Theorem seglecgr12im 24143

Description: Substitution law for segment comparison under congruence. Theorem 5.6 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)

Assertion

Ref	Expression
seglecgr12im	|- ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. /\ <. A , B >. Seg<_ <. C , D >. ) -> <. E , F >. Seg<_ <. G , H >. ) )

Proof of Theorem seglecgr12im

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

Step	Hyp	Ref	Expression
1		simprrl 740	. . . . . . . . 9 |- ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) ) ) -> y Btwn <. C , D >. )
2		simprlr 739	. . . . . . . . 9 |- ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) ) ) -> <. C , D >.Cgr <. G , H >. )
3		simpl11 1030	. . . . . . . . . . 11 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) -> N e. NN )
4		simpl21 1033	. . . . . . . . . . 11 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) -> C e. ( EE ` N ) )
5		simpr 447	. . . . . . . . . . 11 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) -> y e. ( EE ` N ) )
6		simpl22 1034	. . . . . . . . . . 11 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) -> D e. ( EE ` N ) )
7		simpl32 1037	. . . . . . . . . . 11 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) -> G e. ( EE ` N ) )
8		simpl33 1038	. . . . . . . . . . 11 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) -> H e. ( EE ` N ) )
9		cgrxfr 24088	. . . . . . . . . . 11 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ y e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( y Btwn <. C , D >. /\ <. C , D >.Cgr <. G , H >. ) -> E. z e. ( EE ` N ) ( z Btwn <. G , H >. /\ <. C , <. y , D >. >.Cgr3 <. G , <. z , H >. >. ) ) )
10	3, 4, 5, 6, 7, 8, 9	syl132anc 1200	. . . . . . . . . 10 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) -> ( ( y Btwn <. C , D >. /\ <. C , D >.Cgr <. G , H >. ) -> E. z e. ( EE ` N ) ( z Btwn <. G , H >. /\ <. C , <. y , D >. >.Cgr3 <. G , <. z , H >. >. ) ) )
11	10	adantr 451	. . . . . . . . 9 |- ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) ) ) -> ( ( y Btwn <. C , D >. /\ <. C , D >.Cgr <. G , H >. ) -> E. z e. ( EE ` N ) ( z Btwn <. G , H >. /\ <. C , <. y , D >. >.Cgr3 <. G , <. z , H >. >. ) ) )
12	1, 2, 11	mp2and 660	. . . . . . . 8 |- ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) ) ) -> E. z e. ( EE ` N ) ( z Btwn <. G , H >. /\ <. C , <. y , D >. >.Cgr3 <. G , <. z , H >. >. ) )
13		anass 630	. . . . . . . . . . 11 |- ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ z e. ( EE ` N ) ) <-> ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) )
14		simpl11 1030	. . . . . . . . . . . . . . 15 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) -> N e. NN )
15		simpl21 1033	. . . . . . . . . . . . . . 15 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
16		simprl 732	. . . . . . . . . . . . . . 15 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) -> y e. ( EE ` N ) )
17		simpl22 1034	. . . . . . . . . . . . . . 15 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
18		simpl32 1037	. . . . . . . . . . . . . . 15 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) -> G e. ( EE ` N ) )
19		simprr 733	. . . . . . . . . . . . . . 15 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) -> z e. ( EE ` N ) )
20		simpl33 1038	. . . . . . . . . . . . . . 15 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) -> H e. ( EE ` N ) )
21		brcgr3 24079	. . . . . . . . . . . . . . 15 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ y e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( G e. ( EE ` N ) /\ z e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. C , <. y , D >. >.Cgr3 <. G , <. z , H >. >. <-> ( <. C , y >.Cgr <. G , z >. /\ <. C , D >.Cgr <. G , H >. /\ <. y , D >.Cgr <. z , H >. ) ) )
22	14, 15, 16, 17, 18, 19, 20, 21	syl133anc 1205	. . . . . . . . . . . . . 14 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) -> ( <. C , <. y , D >. >.Cgr3 <. G , <. z , H >. >. <-> ( <. C , y >.Cgr <. G , z >. /\ <. C , D >.Cgr <. G , H >. /\ <. y , D >.Cgr <. z , H >. ) ) )
23	22	adantr 451	. . . . . . . . . . . . 13 |- ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) /\ ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) ) ) -> ( <. C , <. y , D >. >.Cgr3 <. G , <. z , H >. >. <-> ( <. C , y >.Cgr <. G , z >. /\ <. C , D >.Cgr <. G , H >. /\ <. y , D >.Cgr <. z , H >. ) ) )
