Theorem cgratr 25715

Description: Angle congruence is transitive. Theorem 11.8 of [Schwabhauser] p. 97. (Contributed by Thierry Arnoux, 5-Mar-2020.)

Hypotheses

Ref	Expression
cgraid.p	|- P = ( Base ` G )
cgraid.i	|- I = (Itv ` G )
cgraid.g	|- ( ph -> G e. TarskiG )
cgraid.k	|- K = (hlG ` G )
cgraid.a	|- ( ph -> A e. P )
cgraid.b	|- ( ph -> B e. P )
cgraid.c	|- ( ph -> C e. P )
cgracom.d	|- ( ph -> D e. P )
cgracom.e	|- ( ph -> E e. P )
cgracom.f	|- ( ph -> F e. P )
cgracom.1	|- ( ph -> <" A B C "> (cgrA ` G ) <" D E F "> )
cgratr.h	|- ( ph -> H e. P )
cgratr.i	|- ( ph -> U e. P )
cgratr.j	|- ( ph -> J e. P )
cgratr.1	|- ( ph -> <" D E F "> (cgrA ` G ) <" H U J "> )

Assertion

Ref	Expression
cgratr	|- ( ph -> <" A B C "> (cgrA ` G ) <" H U J "> )

Proof of Theorem cgratr

Dummy variables x y u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cgraid.p	. . . . 5 |- P = ( Base ` G )
2		eqid 2622	. . . . 5 |- ( dist ` G ) = ( dist ` G )
3		eqid 2622	. . . . 5 |- (cgrG ` G ) = (cgrG ` G )
4		cgraid.g	. . . . . 6 |- ( ph -> G e. TarskiG )
5	4	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> G e. TarskiG )
6		cgraid.a	. . . . . 6 |- ( ph -> A e. P )
7	6	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> A e. P )
8		cgraid.b	. . . . . 6 |- ( ph -> B e. P )
9	8	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> B e. P )
10		cgraid.c	. . . . . 6 |- ( ph -> C e. P )
11	10	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> C e. P )
12		simpllr 799	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> x e. P )
13		cgratr.i	. . . . . 6 |- ( ph -> U e. P )
14	13	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> U e. P )
15		simplr 792	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> y e. P )
16		cgraid.i	. . . . . 6 |- I = (Itv ` G )
17		simprlr 803	. . . . . . 7 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) )
18	17	eqcomd 2628	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( B ( dist ` G ) A ) = ( U ( dist ` G ) x ) )
19	1, 2, 16, 5, 9, 7, 14, 12, 18	tgcgrcomlr 25375	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( A ( dist ` G ) B ) = ( x ( dist ` G ) U ) )
20		simprrr 805	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) )
21	20	eqcomd 2628	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( B ( dist ` G ) C ) = ( U ( dist ` G ) y ) )
22	5	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> G e. TarskiG )
23	7	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> A e. P )
24	9	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> B e. P )
25	11	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> C e. P )
26		simpllr 799	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> u e. P )
27		cgracom.e	. . . . . . . . 9 |- ( ph -> E e. P )
28	27	ad6antr 772	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> E e. P )
29		simplr 792	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> v e. P )
30		simpr1 1067	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> <" A B C "> (cgrG ` G ) <" u E v "> )
31	1, 2, 16, 3, 22, 23, 24, 25, 26, 28, 29, 30	cgr3simp3 25417	. . . . . . 7 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> ( C ( dist ` G ) A ) = ( v ( dist ` G ) u ) )
32	12	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> x e. P )
33	15	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> y e. P )
34		cgraid.k	. . . . . . . . 9 |- K = (hlG ` G )
35		cgracom.d	. . . . . . . . . 10 |- ( ph -> D e. P )
36	35	ad6antr 772	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> D e. P )
37		cgracom.f	. . . . . . . . . 10 |- ( ph -> F e. P )
38	37	ad6antr 772	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> F e. P )
39	14	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> U e. P )
40		cgratr.j	. . . . . . . . . . 11 |- ( ph -> J e. P )
41	40	ad6antr 772	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> J e. P )
42		cgratr.h	. . . . . . . . . . . 12 |- ( ph -> H e. P )
43	42	ad6antr 772	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> H e. P )
44		cgratr.1	. . . . . . . . . . . 12 |- ( ph -> <" D E F "> (cgrA ` G ) <" H U J "> )
45	44	ad6antr 772	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> <" D E F "> (cgrA ` G ) <" H U J "> )
46		simprll 802	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> x ( K ` U ) H )
47	46	ad3antrrr 766	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> x ( K ` U ) H )
48	1, 16, 34, 22, 36, 28, 38, 43, 39, 41, 45, 32, 47	cgrahl1 25708	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> <" D E F "> (cgrA ` G ) <" x U J "> )
49		simprrl 804	. . . . . . . . . . 11 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> y ( K ` U ) J )
50	49	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> y ( K ` U ) J )
51	1, 16, 34, 22, 36, 28, 38, 32, 39, 41, 48, 33, 50	cgrahl2 25709	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> <" D E F "> (cgrA ` G ) <" x U y "> )
52		simpr2 1068	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> u ( K ` E ) D )
