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

Theorem endofsegid 30637
Description: If  A,  B, and  C fall in order on a line, and  A B and  A C are congruent, then  C  =  B. (Contributed by Scott Fenton, 7-Oct-2013.)
Assertion
Ref Expression
endofsegid  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. A ,  B >. )  ->  C  =  B ) )

Proof of Theorem endofsegid
StepHypRef Expression
1 simpl 458 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  N  e.  NN )
2 simpr1 1011 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  A  e.  ( EE `  N ) )
3 simpr3 1013 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  C  e.  ( EE `  N ) )
4 simpr2 1012 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  B  e.  ( EE `  N ) )
5 cgrcom 30542 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) ) )  ->  ( <. A ,  C >.Cgr <. A ,  B >.  <->  <. A ,  B >.Cgr <. A ,  C >. ) )
61, 2, 3, 2, 4, 5syl122anc 1273 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( <. A ,  C >.Cgr
<. A ,  B >.  <->  <. A ,  B >.Cgr <. A ,  C >. ) )
76biimpd 210 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( <. A ,  C >.Cgr
<. A ,  B >.  ->  <. A ,  B >.Cgr <. A ,  C >. ) )
8 idd 25 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( <. A ,  C >.Cgr
<. A ,  B >.  ->  <. A ,  C >.Cgr <. A ,  B >. ) )
9 axcgrrflx 24790 . . . . . . . 8  |-  ( ( N  e.  NN  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  ->  <. B ,  C >.Cgr <. C ,  B >. )
1093adant3r1 1214 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  ->  <. B ,  C >.Cgr <. C ,  B >. )
1110a1d 26 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( <. A ,  C >.Cgr
<. A ,  B >.  ->  <. B ,  C >.Cgr <. C ,  B >. ) )
127, 8, 113jcad 1186 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( <. A ,  C >.Cgr
<. A ,  B >.  -> 
( <. A ,  B >.Cgr
<. A ,  C >.  /\ 
<. A ,  C >.Cgr <. A ,  B >.  /\ 
<. B ,  C >.Cgr <. C ,  B >. ) ) )
13 3ancomb 991 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  C  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  <->  ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )
14 brcgr3 30598 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. C ,  B >. >.  <->  ( <. A ,  B >.Cgr <. A ,  C >.  /\  <. A ,  C >.Cgr <. A ,  B >.  /\  <. B ,  C >.Cgr
<. C ,  B >. ) ) )
1513, 14syl3an3br 1305 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. C ,  B >. >.  <->  ( <. A ,  B >.Cgr <. A ,  C >.  /\  <. A ,  C >.Cgr <. A ,  B >.  /\  <. B ,  C >.Cgr
<. C ,  B >. ) ) )
16153anidm23 1323 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. C ,  B >. >.  <->  (
<. A ,  B >.Cgr <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. A ,  B >.  /\ 
<. B ,  C >.Cgr <. C ,  B >. ) ) )
1712, 16sylibrd 237 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( <. A ,  C >.Cgr
<. A ,  B >.  ->  <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. C ,  B >. >. )
)
18 btwnxfr 30608 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( A  e.  ( EE `  N )  /\  C  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) ) )  ->  (
( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. C ,  B >. >. )  ->  C  Btwn  <. A ,  B >. ) )
1913, 18syl3an3br 1305 . . . . 5  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N ) ) )  ->  (
( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. C ,  B >. >. )  ->  C  Btwn  <. A ,  B >. ) )
20193anidm23 1323 . . . 4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. A ,  <. C ,  B >. >. )  ->  C  Btwn  <. A ,  B >. ) )
2117, 20sylan2d 484 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. A ,  B >. )  ->  C  Btwn  <. A ,  B >. ) )
22 simpl 458 . . . 4  |-  ( ( B  Btwn  <. A ,  C >.  /\  <. A ,  C >.Cgr <. A ,  B >. )  ->  B  Btwn  <. A ,  C >. )
2322a1i 11 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. A ,  B >. )  ->  B  Btwn  <. A ,  C >. ) )
2421, 23jcad 535 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. A ,  B >. )  ->  ( C  Btwn  <. A ,  B >.  /\  B  Btwn  <. A ,  C >. ) ) )
25 3anrot 987 . . 3  |-  ( ( C  e.  ( EE
`  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) )  <->  ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) ) )
26 btwnswapid2 30570 . . 3  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  B  e.  ( EE `  N
) ) )  -> 
( ( C  Btwn  <. A ,  B >.  /\  B  Btwn  <. A ,  C >. )  ->  C  =  B ) )
2725, 26sylan2br 478 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( C  Btwn  <. A ,  B >.  /\  B  Btwn  <. A ,  C >. )  ->  C  =  B ) )
2824, 27syld 45 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( B  Btwn  <. A ,  C >.  /\ 
<. A ,  C >.Cgr <. A ,  B >. )  ->  C  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   <.cop 4008   class class class wbr 4426   ` cfv 5601   NNcn 10609   EEcee 24764    Btwn cbtwn 24765  Cgrccgr 24766  Cgr3ccgr3 30588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-ee 24767  df-btwn 24768  df-cgr 24769  df-ofs 30535  df-ifs 30592  df-cgr3 30593
This theorem is referenced by:  endofsegidand  30638
  Copyright terms: Public domain W3C validator