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Theorem btwnouttr 24054
Description: Outer transitivity law for betweenness. Right hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 14-Jun-2013.)
Assertion
Ref Expression
btwnouttr  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  ->  B  Btwn  <. A ,  D >. ) )

Proof of Theorem btwnouttr
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 simp2r 984 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N ) )
3 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 ) )
4 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 ) )
5 necom 2528 . . . . . . . 8  |-  ( B  =/=  C  <->  C  =/=  B )
65a1i 12 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( B  =/= 
C  <->  C  =/=  B
) )
7 simp3l 985 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N ) )
8 btwncom 24044 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. A ,  C >. 
<->  B  Btwn  <. C ,  A >. ) )
91, 2, 4, 7, 8syl13anc 1186 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( B  Btwn  <. A ,  C >.  <->  B  Btwn  <. C ,  A >. ) )
10 btwncom 24044 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  -> 
( C  Btwn  <. B ,  D >. 
<->  C  Btwn  <. D ,  B >. ) )
111, 7, 2, 3, 10syl13anc 1186 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( C  Btwn  <. B ,  D >.  <->  C  Btwn  <. D ,  B >. ) )
126, 9, 113anbi123d 1254 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  <->  ( C  =/=  B  /\  B  Btwn  <. C ,  A >.  /\  C  Btwn  <. D ,  B >. ) ) )
13 3ancomb 945 . . . . . 6  |-  ( ( C  =/=  B  /\  B  Btwn  <. C ,  A >.  /\  C  Btwn  <. D ,  B >. )  <->  ( C  =/=  B  /\  C  Btwn  <. D ,  B >.  /\  B  Btwn  <. C ,  A >. ) )
1412, 13syl6bb 254 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  <->  ( C  =/=  B  /\  C  Btwn  <. D ,  B >.  /\  B  Btwn  <. C ,  A >. ) ) )
1514biimpa 472 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) )  ->  ( C  =/=  B  /\  C  Btwn  <. D ,  B >.  /\  B  Btwn  <. C ,  A >. ) )
16 btwnouttr2 24052 . . . . . 6  |-  ( ( N  e.  NN  /\  ( D  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) ) )  ->  ( ( C  =/=  B  /\  C  Btwn  <. D ,  B >.  /\  B  Btwn  <. C ,  A >. )  ->  B  Btwn  <. D ,  A >. ) )
171, 3, 7, 2, 4, 16syl122anc 1193 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( C  =/=  B  /\  C  Btwn  <. D ,  B >.  /\  B  Btwn  <. C ,  A >. )  ->  B  Btwn  <. D ,  A >. ) )
1817adantr 453 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) )  ->  ( ( C  =/=  B  /\  C  Btwn  <. D ,  B >.  /\  B  Btwn  <. C ,  A >. )  ->  B  Btwn  <. D ,  A >. ) )
1915, 18mpd 16 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) )  ->  B  Btwn  <. D ,  A >. )
201, 2, 3, 4, 19btwncomand 24045 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  /\  ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. ) )  ->  B  Btwn  <. A ,  D >. )
2120ex 425 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( ( B  =/=  C  /\  B  Btwn  <. A ,  C >.  /\  C  Btwn  <. B ,  D >. )  ->  B  Btwn  <. A ,  D >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 936    e. wcel 1685    =/= wne 2447   <.cop 3644   class class class wbr 4024   ` cfv 5221   NNcn 9741   EEcee 23923    Btwn cbtwn 23924
This theorem is referenced by:  lineunray  24177  lineelsb2  24178
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-sup 7189  df-oi 7220  df-card 7567  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-n0 9961  df-z 10020  df-uz 10226  df-rp 10350  df-ico 10656  df-icc 10657  df-fz 10777  df-fzo 10865  df-seq 11041  df-exp 11099  df-hash 11332  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-clim 11956  df-sum 12153  df-ee 23926  df-btwn 23927  df-cgr 23928  df-ofs 24013
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