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Theorem btwnconn3 30409
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 995 . . 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 1024 . . 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 1021 . . 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 30333 . . 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 1228 . 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 763 . . . . . . . 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 2672 . . . . . . 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 998 . . . . . . . 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 1048 . . . . . . . 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 1049 . . . . . . . 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 459 . . . . . . . 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 1051 . . . . . . . . 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 764 . . . . . . . . . 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 30321 . . . . . . . . 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 762 . . . . . . . . 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 30327 . . . . . . . 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 30321 . . . . . . 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 1050 . . . . . . . 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 765 . . . . . . . . . 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 30321 . . . . . . . . 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 30327 . . . . . . . 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 30321 . . . . . . 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 1175 . . . . . 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 432 . . . . 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 30408 . . . . . 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 1237 . . . . 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 >. ) ) )
2827expd 434 . . 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 2893 . 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 setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    /\ w3a 972    e. wcel 1840    =/= wne 2596   E.wrex 2752   <.cop 3975   class class class wbr 4392   ` cfv 5523   NNcn 10494   EEcee 24490    Btwn cbtwn 24491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  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 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-map 7377  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-sup 7853  df-oi 7887  df-card 8270  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-n0 10755  df-z 10824  df-uz 11044  df-rp 11182  df-ico 11504  df-icc 11505  df-fz 11642  df-fzo 11766  df-seq 12060  df-exp 12119  df-hash 12358  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-clim 13365  df-sum 13563  df-ee 24493  df-btwn 24494  df-cgr 24495  df-ofs 30289  df-colinear 30345  df-ifs 30346  df-cgr3 30347  df-fs 30348
This theorem is referenced by:  midofsegid  30410  outsideoftr  30435  lineelsb2  30454
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