Theorem sacgr 25722

Description: Supplementary angles of congruent angles are themselves congruent. Theorem 11.13 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 30-Sep-2020.)

Hypotheses

Ref	Expression
dfcgra2.p	|- P = ( Base ` G )
dfcgra2.i	|- I = (Itv ` G )
dfcgra2.m	|- .- = ( dist ` G )
dfcgra2.g	|- ( ph -> G e. TarskiG )
dfcgra2.a	|- ( ph -> A e. P )
dfcgra2.b	|- ( ph -> B e. P )
dfcgra2.c	|- ( ph -> C e. P )
dfcgra2.d	|- ( ph -> D e. P )
dfcgra2.e	|- ( ph -> E e. P )
dfcgra2.f	|- ( ph -> F e. P )
sacgr.x	|- ( ph -> X e. P )
sacgr.y	|- ( ph -> Y e. P )
sacgr.1	|- ( ph -> <" A B C "> (cgrA ` G ) <" D E F "> )
sacgr.2	|- ( ph -> B e. ( A I X ) )
sacgr.3	|- ( ph -> E e. ( D I Y ) )
sacgr.4	|- ( ph -> B =/= X )
sacgr.5	|- ( ph -> E =/= Y )

Assertion

Ref	Expression
sacgr	|- ( ph -> <" X B C "> (cgrA ` G ) <" Y E F "> )

