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Theorem btwnouttr 23821
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 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 simp2r 987 . . 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 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 ) )
4 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 ) )
5 necom 2493 . . . . . . . 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 988 . . . . . . . 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 23811 . . . . . . . 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 1189 . . . . . . 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 23811 . . . . . . . 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 1189 . . . . . . 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 1257 . . . . . 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 948 . . . . . 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 23819 . . . . . 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 1196 . . . . 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 23812 . 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 939    e. wcel 1621    =/= wne 2412   <.cop 3547   class class class wbr 3920   ` cfv 4592   NNcn 9626   EEcee 23690    Btwn cbtwn 23691
This theorem is referenced by:  lineunray  23944  lineelsb2  23945
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-ico 10540  df-icc 10541  df-fz 10661  df-fzo 10749  df-seq 10925  df-exp 10983  df-hash 11216  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-clim 11839  df-sum 12036  df-ee 23693  df-btwn 23694  df-cgr 23695  df-ofs 23780
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