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Theorem midofsegid 28048
Description: If two points fall in the same place in the middle of a segment, then they are identical. (Contributed by Scott Fenton, 16-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
midofsegid  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  ->  ( ( D 
Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\  <. A ,  D >.Cgr <. A ,  E >. )  ->  D  =  E ) )

Proof of Theorem midofsegid
StepHypRef Expression
1 simp1 983 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  ->  N  e.  NN )
2 simp2l 1009 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N ) )
3 simp3r 1012 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  ->  E  e.  ( EE `  N ) )
4 simp3l 1011 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N ) )
5 simprr 751 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\  <. A ,  D >.Cgr <. A ,  E >. )  /\  D  Btwn  <. A ,  E >. ) )  ->  D  Btwn  <. A ,  E >. )
6 simprl3 1030 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\  <. A ,  D >.Cgr <. A ,  E >. )  /\  D  Btwn  <. A ,  E >. ) )  ->  <. A ,  D >.Cgr <. A ,  E >. )
71, 2, 4, 2, 3, 6cgrcomand 27935 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\  <. A ,  D >.Cgr <. A ,  E >. )  /\  D  Btwn  <. A ,  E >. ) )  ->  <. A ,  E >.Cgr <. A ,  D >. )
81, 2, 3, 4, 5, 7endofsegidand 28030 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\  <. A ,  D >.Cgr <. A ,  E >. )  /\  D  Btwn  <. A ,  E >. ) )  ->  E  =  D )
98eqcomd 2446 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\  <. A ,  D >.Cgr <. A ,  E >. )  /\  D  Btwn  <. A ,  E >. ) )  ->  D  =  E )
109expr 612 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\ 
<. A ,  D >.Cgr <. A ,  E >. ) )  ->  ( D  Btwn  <. A ,  E >.  ->  D  =  E ) )
11 simprr 751 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\  <. A ,  D >.Cgr <. A ,  E >. )  /\  E  Btwn  <. A ,  D >. ) )  ->  E  Btwn  <. A ,  D >. )
12 simprl3 1030 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\  <. A ,  D >.Cgr <. A ,  E >. )  /\  E  Btwn  <. A ,  D >. ) )  ->  <. A ,  D >.Cgr <. A ,  E >. )
131, 2, 4, 3, 11, 12endofsegidand 28030 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\  <. A ,  D >.Cgr <. A ,  E >. )  /\  E  Btwn  <. A ,  D >. ) )  ->  D  =  E )
1413expr 612 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\ 
<. A ,  D >.Cgr <. A ,  E >. ) )  ->  ( E  Btwn  <. A ,  D >.  ->  D  =  E ) )
15 3simpa 980 . . . . 5  |-  ( ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\ 
<. A ,  D >.Cgr <. A ,  E >. )  ->  ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >. ) )
1615adantl 463 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\ 
<. A ,  D >.Cgr <. A ,  E >. ) )  ->  ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >. ) )
17 simp2r 1010 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N ) )
18 btwnconn3 28047 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) )  /\  ( E  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( ( D 
Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >. )  ->  ( D  Btwn  <. A ,  E >.  \/  E  Btwn  <. A ,  D >. ) ) )
191, 2, 4, 3, 17, 18syl122anc 1222 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  ->  ( ( D 
Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >. )  ->  ( D  Btwn  <. A ,  E >.  \/  E  Btwn  <. A ,  D >. ) ) )
2019adantr 462 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\ 
<. A ,  D >.Cgr <. A ,  E >. ) )  ->  ( ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >. )  ->  ( D  Btwn  <. A ,  E >.  \/  E  Btwn  <. A ,  D >. ) ) )
2116, 20mpd 15 . . 3  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\ 
<. A ,  D >.Cgr <. A ,  E >. ) )  ->  ( D  Btwn  <. A ,  E >.  \/  E  Btwn  <. A ,  D >. ) )
2210, 14, 21mpjaod 381 . 2  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  /\  ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\ 
<. A ,  D >.Cgr <. A ,  E >. ) )  ->  D  =  E )
2322ex 434 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  ->  ( ( D 
Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\  <. A ,  D >.Cgr <. A ,  E >. )  ->  D  =  E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   <.cop 3880   class class class wbr 4289   ` cfv 5415   NNcn 10318   EEcee 23053    Btwn cbtwn 23054  Cgrccgr 23055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-ee 23056  df-btwn 23057  df-cgr 23058  df-ofs 27927  df-colinear 27983  df-ifs 27984  df-cgr3 27985  df-fs 27986
This theorem is referenced by:  outsideofeq  28074
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