Proof of Theorem sacgr

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

Step	Hyp	Ref	Expression
1		dfcgra2.p	. . 3 |- P = ( Base ` G )
2		dfcgra2.i	. . 3 |- I = (Itv ` G )
3		eqid 2622	. . 3 |- (hlG ` G ) = (hlG ` G )
4		dfcgra2.g	. . . 4 |- ( ph -> G e. TarskiG )
5	4	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> G e. TarskiG )
6		sacgr.x	. . . 4 |- ( ph -> X e. P )
7	6	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> X e. P )
8		dfcgra2.b	. . . 4 |- ( ph -> B e. P )
9	8	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> B e. P )
10		dfcgra2.c	. . . 4 |- ( ph -> C e. P )
11	10	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> C e. P )
12		sacgr.y	. . . 4 |- ( ph -> Y e. P )
13	12	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> Y e. P )
14		dfcgra2.e	. . . 4 |- ( ph -> E e. P )
15	14	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> E e. P )
16		dfcgra2.f	. . . 4 |- ( ph -> F e. P )
17	16	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> F e. P )
18		dfcgra2.m	. . . 4 |- .- = ( dist ` G )
19		eqid 2622	. . . 4 |- (LineG ` G ) = (LineG ` G )
20		eqid 2622	. . . 4 |- (pInvG ` G ) = (pInvG ` G )
21		eqid 2622	. . . 4 |- ( (pInvG ` G ) ` E ) = ( (pInvG ` G ) ` E )
22		simpllr 799	. . . 4 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> x e. P )
23	1, 18, 2, 19, 20, 5, 15, 21, 22	mircl 25556	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> ( ( (pInvG ` G ) ` E ) ` x ) e. P )
24		simplr 792	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> y e. P )
25		eqid 2622	. . . 4 |- (cgrG ` G ) = (cgrG ` G )
26	1, 18, 2, 19, 20, 5, 15, 21, 22	mircgr 25552	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> ( E .- ( ( (pInvG ` G ) ` E ) ` x ) ) = ( E .- x ) )
27	1, 18, 2, 5, 15, 23, 15, 22, 26	tgcgrcomlr 25375	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> ( ( ( (pInvG ` G ) ` E ) ` x ) .- E ) = ( x .- E ) )
28		eqid 2622	. . . . . . . 8 |- ( (pInvG ` G ) ` B ) = ( (pInvG ` G ) ` B )
29	1, 18, 2, 19, 20, 4, 8, 28, 6	mircl 25556	. . . . . . 7 |- ( ph -> ( ( (pInvG ` G ) ` B ) ` X ) e. P )
30	29	ad3antrrr 766	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> ( ( (pInvG ` G ) ` B ) ` X ) e. P )
31		simpr1 1067	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> )
32	1, 18, 2, 25, 5, 30, 9, 11, 22, 15, 24, 31	cgr3simp1 25415	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> ( ( ( (pInvG ` G ) ` B ) ` X ) .- B ) = ( x .- E ) )
33	1, 18, 2, 19, 20, 4, 8, 28, 6	mircgr 25552	. . . . . . 7 |- ( ph -> ( B .- ( ( (pInvG ` G ) ` B ) ` X ) ) = ( B .- X ) )
34	1, 18, 2, 4, 8, 29, 8, 6, 33	tgcgrcomlr 25375	. . . . . 6 |- ( ph -> ( ( ( (pInvG ` G ) ` B ) ` X ) .- B ) = ( X .- B ) )
35	34	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> ( ( ( (pInvG ` G ) ` B ) ` X ) .- B ) = ( X .- B ) )
36	27, 32, 35	3eqtr2rd 2663	. . . 4 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> ( X .- B ) = ( ( ( (pInvG ` G ) ` E ) ` x ) .- E ) )
37	1, 18, 2, 25, 5, 30, 9, 11, 22, 15, 24, 31	cgr3simp2 25416	. . . 4 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> ( B .- C ) = ( E .- y ) )
38	1, 18, 2, 19, 20, 4, 8, 28, 6	mirmir 25557	. . . . . . . . . 10 |- ( ph -> ( ( (pInvG ` G ) ` B ) ` ( ( (pInvG ` G ) ` B ) ` X ) ) = X )
39		eqidd 2623	. . . . . . . . . 10 |- ( ph -> B = B )
40		eqidd 2623	. . . . . . . . . 10 |- ( ph -> C = C )
41	38, 39, 40	s3eqd 13609	. . . . . . . . 9 |- ( ph -> <" ( ( (pInvG ` G ) ` B ) ` ( ( (pInvG ` G ) ` B ) ` X ) ) B C "> = <" X B C "> )
42	41	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> <" ( ( (pInvG ` G ) ` B ) ` ( ( (pInvG ` G ) ` B ) ` X ) ) B C "> = <" X B C "> )
43		sacgr.4	. . . . . . . . . . . 12 |- ( ph -> B =/= X )
44	43	necomd 2849	. . . . . . . . . . 11 |- ( ph -> X =/= B )
45	1, 18, 2, 19, 20, 4, 8, 28, 6, 44	mirne 25562	. . . . . . . . . 10 |- ( ph -> ( ( (pInvG ` G ) ` B ) ` X ) =/= B )
46	45	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> ( ( (pInvG ` G ) ` B ) ` X ) =/= B )
47	1, 18, 2, 19, 20, 5, 25, 28, 21, 30, 9, 22, 15, 11, 24, 46, 31	mirtrcgr 25578	. . . . . . . 8 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> <" ( ( (pInvG ` G ) ` B ) ` ( ( (pInvG ` G ) ` B ) ` X ) ) B C "> (cgrG ` G ) <" ( ( (pInvG ` G ) ` E ) ` x ) E y "> )
48	42, 47	eqbrtrrd 4677	. . . . . . 7 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> <" X B C "> (cgrG ` G ) <" ( ( (pInvG ` G ) ` E ) ` x ) E y "> )
49	1, 18, 2, 25, 5, 7, 9, 11, 23, 15, 24, 48	cgr3swap13 25420	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> <" C B X "> (cgrG ` G ) <" y E ( ( (pInvG ` G ) ` E ) ` x ) "> )
50	1, 18, 2, 25, 5, 11, 9, 7, 24, 15, 23, 49	cgr3simp3 25417	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> ( X .- C ) = ( ( ( (pInvG ` G ) ` E ) ` x ) .- y ) )
