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Theorem btwnouttr2 32129
Description: Outer transitivity law for betweenness. Left-hand side of Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Scott Fenton, 12-Jun-2013.)
Assertion
Ref Expression
btwnouttr2 |- ( ( 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 Btwn <. A , D >. ) )
Proof of Theorem btwnouttr2
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 simp1 1061 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> N e. NN )
2 simp2l 1087 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
3 simp3l 1089 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
4 simp3r 1090 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
5 axsegcon 25807 . . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> E. x e. ( EE ` N ) ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) )
61, 2, 3, 3, 4, 5syl122anc 1335 . . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> E. x e. ( EE ` N ) ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) )
76adantr 481 . . 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 >. ) ) -> E. x e. ( EE ` N ) ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) )
8 simprrl 804 . . . . . . 7 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) /\ ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) ) ) -> C Btwn <. A , x >. )
9 simprl1 1106 . . . . . . . . 9 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) /\ ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) ) ) -> B =/= C )
10 simpl2 1065 . . . . . . . . . . . . 13 |- ( ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) /\ ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) ) -> B Btwn <. A , C >. )
11 simprl 794 . . . . . . . . . . . . 13 |- ( ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) /\ ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) ) -> C Btwn <. A , x >. )
1210, 11jca 554 . . . . . . . . . . . 12 |- ( ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) /\ ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) ) -> ( B Btwn <. A , C >. /\ C Btwn <. A , x >. ) )
1312adantl 482 . . . . . . . . . . 11 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) /\ ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) ) ) -> ( B Btwn <. A , C >. /\ C Btwn <. A , x >. ) )
14 simpl1 1064 . . . . . . . . . . . . 13 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> N e. NN )
15 simpl2l 1114 . . . . . . . . . . . . 13 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> A e. ( EE ` N ) )
16 simpl2r 1115 . . . . . . . . . . . . 13 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> B e. ( EE ` N ) )
17 simpl3l 1116 . . . . . . . . . . . . 13 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> C e. ( EE ` N ) )
18 simpr 477 . . . . . . . . . . . . 13 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> x e. ( EE ` N ) )
19 btwnexch3 32127 . . . . . . . . . . . . 13 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ x e. ( EE ` N ) ) ) -> ( ( B Btwn <. A , C >. /\ C Btwn <. A , x >. ) -> C Btwn <. B , x >. ) )
2014, 15, 16, 17, 18, 19syl122anc 1335 . . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( ( B Btwn <. A , C >. /\ C Btwn <. A , x >. ) -> C Btwn <. B , x >. ) )
2120adantr 481 . . . . . . . . . . 11 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) /\ ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) ) ) -> ( ( B Btwn <. A , C >. /\ C Btwn <. A , x >. ) -> C Btwn <. B , x >. ) )
2213, 21mpd 15 . . . . . . . . . 10 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) /\ ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) ) ) -> C Btwn <. B , x >. )
23 simprrr 805 . . . . . . . . . 10 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) /\ ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) ) ) -> <. C , x >.Cgr <. C , D >. )
2422, 23jca 554 . . . . . . . . 9 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) /\ ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) ) ) -> ( C Btwn <. B , x >. /\ <. C , x >.Cgr <. C , D >. ) )
25 simprl3 1108 . . . . . . . . . 10 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) /\ ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) ) ) -> C Btwn <. B , D >. )
26 simpl3r 1117 . . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> D e. ( EE ` N ) )
2714, 17, 26cgrrflxd 32095 . . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> <. C , D >.Cgr <. C , D >. )
2827adantr 481 . . . . . . . . . 10 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) /\ ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) ) ) -> <. C , D >.Cgr <. C , D >. )
2925, 28jca 554 . . . . . . . . 9 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) /\ ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) ) ) -> ( C Btwn <. B , D >. /\ <. C , D >.Cgr <. C , D >. ) )
30 segconeq 32117 . . . . . . . . . . 11 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( B e. ( EE ` N ) /\ x e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( B =/= C /\ ( C Btwn <. B , x >. /\ <. C , x >.Cgr <. C , D >. ) /\ ( C Btwn <. B , D >. /\ <. C , D >.Cgr <. C , D >. ) ) -> x = D ) )
3114, 17, 17, 26, 16, 18, 26, 30syl133anc 1349 . . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) -> ( ( B =/= C /\ ( C Btwn <. B , x >. /\ <. C , x >.Cgr <. C , D >. ) /\ ( C Btwn <. B , D >. /\ <. C , D >.Cgr <. C , D >. ) ) -> x = D ) )
3231adantr 481 . . . . . . . . 9 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) /\ ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) ) ) -> ( ( B =/= C /\ ( C Btwn <. B , x >. /\ <. C , x >.Cgr <. C , D >. ) /\ ( C Btwn <. B , D >. /\ <. C , D >.Cgr <. C , D >. ) ) -> x = D ) )
339, 24, 29, 32mp3and 1427 . . . . . . . 8 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) /\ ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) ) ) -> x = D )
3433opeq2d 4409 . . . . . . 7 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) /\ ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) ) ) -> <. A , x >. = <. A , D >. )
358, 34breqtrd 4679 . . . . . 6 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) /\ ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) ) ) -> C Btwn <. A , D >. )
3635expr 643 . . . . 5 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ x e. ( EE ` N ) ) /\ ( B =/= C /\ B Btwn <. A , C >. /\ C Btwn <. B , D >. ) ) -> ( ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) -> C Btwn <. A , D >. ) )
3736an32s 846 . . . 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 >. ) ) /\ x e. ( EE ` N ) ) -> ( ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) -> C Btwn <. A , D >. ) )
3837rexlimdva 3031 . . 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 >. ) ) -> ( E. x e. ( EE ` N ) ( C Btwn <. A , x >. /\ <. C , x >.Cgr <. C , D >. ) -> C Btwn <. A , D >. ) )
397, 38mpd 15 . 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 >. ) ) -> C Btwn <. A , D >. )
4039ex 450 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 >. ) -> C Btwn <. A , D >. ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-ee 25771  df-btwn 25772  df-cgr 25773  df-ofs 32090
This theorem is referenced by:  btwnexch2  32130  btwnouttr  32131  btwnoutside  32232  lineelsb2  32255
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