Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  midofsegid Structured version   Unicode version

Theorem midofsegid 29649
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 996 . . . . . 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 1022 . . . . . 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 1025 . . . . . 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 1024 . . . . . 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 756 . . . . . 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 1043 . . . . . . 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 29536 . . . . . 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 29631 . . . . 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 2475 . . . 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 615 . . 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 756 . . . . 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 1043 . . . . 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 29631 . . . 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 615 . . 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 993 . . . . 5  |-  ( ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >.  /\ 
<. A ,  D >.Cgr <. A ,  E >. )  ->  ( D  Btwn  <. A ,  B >.  /\  E  Btwn  <. A ,  B >. ) )
1615adantl 466 . . . 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 1023 . . . . . 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 29648 . . . . . 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 1237 . . . . 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 465 . . . 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 973    = wceq 1379    e. wcel 1767   <.cop 4038   class class class wbr 4452   ` cfv 5593   NNcn 10546   EEcee 23982    Btwn cbtwn 23983  Cgrccgr 23984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-er 7321  df-map 7432  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-sup 7911  df-oi 7945  df-card 8330  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-n0 10806  df-z 10875  df-uz 11093  df-rp 11231  df-ico 11545  df-icc 11546  df-fz 11683  df-fzo 11803  df-seq 12086  df-exp 12145  df-hash 12384  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-clim 13286  df-sum 13484  df-ee 23985  df-btwn 23986  df-cgr 23987  df-ofs 29528  df-colinear 29584  df-ifs 29585  df-cgr3 29586  df-fs 29587
This theorem is referenced by:  outsideofeq  29675
  Copyright terms: Public domain W3C validator