24		df-3an 936	. . . . . . . . . . . . . . 15 |- ( ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( <. C , y >.Cgr <. G , z >. /\ <. C , D >.Cgr <. G , H >. /\ <. y , D >.Cgr <. z , H >. ) ) <-> ( ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) ) /\ ( <. C , y >.Cgr <. G , z >. /\ <. C , D >.Cgr <. G , H >. /\ <. y , D >.Cgr <. z , H >. ) ) )
25		simpl23 1035	. . . . . . . . . . . . . . . 16 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) -> E e. ( EE ` N ) )
26		simpl31 1036	. . . . . . . . . . . . . . . 16 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) -> F e. ( EE ` N ) )
27		simpl12 1031	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
28		simpl13 1032	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
29		simpr1l 1012	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) /\ ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( <. C , y >.Cgr <. G , z >. /\ <. C , D >.Cgr <. G , H >. /\ <. y , D >.Cgr <. z , H >. ) ) ) -> <. A , B >.Cgr <. E , F >. )
30		simpr2r 1015	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) /\ ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( <. C , y >.Cgr <. G , z >. /\ <. C , D >.Cgr <. G , H >. /\ <. y , D >.Cgr <. z , H >. ) ) ) -> <. A , B >.Cgr <. C , y >. )
31	14, 27, 28, 25, 26, 15, 16, 29, 30	cgrtr4and 24019	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) /\ ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( <. C , y >.Cgr <. G , z >. /\ <. C , D >.Cgr <. G , H >. /\ <. y , D >.Cgr <. z , H >. ) ) ) -> <. E , F >.Cgr <. C , y >. )
32		simpr31 1045	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) /\ ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( <. C , y >.Cgr <. G , z >. /\ <. C , D >.Cgr <. G , H >. /\ <. y , D >.Cgr <. z , H >. ) ) ) -> <. C , y >.Cgr <. G , z >. )
33	14, 25, 26, 15, 16, 18, 19, 31, 32	cgrtrand 24026	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) /\ ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) /\ ( <. C , y >.Cgr <. G , z >. /\ <. C , D >.Cgr <. G , H >. /\ <. y , D >.Cgr <. z , H >. ) ) ) -> <. E , F >.Cgr <. G , z >. )
34	24, 33	sylan2br 462	. . . . . . . . . . . . . 14 |- ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) /\ ( ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) ) /\ ( <. C , y >.Cgr <. G , z >. /\ <. C , D >.Cgr <. G , H >. /\ <. y , D >.Cgr <. z , H >. ) ) ) -> <. E , F >.Cgr <. G , z >. )
35	34	expr 598	. . . . . . . . . . . . 13 |- ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) /\ ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) ) ) -> ( ( <. C , y >.Cgr <. G , z >. /\ <. C , D >.Cgr <. G , H >. /\ <. y , D >.Cgr <. z , H >. ) -> <. E , F >.Cgr <. G , z >. ) )
36	23, 35	sylbid 206	. . . . . . . . . . . 12 |- ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) /\ ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) ) ) -> ( <. C , <. y , D >. >.Cgr3 <. G , <. z , H >. >. -> <. E , F >.Cgr <. G , z >. ) )
37	36	anim2d 548	. . . . . . . . . . 11 |- ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( y e. ( EE ` N ) /\ z e. ( EE ` N ) ) ) /\ ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) ) ) -> ( ( z Btwn <. G , H >. /\ <. C , <. y , D >. >.Cgr3 <. G , <. z , H >. >. ) -> ( z Btwn <. G , H >. /\ <. E , F >.Cgr <. G , z >. ) ) )
38	13, 37	sylanb 458	. . . . . . . . . 10 |- ( ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ z e. ( EE ` N ) ) /\ ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) ) ) -> ( ( z Btwn <. G , H >. /\ <. C , <. y , D >. >.Cgr3 <. G , <. z , H >. >. ) -> ( z Btwn <. G , H >. /\ <. E , F >.Cgr <. G , z >. ) ) )
39	38	an32s 779	. . . . . . . . 9 |- ( ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) ) ) /\ z e. ( EE ` N ) ) -> ( ( z Btwn <. G , H >. /\ <. C , <. y , D >. >.Cgr3 <. G , <. z , H >. >. ) -> ( z Btwn <. G , H >. /\ <. E , F >.Cgr <. G , z >. ) ) )
40	39	reximdva 2656	. . . . . . . 8 |- ( ( ( ( ( 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) ) ) -> ( E. z e. ( EE ` N ) ( z Btwn <. G , H >. /\ <. C , <. y , D >. >.Cgr3 <. G , <. z , H >. >. ) -> E. z e. ( EE ` N ) ( z Btwn <. G , H >. /\ <. E , F >.Cgr <. G , z >. ) ) )
