  B >.  /\  A  Btwn  <. p ,  C >. ) )
2423ex 425 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  -> 
( ( ( A 
Btwn  <. D ,  p >.  /\  A  =/=  p
)  /\  ( B  Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. ) )  -> 
( p  =/=  A  /\  A  Btwn  <. p ,  B >.  /\  A  Btwn  <.
p ,  C >. ) ) )
25 btwnconn2 24086 . . . . . 6  |-  ( ( N  e.  NN  /\  ( p  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( ( p  =/=  A  /\  A  Btwn  <. p ,  B >.  /\  A  Btwn  <. p ,  C >. )  ->  ( B  Btwn  <. A ,  C >.  \/  C  Btwn  <. A ,  B >. ) ) )
268, 11, 9, 10, 18, 25syl122anc 1196 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  -> 
( ( p  =/= 
A  /\  A  Btwn  <.
p ,  B >.  /\  A  Btwn  <. p ,  C >. )  ->  ( B  Btwn  <. A ,  C >.  \/  C  Btwn  <. A ,  B >. ) ) )
2724, 26syld 42 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  -> 
( ( ( A 
Btwn  <. D ,  p >.  /\  A  =/=  p
)  /\  ( B  Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. ) )  -> 
( B  Btwn  <. A ,  C >.  \/  C  Btwn  <. A ,  B >. ) ) )
2827exp3a 427 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  p  e.  ( EE `  N ) )  -> 
( ( A  Btwn  <. D ,  p >.  /\  A  =/=  p )  ->  ( ( B 
Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. )  ->  ( B  Btwn  <. A ,  C >.  \/  C  Btwn  <. A ,  B >. ) ) ) )
2928rexlimdva 2640 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( E. p  e.  ( EE `  N
) ( A  Btwn  <. D ,  p >.  /\  A  =/=  p )  ->  ( ( B 
Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. )  ->  ( B  Btwn  <. A ,  C >.  \/  C  Btwn  <. A ,  B >. ) ) ) )
305, 29mpd 16 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( B 
Btwn  <. A ,  D >.  /\  C  Btwn  <. A ,  D >. )  ->  ( B  Btwn  <. A ,  C >.  \/  C  Btwn  <. A ,  B >. ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ wa 360    /\ w3a 939    e. wcel 1621    =/= wne 2419   E.wrex 2517   <.cop 3603   class class class wbr 3983   ` cfv 4659   NNcn 9700   EEcee 23877    Btwn cbtwn 23878
This theorem is referenced by:  midofsegid  24088  outsideoftr  24113  lineelsb2  24132
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-oadd 6437  df-er 6614  df-map 6728  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-sup 7148  df-oi 7179  df-card 7526  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-n0 9919  df-z 9978  df-uz 10184  df-rp 10308  df-ico 10614  df-icc 10615  df-fz 10735  df-fzo 10823  df-seq 10999  df-exp 11057  df-hash 11290  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-clim 11913  df-sum 12110  df-ee 23880  df-btwn 23881  df-cgr 23882  df-ofs 23967  df-ifs 24023  df-cgr3 24024  df-colinear 24025  df-fs 24026
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