51	1, 18, 2, 5, 7, 11, 23, 24, 50	tgcgrcomlr 25375	. . . 4 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> ( C .- X ) = ( y .- ( ( (pInvG ` G ) ` E ) ` x ) ) )
52	1, 18, 25, 5, 7, 9, 11, 23, 15, 24, 36, 37, 51	trgcgr 25411	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> <" X B C "> (cgrG ` G ) <" ( ( (pInvG ` G ) ` E ) ` x ) E y "> )
53		sacgr.5	. . . . . . 7 |- ( ph -> E =/= Y )
54	53	ad3antrrr 766	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> E =/= Y )
55	54	necomd 2849	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> Y =/= E )
56		dfcgra2.d	. . . . . . . 8 |- ( ph -> D e. P )
57	56	ad3antrrr 766	. . . . . . 7 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> D e. P )
58		simpr2 1068	. . . . . . 7 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> x ( (hlG ` G ) ` E ) D )
59	1, 2, 3, 22, 57, 15, 5, 58	hlne1 25500	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> x =/= E )
60	1, 18, 2, 19, 20, 5, 15, 21, 22, 59	mirne 25562	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> ( ( (pInvG ` G ) ` E ) ` x ) =/= E )
61	1, 2, 3, 22, 57, 15, 5, 58	hlcomd 25499	. . . . . . . 8 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> D ( (hlG ` G ) ` E ) x )
62		sacgr.3	. . . . . . . . 9 |- ( ph -> E e. ( D I Y ) )
63	62	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> E e. ( D I Y ) )
64	1, 2, 3, 57, 22, 13, 5, 15, 61, 63	btwnhl 25509	. . . . . . 7 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> E e. ( x I Y ) )
65	1, 18, 2, 5, 22, 15, 13, 64	tgbtwncom 25383	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> E e. ( Y I x ) )
66	1, 18, 2, 19, 20, 5, 15, 21, 22	mirmir 25557	. . . . . . 7 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> ( ( (pInvG ` G ) ` E ) ` ( ( (pInvG ` G ) ` E ) ` x ) ) = x )
67	66	oveq2d 6666	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> ( Y I ( ( (pInvG ` G ) ` E ) ` ( ( (pInvG ` G ) ` E ) ` x ) ) ) = ( Y I x ) )
68	65, 67	eleqtrrd 2704	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> E e. ( Y I ( ( (pInvG ` G ) ` E ) ` ( ( (pInvG ` G ) ` E ) ` x ) ) ) )
69	1, 18, 2, 19, 20, 5, 21, 3, 15, 13, 23, 15, 55, 60, 68	mirhl2 25576	. . . 4 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> Y ( (hlG ` G ) ` E ) ( ( (pInvG ` G ) ` E ) ` x ) )
70	1, 2, 3, 13, 23, 15, 5, 69	hlcomd 25499	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> ( ( (pInvG ` G ) ` E ) ` x ) ( (hlG ` G ) ` E ) Y )
71		simpr3 1069	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> y ( (hlG ` G ) ` E ) F )
72	1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 23, 24, 52, 70, 71	iscgrad 25703	. 2 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) -> <" X B C "> (cgrA ` G ) <" Y E F "> )
73		dfcgra2.a	. . . 4 |- ( ph -> A e. P )
74		sacgr.1	. . . . . . 7 |- ( ph -> <" A B C "> (cgrA ` G ) <" D E F "> )
75	1, 2, 3, 4, 73, 8, 10, 56, 14, 16, 74	cgrane2 25705	. . . . . 6 |- ( ph -> B =/= C )
76	1, 2, 4, 3, 29, 8, 10, 45, 75	cgraid 25711	. . . . 5 |- ( ph -> <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrA ` G ) <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> )
77	1, 2, 3, 4, 73, 8, 10, 56, 14, 16, 74	cgrane1 25704	. . . . . 6 |- ( ph -> A =/= B )
78		sacgr.2	. . . . . . 7 |- ( ph -> B e. ( A I X ) )
79	38	oveq2d 6666	. . . . . . 7 |- ( ph -> ( A I ( ( (pInvG ` G ) ` B ) ` ( ( (pInvG ` G ) ` B ) ` X ) ) ) = ( A I X ) )
80	78, 79	eleqtrrd 2704	. . . . . 6 |- ( ph -> B e. ( A I ( ( (pInvG ` G ) ` B ) ` ( ( (pInvG ` G ) ` B ) ` X ) ) ) )
81	1, 18, 2, 19, 20, 4, 28, 3, 8, 73, 29, 73, 77, 45, 80	mirhl2 25576	. . . . 5 |- ( ph -> A ( (hlG ` G ) ` B ) ( ( (pInvG ` G ) ` B ) ` X ) )
82	1, 2, 3, 4, 29, 8, 10, 29, 8, 10, 76, 73, 81	cgrahl1 25708	. . . 4 |- ( ph -> <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrA ` G ) <" A B C "> )
83	1, 2, 4, 3, 29, 8, 10, 73, 8, 10, 82, 56, 14, 16, 74	cgratr 25715	. . 3 |- ( ph -> <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrA ` G ) <" D E F "> )
84	1, 2, 3, 4, 29, 8, 10, 56, 14, 16	iscgra 25701	. . 3 |- ( ph -> ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrA ` G ) <" D E F "> <-> E. x e. P E. y e. P ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) ) )
85	83, 84	mpbid 222	. 2 |- ( ph -> E. x e. P E. y e. P ( <" ( ( (pInvG ` G ) ` B ) ` X ) B C "> (cgrG ` G ) <" x E y "> /\ x ( (hlG ` G ) ` E ) D /\ y ( (hlG ` G ) ` E ) F ) )
86	72, 85	r19.29vva 3081	1 |- ( ph -> <" X B C "> (cgrA ` G ) <" Y E F "> )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   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  LineGclng 25336  cgrGccgrg 25405  hlGchlg 25495  pInvGcmir 25547  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-mir 25548  df-cgra 25700

This theorem is referenced by:  oacgr  25723