41	12, 40	mpd 14	. . . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) /\ ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) ) ) -> E. z e. ( EE ` N ) ( z Btwn <. G , H >. /\ <. E , F >.Cgr <. G , z >. ) )
42	41	expr 598	. . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ y e. ( EE ` N ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) ) -> ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) -> E. z e. ( EE ` N ) ( z Btwn <. G , H >. /\ <. E , F >.Cgr <. G , z >. ) ) )
43	42	an32s 779	. . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) ) /\ y e. ( EE ` N ) ) -> ( ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) -> E. z e. ( EE ` N ) ( z Btwn <. G , H >. /\ <. E , F >.Cgr <. G , z >. ) ) )
44	43	rexlimdva 2668	. . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) ) -> ( E. y e. ( EE ` N ) ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) -> E. z e. ( EE ` N ) ( z Btwn <. G , H >. /\ <. E , F >.Cgr <. G , z >. ) ) )
45		simp11 985	. . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> N e. NN )
46		simp12 986	. . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> A e. ( EE ` N ) )
47		simp13 987	. . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> B e. ( EE ` N ) )
48		simp21 988	. . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
49		simp22 989	. . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
50		brsegle 24141	. . . . . 6 |- ( ( 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 >. ) ) )
51	45, 46, 47, 48, 49, 50	syl122anc 1191	. . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. A , B >. Seg<_ <. C , D >. <-> E. y e. ( EE ` N ) ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) ) )
52	51	adantr 451	. . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) ) -> ( <. A , B >. Seg<_ <. C , D >. <-> E. y e. ( EE ` N ) ( y Btwn <. C , D >. /\ <. A , B >.Cgr <. C , y >. ) ) )
53		simp23 990	. . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> E e. ( EE ` N ) )
54		simp31 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> F e. ( EE ` N ) )
55		simp32 992	. . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> G e. ( EE ` N ) )
56		simp33 993	. . . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> H e. ( EE ` N ) )
57		brsegle 24141	. . . . . 6 |- ( ( N e. NN /\ ( E e. ( EE ` N ) /\ F e. ( EE ` N ) ) /\ ( G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. E , F >. Seg<_ <. G , H >. <-> E. z e. ( EE ` N ) ( z Btwn <. G , H >. /\ <. E , F >.Cgr <. G , z >. ) ) )
58	45, 53, 54, 55, 56, 57	syl122anc 1191	. . . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. E , F >. Seg<_ <. G , H >. <-> E. z e. ( EE ` N ) ( z Btwn <. G , H >. /\ <. E , F >.Cgr <. G , z >. ) ) )
59	58	adantr 451	. . . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) ) -> ( <. E , F >. Seg<_ <. G , H >. <-> E. z e. ( EE ` N ) ( z Btwn <. G , H >. /\ <. E , F >.Cgr <. G , z >. ) ) )
60	44, 52, 59	3imtr4d 259	. . 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. ) ) -> ( <. A , B >. Seg<_ <. C , D >. -> <. E , F >. Seg<_ <. G , H >. ) )
61	60	exp32 588	. 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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( <. A , B >.Cgr <. E , F >. -> ( <. C , D >.Cgr <. G , H >. -> ( <. A , B >. Seg<_ <. C , D >. -> <. E , F >. Seg<_ <. G , H >. ) ) ) )
62	61	3impd 1165	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 ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( <. A , B >.Cgr <. E , F >. /\ <. C , D >.Cgr <. G , H >. /\ <. A , B >. Seg<_ <. C , D >. ) -> <. E , F >. Seg<_ <. G , H >. ) )

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934   e. wcel 1685   E.wrex 2545   <.cop 3644   class class class wbr 4024   ` cfv 5221   NNcn 9742   EEcee 23926   Btwn cbtwn 23927  Cgrccgr 23928  Cgr3ccgr3 24069   Seg<_ csegle 24139

This theorem is referenced by:  seglecgr12  24144

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-sup 7190  df-oi 7221  df-card 7568  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-ico 10658  df-icc 10659  df-fz 10779  df-fzo 10867  df-seq 11043  df-exp 11101  df-hash 11334  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-clim 11958  df-sum 12155  df-ee 23929  df-btwn 23930  df-cgr 23931  df-ofs 24016  df-cgr3 24073  df-segle 24140