53		simpr3 1069	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> v ( K ` E ) F )
54	1, 2, 16, 3, 22, 23, 24, 25, 26, 28, 29, 30	cgr3simp1 25415	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> ( A ( dist ` G ) B ) = ( u ( dist ` G ) E ) )
55	54	eqcomd 2628	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> ( u ( dist ` G ) E ) = ( A ( dist ` G ) B ) )
56	1, 2, 16, 22, 26, 28, 23, 24, 55	tgcgrcomlr 25375	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> ( E ( dist ` G ) u ) = ( B ( dist ` G ) A ) )
57	18	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> ( B ( dist ` G ) A ) = ( U ( dist ` G ) x ) )
58	56, 57	eqtrd 2656	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> ( E ( dist ` G ) u ) = ( U ( dist ` G ) x ) )
59	1, 2, 16, 3, 22, 23, 24, 25, 26, 28, 29, 30	cgr3simp2 25416	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> ( B ( dist ` G ) C ) = ( E ( dist ` G ) v ) )
60	59	eqcomd 2628	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> ( E ( dist ` G ) v ) = ( B ( dist ` G ) C ) )
61	21	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> ( B ( dist ` G ) C ) = ( U ( dist ` G ) y ) )
62	60, 61	eqtrd 2656	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> ( E ( dist ` G ) v ) = ( U ( dist ` G ) y ) )
63	1, 16, 34, 22, 36, 28, 38, 32, 39, 33, 51, 26, 2, 29, 52, 53, 58, 62	cgracgr 25710	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> ( u ( dist ` G ) v ) = ( x ( dist ` G ) y ) )
64	1, 2, 16, 22, 26, 29, 32, 33, 63	tgcgrcomlr 25375	. . . . . . 7 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> ( v ( dist ` G ) u ) = ( y ( dist ` G ) x ) )
65	31, 64	eqtrd 2656	. . . . . 6 |- ( ( ( ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) /\ u e. P ) /\ v e. P ) /\ ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) -> ( C ( dist ` G ) A ) = ( y ( dist ` G ) x ) )
66		cgracom.1	. . . . . . . 8 |- ( ph -> <" A B C "> (cgrA ` G ) <" D E F "> )
67	1, 16, 34, 4, 6, 8, 10, 35, 27, 37	iscgra 25701	. . . . . . . 8 |- ( ph -> ( <" A B C "> (cgrA ` G ) <" D E F "> <-> E. u e. P E. v e. P ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) ) )
68	66, 67	mpbid 222	. . . . . . 7 |- ( ph -> E. u e. P E. v e. P ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) )
69	68	ad3antrrr 766	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> E. u e. P E. v e. P ( <" A B C "> (cgrG ` G ) <" u E v "> /\ u ( K ` E ) D /\ v ( K ` E ) F ) )
70	65, 69	r19.29vva 3081	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( C ( dist ` G ) A ) = ( y ( dist ` G ) x ) )
71	1, 2, 3, 5, 7, 9, 11, 12, 14, 15, 19, 21, 70	trgcgr 25411	. . . 4 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> <" A B C "> (cgrG ` G ) <" x U y "> )
72	71, 46, 49	3jca 1242	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) ) -> ( <" A B C "> (cgrG ` G ) <" x U y "> /\ x ( K ` U ) H /\ y ( K ` U ) J ) )
73	1, 16, 34, 4, 35, 27, 37, 42, 13, 40, 44	cgrane3 25706	. . . . . 6 |- ( ph -> U =/= H )
74	73	necomd 2849	. . . . 5 |- ( ph -> H =/= U )
75	1, 16, 34, 4, 6, 8, 10, 35, 27, 37, 66	cgrane1 25704	. . . . . 6 |- ( ph -> A =/= B )
76	75	necomd 2849	. . . . 5 |- ( ph -> B =/= A )
77	1, 16, 34, 13, 8, 6, 4, 42, 2, 74, 76	hlcgrex 25511	. . . 4 |- ( ph -> E. x e. P ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) )
78	1, 16, 34, 4, 35, 27, 37, 42, 13, 40, 44	cgrane4 25707	. . . . . 6 |- ( ph -> U =/= J )
79	78	necomd 2849	. . . . 5 |- ( ph -> J =/= U )
80	1, 16, 34, 4, 6, 8, 10, 35, 27, 37, 66	cgrane2 25705	. . . . 5 |- ( ph -> B =/= C )
81	1, 16, 34, 13, 8, 10, 4, 40, 2, 79, 80	hlcgrex 25511	. . . 4 |- ( ph -> E. y e. P ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) )
82		reeanv 3107	. . . 4 |- ( E. x e. P E. y e. P ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) <-> ( E. x e. P ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ E. y e. P ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) )
83	77, 81, 82	sylanbrc 698	. . 3 |- ( ph -> E. x e. P E. y e. P ( ( x ( K ` U ) H /\ ( U ( dist ` G ) x ) = ( B ( dist ` G ) A ) ) /\ ( y ( K ` U ) J /\ ( U ( dist ` G ) y ) = ( B ( dist ` G ) C ) ) ) )
84	72, 83	reximddv2 3020	. 2 |- ( ph -> E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x U y "> /\ x ( K ` U ) H /\ y ( K ` U ) J ) )
85	1, 16, 34, 4, 6, 8, 10, 42, 13, 40	iscgra 25701	. 2 |- ( ph -> ( <" A B C "> (cgrA ` G ) <" H U J "> <-> E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x U y "> /\ x ( K ` U ) H /\ y ( K ` U ) J ) ) )
86	84, 85	mpbird 247	1 |- ( ph -> <" A B C "> (cgrA ` G ) <" H U J "> )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  cgrGccgrg 25405  hlGchlg 25495  cgrAccgra 25699

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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-cgrg 25406  df-leg 25478  df-hlg 25496  df-cgra 25700

This theorem is referenced by:  cgraswaplr  25716  sacgr  25722  oacgr  25723  tgasa1  25739  isoas  25744