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Theorem cgrsub 29663
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 998 . . . . . 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 1073 . . . . . 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 1076 . . . . . 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 29620 . . . . . 6 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> <. A , A >.Cgr <. D , D >. )
73, 4, 5, 6syl3anc 1227 . . . . 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 1075 . . . . . . 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 1078 . . . . . . 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 29616 . . . . . . 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 1236 . . . . . 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 1074 . . . . 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 1077 . . . . 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 29661 . . . . . 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 29662 . . . . . 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 1259 . . . 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 1326 . . 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 29616 . . . 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 1236 . . 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 972   e. wcel 1802   <.cop 4016   class class class wbr 4433   ` cfv 5574   NNcn 10537   EEcee 24056   Btwn cbtwn 24057  Cgrccgr 24058   InnerFiveSeg cifs 29653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-ico 11539  df-icc 11540  df-fz 11677  df-fzo 11799  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-sum 13483  df-ee 24059  df-btwn 24060  df-cgr 24061  df-ofs 29601  df-ifs 29658
This theorem is referenced by:  brifs2  29696
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