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Theorem btwnconn3 25985
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
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 simp1 957 . . 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 986 . . 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 983 . . 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 25909 . . 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 1184 . 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 740 . . . . . . . 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 2681 . . . . . . 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 960 . . . . . . . 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 1010 . . . . . . . 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 1011 . . . . . . . 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 448 . . . . . . . 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 1013 . . . . . . . . 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 741 . . . . . . . . . 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 25897 . . . . . . . . 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 739 . . . . . . . . 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 25903 . . . . . . . 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 25897 . . . . . . 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 1012 . . . . . . . 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 742 . . . . . . . . . 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 25897 . . . . . . . . 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 25903 . . . . . . . 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 25897 . . . . . . 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 1134 . . . . . 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 424 . . . . 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 25984 . . . . . 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 1193 . . . . 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 426 . . 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 2822 . 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 15 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 4    \/ wo 358    /\ wa 359    /\ w3a 936    e. wcel 1725    =/= wne 2598   E.wrex 2698   <.cop 3809   class class class wbr 4204   ` cfv 5445   NNcn 9989   EEcee 25775    Btwn cbtwn 25776
This theorem is referenced by:  midofsegid  25986  outsideoftr  26011  lineelsb2  26030
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-map 7011  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-sup 7437  df-oi 7468  df-card 7815  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-n0 10211  df-z 10272  df-uz 10478  df-rp 10602  df-ico 10911  df-icc 10912  df-fz 11033  df-fzo 11124  df-seq 11312  df-exp 11371  df-hash 11607  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-clim 12270  df-sum 12468  df-ee 25778  df-btwn 25779  df-cgr 25780  df-ofs 25865  df-ifs 25921  df-cgr3 25922  df-colinear 25923  df-fs 25924
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