Theorem acopyeu 24923

Description: Angle construction. Theorem 11.15 of [Schwabhauser] p. 98. This is Hilbert's axiom III.4 for geometry. Akin to a uniqueness theorem, this states that if two points X and Y both fulfill the conditions, then they are on the same half-line. (Contributed by Thierry Arnoux, 9-Aug-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 )
acopy.l	|- L = (LineG ` G )
acopy.1	|- ( ph -> -. ( A e. ( B L C ) \/ B = C ) )
acopy.2	|- ( ph -> -. ( D e. ( E L F ) \/ E = F ) )
acopyeu.x	|- ( ph -> X e. P )
acopyeu.y	|- ( ph -> Y e. P )
acopyeu.k	|- K = (hlG ` G )
acopyeu.1	|- ( ph -> <" A B C "> (cgrA ` G ) <" D E X "> )
acopyeu.2	|- ( ph -> <" A B C "> (cgrA ` G ) <" D E Y "> )
acopyeu.3	|- ( ph -> X ( (hpG ` G ) ` ( D L E ) ) F )
acopyeu.4	|- ( ph -> Y ( (hpG ` G ) ` ( D L E ) ) F )

Assertion

Ref	Expression
acopyeu	|- ( ph -> X ( K ` E ) Y )

Proof of Theorem acopyeu

Dummy variables a d t x y b u v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcgra2.p	. . . 4 |- P = ( Base ` G )
2		dfcgra2.i	. . . 4 |- I = (Itv ` G )
3		acopyeu.k	. . . 4 |- K = (hlG ` G )
4		acopyeu.x	. . . . . 6 |- ( ph -> X e. P )
5	4	ad2antrr 737	. . . . 5 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> X e. P )
6	5	ad3antrrr 741	. . . 4 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> X e. P )
7		simplr 767	. . . 4 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> y e. P )
8		acopyeu.y	. . . . . 6 |- ( ph -> Y e. P )
9	8	ad2antrr 737	. . . . 5 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> Y e. P )
10	9	ad3antrrr 741	. . . 4 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> Y e. P )
11		dfcgra2.g	. . . . . 6 |- ( ph -> G e. TarskiG )
12	11	ad2antrr 737	. . . . 5 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> G e. TarskiG )
13	12	ad3antrrr 741	. . . 4 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> G e. TarskiG )
14		dfcgra2.e	. . . . . 6 |- ( ph -> E e. P )
15	14	ad2antrr 737	. . . . 5 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> E e. P )
16	15	ad3antrrr 741	. . . 4 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> E e. P )
17		dfcgra2.m	. . . . . . 7 |- .- = ( dist ` G )
18		acopy.l	. . . . . . 7 |- L = (LineG ` G )
19		dfcgra2.a	. . . . . . . . 9 |- ( ph -> A e. P )
20	19	ad2antrr 737	. . . . . . . 8 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> A e. P )
21	20	ad3antrrr 741	. . . . . . 7 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> A e. P )
22		dfcgra2.b	. . . . . . . . 9 |- ( ph -> B e. P )
23	22	ad2antrr 737	. . . . . . . 8 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> B e. P )
24	23	ad3antrrr 741	. . . . . . 7 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> B e. P )
25		dfcgra2.c	. . . . . . . . 9 |- ( ph -> C e. P )
26	25	ad2antrr 737	. . . . . . . 8 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> C e. P )
27	26	ad3antrrr 741	. . . . . . 7 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> C e. P )
28		simplr 767	. . . . . . . 8 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> d e. P )
29	28	ad3antrrr 741	. . . . . . 7 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> d e. P )
30		dfcgra2.f	. . . . . . . . 9 |- ( ph -> F e. P )
31	30	ad2antrr 737	. . . . . . . 8 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> F e. P )
32	31	ad3antrrr 741	. . . . . . 7 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> F e. P )
33		acopy.1	. . . . . . . . 9 |- ( ph -> -. ( A e. ( B L C ) \/ B = C ) )
34	33	ad2antrr 737	. . . . . . . 8 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> -. ( A e. ( B L C ) \/ B = C ) )
35	34	ad3antrrr 741	. . . . . . 7 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> -. ( A e. ( B L C ) \/ B = C ) )
36		dfcgra2.d	. . . . . . . . . 10 |- ( ph -> D e. P )
37	36	ad2antrr 737	. . . . . . . . 9 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> D e. P )
38		acopy.2	. . . . . . . . . 10 |- ( ph -> -. ( D e. ( E L F ) \/ E = F ) )
39	38	ad2antrr 737	. . . . . . . . 9 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> -. ( D e. ( E L F ) \/ E = F ) )
