Theorem idinside 32191

Description: Law for finding a point inside a segment. Theorem 4.19 of [Schwabhauser] p. 38. (Contributed by Scott Fenton, 7-Oct-2013.)

Assertion

Ref	Expression
idinside	|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( C Btwn <. A , B >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) -> C = D ) )

Proof of Theorem idinside

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> N e. NN )
2		simp3l 1089	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
3		simp3r 1090	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
4		cgrid2 32110	. . . . . 6 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. C , C >.Cgr <. C , D >. -> C = D ) )
5	1, 2, 2, 3, 4	syl13anc 1328	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. C , C >.Cgr <. C , D >. -> C = D ) )
6		simp2l 1087	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
7		axbtwnid 25819	. . . . . 6 |- ( ( N e. NN /\ C e. ( EE ` N ) /\ A e. ( EE ` N ) ) -> ( C Btwn <. A , A >. -> C = A ) )
8	1, 2, 6, 7	syl3anc 1326	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( C Btwn <. A , A >. -> C = A ) )
9		opeq1 4402	. . . . . . . . 9 |- ( C = A -> <. C , C >. = <. A , C >. )
10		opeq1 4402	. . . . . . . . 9 |- ( C = A -> <. C , D >. = <. A , D >. )
11	9, 10	breq12d 4666	. . . . . . . 8 |- ( C = A -> ( <. C , C >.Cgr <. C , D >. <-> <. A , C >.Cgr <. A , D >. ) )
12	11	imbi1d 331	. . . . . . 7 |- ( C = A -> ( ( <. C , C >.Cgr <. C , D >. -> C = D ) <-> ( <. A , C >.Cgr <. A , D >. -> C = D ) ) )
13	12	biimpcd 239	. . . . . 6 |- ( ( <. C , C >.Cgr <. C , D >. -> C = D ) -> ( C = A -> ( <. A , C >.Cgr <. A , D >. -> C = D ) ) )
14		ax-1 6	. . . . . 6 |- ( C = D -> ( <. B , C >.Cgr <. B , D >. -> C = D ) )
15	13, 14	syl8 76	. . . . 5 |- ( ( <. C , C >.Cgr <. C , D >. -> C = D ) -> ( C = A -> ( <. A , C >.Cgr <. A , D >. -> ( <. B , C >.Cgr <. B , D >. -> C = D ) ) ) )
16	5, 8, 15	sylsyld 61	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( C Btwn <. A , A >. -> ( <. A , C >.Cgr <. A , D >. -> ( <. B , C >.Cgr <. B , D >. -> C = D ) ) ) )
17	16	3impd 1281	. . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( C Btwn <. A , A >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) -> C = D ) )
18		opeq2 4403	. . . . . 6 |- ( A = B -> <. A , A >. = <. A , B >. )
19	18	breq2d 4665	. . . . 5 |- ( A = B -> ( C Btwn <. A , A >. <-> C Btwn <. A , B >. ) )
20	19	3anbi1d 1403	. . . 4 |- ( A = B -> ( ( C Btwn <. A , A >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) <-> ( C Btwn <. A , B >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) ) )
21	20	imbi1d 331	. . 3 |- ( A = B -> ( ( ( C Btwn <. A , A >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) -> C = D ) <-> ( ( C Btwn <. A , B >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) -> C = D ) ) )
22	17, 21	syl5ib 234	. 2 |- ( A = B -> ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( C Btwn <. A , B >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) -> C = D ) ) )
23		simpr1 1067	. . . . . 6 |- ( ( A =/= B /\ ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) ) -> N e. NN )
24		simpr2l 1120	. . . . . 6 |- ( ( A =/= B /\ ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) ) -> A e. ( EE ` N ) )
25		simpr2r 1121	. . . . . 6 |- ( ( A =/= B /\ ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) ) -> B e. ( EE ` N ) )
26		simpr3l 1122	. . . . . 6 |- ( ( A =/= B /\ ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) ) -> C e. ( EE ` N ) )
27		btwncolinear1 32176	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( C Btwn <. A , B >. -> A Colinear <. B , C >. ) )
28	23, 24, 25, 26, 27	syl13anc 1328	. . . . 5 |- ( ( A =/= B /\ ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) ) -> ( C Btwn <. A , B >. -> A Colinear <. B , C >. ) )
29		idd 24	. . . . 5 |- ( ( A =/= B /\ ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) ) -> ( <. A , C >.Cgr <. A , D >. -> <. A , C >.Cgr <. A , D >. ) )
30		idd 24	. . . . 5 |- ( ( A =/= B /\ ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) ) -> ( <. B , C >.Cgr <. B , D >. -> <. B , C >.Cgr <. B , D >. ) )
31	28, 29, 30	3anim123d 1406	. . . 4 |- ( ( A =/= B /\ ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) ) -> ( ( C Btwn <. A , B >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) -> ( A Colinear <. B , C >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) ) )
32		simp1 1061	. . . . . . . . 9 |- ( ( A Colinear <. B , C >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) -> A Colinear <. B , C >. )
33	32	anim2i 593	. . . . . . . 8 |- ( ( A =/= B /\ ( A Colinear <. B , C >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) ) -> ( A =/= B /\ A Colinear <. B , C >. ) )
34		3simpc 1060	. . . . . . . . 9 |- ( ( A Colinear <. B , C >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) -> ( <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) )
35	34	adantl 482	. . . . . . . 8 |- ( ( A =/= B /\ ( A Colinear <. B , C >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) ) -> ( <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) )
36	33, 35	jca 554	. . . . . . 7 |- ( ( A =/= B /\ ( A Colinear <. B , C >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) ) -> ( ( A =/= B /\ A Colinear <. B , C >. ) /\ ( <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) ) )
37		lineid 32190	. . . . . . 7 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( ( A =/= B /\ A Colinear <. B , C >. ) /\ ( <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) ) -> C = D ) )
38	36, 37	syl5 34	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( A =/= B /\ ( A Colinear <. B , C >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) ) -> C = D ) )
39	38	expd 452	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( A =/= B -> ( ( A Colinear <. B , C >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) -> C = D ) ) )
40	39	impcom 446	. . . 4 |- ( ( A =/= B /\ ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) ) -> ( ( A Colinear <. B , C >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) -> C = D ) )
41	31, 40	syld 47	. . 3 |- ( ( A =/= B /\ ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) ) -> ( ( C Btwn <. A , B >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) -> C = D ) )
42	41	ex 450	. 2 |- ( A =/= B -> ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( C Btwn <. A , B >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) -> C = D ) ) )
43	22, 42	pm2.61ine 2877	1 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( C Btwn <. A , B >. /\ <. A , C >.Cgr <. A , D >. /\ <. B , C >.Cgr <. B , D >. ) -> C = D ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770   Colinear ccolin 32144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-ee 25771  df-btwn 25772  df-cgr 25773  df-ofs 32090  df-colinear 32146  df-ifs 32147  df-cgr3 32148  df-fs 32149

This theorem is referenced by: (None)