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Theorem btwnconn3 24726
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 955 . . 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 984 . . 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 981 . . 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 24650 . . 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 1182 . 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 739 . . . . . . . 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 2529 . . . . . . 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 958 . . . . . . . 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 1008 . . . . . . . 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 1009 . . . . . . . 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 447 . . . . . . . 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 1011 . . . . . . . . 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 740 . . . . . . . . . 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 24638 . . . . . . . . 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 738 . . . . . . . . 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 24644 . . . . . . . 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 24638 . . . . . . 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 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 ) )  ->  C  e.  ( EE `  N ) )
19 simprrr 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 >. ) ) )  ->  C  Btwn  <. A ,  D >. )
208, 18, 9, 12, 19btwncomand 24638 . . . . . . . . 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 24644 . . . . . . . 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 24638 . . . . . . 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 1132 . . . . . 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 423 . . . . 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 24725 . . . . . 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 1191 . . . . 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 40 . . . 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 425 . . 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 2667 . 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 14 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 357    /\ wa 358    /\ w3a 934    e. wcel 1684    =/= wne 2446   E.wrex 2544   <.cop 3643   class class class wbr 4023   ` cfv 5255   NNcn 9746   EEcee 24516    Btwn cbtwn 24517
This theorem is referenced by:  midofsegid  24727  outsideoftr  24752  lineelsb2  24771
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-ee 24519  df-btwn 24520  df-cgr 24521  df-ofs 24606  df-ifs 24662  df-cgr3 24663  df-colinear 24664  df-fs 24665
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