Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cgrsub Structured version   Unicode version
Theorem cgrsub 28213
Description: Removing identical parts from the end of a line segment preserves congruence. Theorem 4.3 of [Schwabhauser] p. 35. (Contributed by Scott Fenton, 4-Oct-2013.)
Assertion
Ref Expression
cgrsub |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) -> ( ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) -> <. A , B >.Cgr <. D , E >. ) )
Proof of Theorem cgrsub
StepHypRef Expression
1 simprl 755 . . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) )
2 simprr 756 . . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) )
3 simpl1 991 . . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> N e. NN )
4 simpl21 1066 . . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> A e. ( EE ` N ) )
5 simpl31 1069 . . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> D e. ( EE ` N ) )
6 cgrtriv 28170 . . . . . 6 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> <. A , A >.Cgr <. D , D >. )
73, 4, 5, 6syl3anc 1219 . . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> <. A , A >.Cgr <. D , D >. )
8 simprrl 763 . . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> <. A , C >.Cgr <. D , F >. )
9 simpl23 1068 . . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> C e. ( EE ` N ) )
10 simpl33 1071 . . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> F e. ( EE ` N ) )
11 cgrcomlr 28166 . . . . . . 7 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) -> ( <. A , C >.Cgr <. D , F >. <-> <. C , A >.Cgr <. F , D >. ) )
123, 4, 9, 5, 10, 11syl122anc 1228 . . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> ( <. A , C >.Cgr <. D , F >. <-> <. C , A >.Cgr <. F , D >. ) )
138, 12mpbid 210 . . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> <. C , A >.Cgr <. F , D >. )
147, 13jca 532 . . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> ( <. A , A >.Cgr <. D , D >. /\ <. C , A >.Cgr <. F , D >. ) )
15 simpl22 1067 . . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> B e. ( EE ` N ) )
16 simpl32 1070 . . . . 5 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> E e. ( EE ` N ) )
17 brifs 28211 . . . . . 6 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. <. A , B >. , <. C , A >. >. InnerFiveSeg <. <. D , E >. , <. F , D >. >. <-> ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) /\ ( <. A , A >.Cgr <. D , D >. /\ <. C , A >.Cgr <. F , D >. ) ) ) )
18 ifscgr 28212 . . . . . 6 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. <. A , B >. , <. C , A >. >. InnerFiveSeg <. <. D , E >. , <. F , D >. >. -> <. B , A >.Cgr <. E , D >. ) )
1917, 18sylbird 235 . . . . 5 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ A e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ F e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) /\ ( <. A , A >.Cgr <. D , D >. /\ <. C , A >.Cgr <. F , D >. ) ) -> <. B , A >.Cgr <. E , D >. ) )
203, 4, 15, 9, 4, 5, 16, 10, 5, 19syl333anc 1251 . . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> ( ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) /\ ( <. A , A >.Cgr <. D , D >. /\ <. C , A >.Cgr <. F , D >. ) ) -> <. B , A >.Cgr <. E , D >. ) )
211, 2, 14, 20mp3and 1318 . . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> <. B , A >.Cgr <. E , D >. )
22 cgrcomlr 28166 . . . 4 |- ( ( N e. NN /\ ( B e. ( EE ` N ) /\ A e. ( EE ` N ) ) /\ ( E e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. B , A >.Cgr <. E , D >. <-> <. A , B >.Cgr <. D , E >. ) )
233, 15, 4, 16, 5, 22syl122anc 1228 . . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> ( <. B , A >.Cgr <. E , D >. <-> <. A , B >.Cgr <. D , E >. ) )
2421, 23mpbid 210 . 2 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) ) -> <. A , B >.Cgr <. D , E >. )
2524ex 434 1 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) -> ( ( ( B Btwn <. A , C >. /\ E Btwn <. D , F >. ) /\ ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) ) -> <. A , B >.Cgr <. D , E >. ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   e. wcel 1758   <.cop 3984   class class class wbr 4393   ` cfv 5519   NNcn 10426   EEcee 23279   Btwn cbtwn 23280  Cgrccgr 23281   InnerFiveSeg cifs 28203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-ico 11410  df-icc 11411  df-fz 11548  df-fzo 11659  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077  df-sum 13275  df-ee 23282  df-btwn 23283  df-cgr 23284  df-ofs 28151  df-ifs 28208
This theorem is referenced by:  brifs2  28246
  Copyright terms: Public domain W3C validator