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Theorem btwnsegle 24812
Description: If  B falls between  A and  C, then 
A B is no longer than  A C. (Contributed by Scott Fenton, 16-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
btwnsegle  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. A ,  C >.  ->  <. A ,  B >.  Seg<_  <. A ,  C >. ) )

Proof of Theorem btwnsegle
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simplr2 998 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  B  Btwn  <. A ,  C >. )  ->  B  e.  ( EE `  N ) )
2 simpr 447 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  B  Btwn  <. A ,  C >. )  ->  B  Btwn  <. A ,  C >. )
3 simpl 443 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  N  e.  NN )
4 simpr1 961 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
5 simpr2 962 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
63, 4, 5cgrrflxd 24683 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  <. A ,  B >.Cgr <. A ,  B >. )
76adantr 451 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  B  Btwn  <. A ,  C >. )  ->  <. A ,  B >.Cgr <. A ,  B >. )
8 breq1 4042 . . . . . 6  |-  ( x  =  B  ->  (
x  Btwn  <. A ,  C >. 
<->  B  Btwn  <. A ,  C >. ) )
9 opeq2 3813 . . . . . . 7  |-  ( x  =  B  ->  <. A ,  x >.  =  <. A ,  B >. )
109breq2d 4051 . . . . . 6  |-  ( x  =  B  ->  ( <. A ,  B >.Cgr <. A ,  x >.  <->  <. A ,  B >.Cgr <. A ,  B >. ) )
118, 10anbi12d 691 . . . . 5  |-  ( x  =  B  ->  (
( x  Btwn  <. A ,  C >.  /\  <. A ,  B >.Cgr <. A ,  x >. )  <->  ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  B >.Cgr <. A ,  B >. ) ) )
1211rspcev 2897 . . . 4  |-  ( ( B  e.  ( EE
`  N )  /\  ( B  Btwn  <. A ,  C >.  /\  <. A ,  B >.Cgr <. A ,  B >. ) )  ->  E. x  e.  ( EE `  N
) ( x  Btwn  <. A ,  C >.  /\ 
<. A ,  B >.Cgr <. A ,  x >. ) )
131, 2, 7, 12syl12anc 1180 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  B  Btwn  <. A ,  C >. )  ->  E. x  e.  ( EE `  N
) ( x  Btwn  <. A ,  C >.  /\ 
<. A ,  B >.Cgr <. A ,  x >. ) )
14 simpr3 963 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  C  e.  ( EE `  N ) )
15 brsegle 24803 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  ->  ( <. A ,  B >.  Seg<_  <. A ,  C >.  <->  E. x  e.  ( EE `  N ) ( x  Btwn  <. A ,  C >.  /\  <. A ,  B >.Cgr <. A ,  x >. ) ) )
163, 4, 5, 4, 14, 15syl122anc 1191 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( <. A ,  B >. 
Seg<_ 
<. A ,  C >.  <->  E. x  e.  ( EE `  N ) ( x 
Btwn  <. A ,  C >.  /\  <. A ,  B >.Cgr
<. A ,  x >. ) ) )
1716adantr 451 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  B  Btwn  <. A ,  C >. )  ->  ( <. A ,  B >.  Seg<_  <. A ,  C >. 
<->  E. x  e.  ( EE `  N ) ( x  Btwn  <. A ,  C >.  /\  <. A ,  B >.Cgr <. A ,  x >. ) ) )
1813, 17mpbird 223 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )  /\  B  Btwn  <. A ,  C >. )  ->  <. A ,  B >.  Seg<_  <. A ,  C >. )
1918ex 423 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. A ,  C >.  ->  <. A ,  B >.  Seg<_  <. A ,  C >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557   <.cop 3656   class class class wbr 4039   ` cfv 5271   NNcn 9762   EEcee 24588    Btwn cbtwn 24589  Cgrccgr 24590    Seg<_ csegle 24801
This theorem is referenced by:  colinbtwnle  24813  outsidele  24827
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-exp 11121  df-sum 12175  df-ee 24591  df-cgr 24593  df-segle 24802
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