Theorem segleantisym 32222

Description: Antisymmetry law for segment comparison. Theorem 5.9 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 14-Oct-2013.)

Assertion

Ref	Expression
segleantisym	|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( <. A , B >. Seg<_ <. C , D >. /\ <. C , D >. Seg<_ <. A , B >. ) -> <. A , B >.Cgr <. C , D >. ) )

Proof of Theorem segleantisym

Dummy variables y t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brsegle 32215	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. A , B >. Seg<_ <. C , D >. <-> E. y e. ( EE ` N ) ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) ) )
2		brsegle2 32216	. . . . 5 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) ) -> ( <. C , D >. Seg<_ <. A , B >. <-> E. t e. ( EE ` N ) ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) )
3	2	3com23 1271	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. C , D >. Seg<_ <. A , B >. <-> E. t e. ( EE ` N ) ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) )
4	1, 3	anbi12d 747	. . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( <. A , B >. Seg<_ <. C , D >. /\ <. C , D >. Seg<_ <. A , B >. ) <-> ( E. y e. ( EE ` N ) ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ E. t e. ( EE ` N ) ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) ) )
5		reeanv 3107	. . 3 |- ( E. y e. ( EE ` N ) E. t e. ( EE ` N ) ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) <-> ( E. y e. ( EE ` N ) ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ E. t e. ( EE ` N ) ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) )
6	4, 5	syl6bbr 278	. 2 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( <. A , B >. Seg<_ <. C , D >. /\ <. C , D >. Seg<_ <. A , B >. ) <-> E. y e. ( EE ` N ) E. t e. ( EE ` N ) ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) ) )
7		simpl1 1064	. . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) -> N e. NN )
8		simpl3l 1116	. . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
9		simprr 796	. . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) -> t e. ( EE ` N ) )
10		simprl 794	. . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) -> y e. ( EE ` N ) )
11		simpl3r 1117	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
12		simprll 802	. . . . . . 7 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) ) -> y Btwn <. C , D >. )
13		simprrl 804	. . . . . . 7 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) ) -> D Btwn <. C , t >. )
14	7, 8, 10, 11, 9, 12, 13	btwnexchand 32133	. . . . . 6 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) ) -> y Btwn <. C , t >. )
15		simpl2l 1114	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
16		simpl2r 1115	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
17		simprrr 805	. . . . . . 7 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) ) -> <. C , t >.Cgr <. A , B >. )
18		simprlr 803	. . . . . . 7 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) ) -> <. A , B >.Cgr <. C , y >. )
19	7, 8, 9, 15, 16, 8, 10, 17, 18	cgrtrand 32100	. . . . . 6 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) ) -> <. C , t >.Cgr <. C , y >. )
20	7, 8, 9, 10, 14, 19	endofsegidand 32193	. . . . 5 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) ) -> t = y )
21		opeq2 4403	. . . . . . . . . 10 |- ( t = y -> <. C , t >. = <. C , y >. )
22	21	breq2d 4665	. . . . . . . . 9 |- ( t = y -> ( D Btwn <. C , t >. <-> D Btwn <. C , y >. ) )
23	21	breq1d 4663	. . . . . . . . 9 |- ( t = y -> ( <. C , t >.Cgr <. A , B >. <-> <. C , y >.Cgr <. A , B >. ) )
24	22, 23	anbi12d 747	. . . . . . . 8 |- ( t = y -> ( ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) <-> ( D Btwn <. C , y >. /\ <. C , y >.Cgr <. A , B >. ) ) )
25	24	anbi2d 740	. . . . . . 7 |- ( t = y -> ( ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) <-> ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , y >. /\ <. C , y >.Cgr <. A , B >. ) ) ) )
26	25	anbi2d 740	. . . . . 6 |- ( t = y -> ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) ) <-> ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , y >. /\ <. C , y >.Cgr <. A , B >. ) ) ) ) )
27		simprrl 804	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , y >. /\ <. C , y >.Cgr <. A , B >. ) ) ) -> D Btwn <. C , y >. )
28	7, 11, 8, 10, 27	btwncomand 32122	. . . . . . . 8 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , y >. /\ <. C , y >.Cgr <. A , B >. ) ) ) -> D Btwn <. y , C >. )
29		simprll 802	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , y >. /\ <. C , y >.Cgr <. A , B >. ) ) ) -> y Btwn <. C , D >. )
30	7, 10, 8, 11, 29	btwncomand 32122	. . . . . . . 8 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , y >. /\ <. C , y >.Cgr <. A , B >. ) ) ) -> y Btwn <. D , C >. )
31		btwnswapid 32124	. . . . . . . . . 10 |- ( ( N e. NN /\ ( D e. ( EE ` N ) /\ y e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( D Btwn <. y , C >. /\ y Btwn <. D , C >. ) -> D = y ) )
32	7, 11, 10, 8, 31	syl13anc 1328	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) -> ( ( D Btwn <. y , C >. /\ y Btwn <. D , C >. ) -> D = y ) )
33	32	adantr 481	. . . . . . . 8 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , y >. /\ <. C , y >.Cgr <. A , B >. ) ) ) -> ( ( D Btwn <. y , C >. /\ y Btwn <. D , C >. ) -> D = y ) )
34	28, 30, 33	mp2and 715	. . . . . . 7 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , y >. /\ <. C , y >.Cgr <. A , B >. ) ) ) -> D = y )
35		simprlr 803	. . . . . . . 8 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , y >. /\ <. C , y >.Cgr <. A , B >. ) ) ) -> <. A , B >.Cgr <. C , y >. )
36		opeq2 4403	. . . . . . . . 9 |- ( D = y -> <. C , D >. = <. C , y >. )
37	36	breq2d 4665	. . . . . . . 8 |- ( D = y -> ( <. A , B >.Cgr <. C , D >. <-> <. A , B >.Cgr <. C , y >. ) )
38	35, 37	syl5ibrcom 237	. . . . . . 7 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , y >. /\ <. C , y >.Cgr <. A , B >. ) ) ) -> ( D = y -> <. A , B >.Cgr <. C , D >. ) )
39	34, 38	mpd 15	. . . . . 6 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , y >. /\ <. C , y >.Cgr <. A , B >. ) ) ) -> <. A , B >.Cgr <. C , D >. )
40	26, 39	syl6bi 243	. . . . 5 |- ( t = y -> ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) ) -> <. A , B >.Cgr <. C , D >. ) )
41	20, 40	mpcom 38	. . . 4 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) ) /\ ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) ) -> <. A , B >.Cgr <. C , D >. )
42	41	exp31 630	. . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( y e. ( EE ` N ) /\ t e. ( EE ` N ) ) -> ( ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) -> <. A , B >.Cgr <. C , D >. ) ) )
43	42	rexlimdvv 3037	. 2 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( E. y e. ( EE ` N ) E. t e. ( EE ` N ) ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( D Btwn <. C , t >. /\ <. C , t >.Cgr <. A , B >. ) ) -> <. A , B >.Cgr <. C , D >. ) )
44	6, 43	sylbid 230	1 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( <. A , B >. Seg<_ <. C , D >. /\ <. C , D >. Seg<_ <. A , B >. ) -> <. A , B >.Cgr <. C , D >. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770   Seg<_ csegle 32213

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-segle 32214

This theorem is referenced by:  colinbtwnle  32225