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Theorem cgrsub 29926
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 754 . . . 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 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 >. ) ) ) -> ( <. A , C >.Cgr <. D , F >. /\ <. B , C >.Cgr <. E , F >. ) )
3 simpl1 997 . . . . . 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 1072 . . . . . 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 1075 . . . . . 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 29883 . . . . . 6 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> <. A , A >.Cgr <. D , D >. )
73, 4, 5, 6syl3anc 1226 . . . . 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 1074 . . . . . . 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 1077 . . . . . . 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 29879 . . . . . . 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 1235 . . . . . 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 530 . . . 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 1073 . . . . 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 1076 . . . . 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 29924 . . . . . 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 29925 . . . . . 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 1258 . . . 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 1325 . . 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 29879 . . . 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 1235 . . 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 432 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 367   /\ w3a 971   e. wcel 1823   <.cop 4022   class class class wbr 4439   ` cfv 5570   NNcn 10531   EEcee 24396   Btwn cbtwn 24397  Cgrccgr 24398   InnerFiveSeg cifs 29916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-seq 12093  df-exp 12152  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-clim 13396  df-sum 13594  df-ee 24399  df-btwn 24400  df-cgr 24401  df-ofs 29864  df-ifs 29921
This theorem is referenced by:  brifs2  29959
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