40		simprl 769	. . . . . . . . . 10 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> d ( K ` E ) D )
41	1, 2, 3, 28, 37, 15, 12, 18, 40	hlln 24700	. . . . . . . . 9 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> d e. ( D L E ) )
42	1, 2, 3, 28, 37, 15, 12, 40	hlne1 24698	. . . . . . . . 9 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> d =/= E )
43	1, 2, 18, 12, 37, 15, 31, 28, 39, 41, 42	ncolncol 24739	. . . . . . . 8 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> -. ( d e. ( E L F ) \/ E = F ) )
44	43	ad3antrrr 741	. . . . . . 7 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> -. ( d e. ( E L F ) \/ E = F ) )
45		simprr 771	. . . . . . . . . 10 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> ( E .- d ) = ( B .- A ) )
46	1, 17, 2, 12, 15, 28, 23, 20, 45	tgcgrcomlr 24572	. . . . . . . . 9 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> ( d .- E ) = ( A .- B ) )
47	46	eqcomd 2467	. . . . . . . 8 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> ( A .- B ) = ( d .- E ) )
48	47	ad3antrrr 741	. . . . . . 7 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> ( A .- B ) = ( d .- E ) )
49		simpl 463	. . . . . . . . . . 11 |- ( ( u = a /\ v = b ) -> u = a )
50	49	eleq1d 2523	. . . . . . . . . 10 |- ( ( u = a /\ v = b ) -> ( u e. ( P \ ( d L E ) ) <-> a e. ( P \ ( d L E ) ) ) )
51		simpr 467	. . . . . . . . . . 11 |- ( ( u = a /\ v = b ) -> v = b )
52	51	eleq1d 2523	. . . . . . . . . 10 |- ( ( u = a /\ v = b ) -> ( v e. ( P \ ( d L E ) ) <-> b e. ( P \ ( d L E ) ) ) )
53	50, 52	anbi12d 722	. . . . . . . . 9 |- ( ( u = a /\ v = b ) -> ( ( u e. ( P \ ( d L E ) ) /\ v e. ( P \ ( d L E ) ) ) <-> ( a e. ( P \ ( d L E ) ) /\ b e. ( P \ ( d L E ) ) ) ) )
54		simpr 467	. . . . . . . . . . 11 |- ( ( ( u = a /\ v = b ) /\ w = t ) -> w = t )
55		simpll 765	. . . . . . . . . . . 12 |- ( ( ( u = a /\ v = b ) /\ w = t ) -> u = a )
56		simplr 767	. . . . . . . . . . . 12 |- ( ( ( u = a /\ v = b ) /\ w = t ) -> v = b )
57	55, 56	oveq12d 6332	. . . . . . . . . . 11 |- ( ( ( u = a /\ v = b ) /\ w = t ) -> ( u I v ) = ( a I b ) )
58	54, 57	eleq12d 2533	. . . . . . . . . 10 |- ( ( ( u = a /\ v = b ) /\ w = t ) -> ( w e. ( u I v ) <-> t e. ( a I b ) ) )
59	58	cbvrexdva 3037	. . . . . . . . 9 |- ( ( u = a /\ v = b ) -> ( E. w e. ( d L E ) w e. ( u I v ) <-> E. t e. ( d L E ) t e. ( a I b ) ) )
60	53, 59	anbi12d 722	. . . . . . . 8 |- ( ( u = a /\ v = b ) -> ( ( ( u e. ( P \ ( d L E ) ) /\ v e. ( P \ ( d L E ) ) ) /\ E. w e. ( d L E ) w e. ( u I v ) ) <-> ( ( a e. ( P \ ( d L E ) ) /\ b e. ( P \ ( d L E ) ) ) /\ E. t e. ( d L E ) t e. ( a I b ) ) ) )
61	60	cbvopabv 4485	. . . . . . 7 |- { <. u , v >. | ( ( u e. ( P \ ( d L E ) ) /\ v e. ( P \ ( d L E ) ) ) /\ E. w e. ( d L E ) w e. ( u I v ) ) } = { <. a , b >. | ( ( a e. ( P \ ( d L E ) ) /\ b e. ( P \ ( d L E ) ) ) /\ E. t e. ( d L E ) t e. ( a I b ) ) }
62		simpllr 774	. . . . . . 7 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> x e. P )
63		simprll 777	. . . . . . 7 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> <" A B C "> (cgrG ` G ) <" d E x "> )
64		simprrl 779	. . . . . . 7 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> <" A B C "> (cgrG ` G ) <" d E y "> )
65	1, 2, 18, 11, 36, 14, 30, 38	ncolne1 24718	. . . . . . . . . . . . 13 |- ( ph -> D =/= E )
66	1, 2, 18, 11, 36, 14, 65	tgelrnln 24723	. . . . . . . . . . . 12 |- ( ph -> ( D L E ) e. ran L )
67	66	ad2antrr 737	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> ( D L E ) e. ran L )
68	1, 2, 18, 11, 36, 14, 65	tglinerflx2 24727	. . . . . . . . . . . 12 |- ( ph -> E e. ( D L E ) )
69	68	ad2antrr 737	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> E e. ( D L E ) )
70	1, 2, 18, 12, 28, 15, 42, 42, 67, 41, 69	tglinethru 24729	. . . . . . . . . 10 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> ( D L E ) = ( d L E ) )
71	70, 67	eqeltrrd 2540	. . . . . . . . 9 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> ( d L E ) e. ran L )
72	71	ad3antrrr 741	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> ( d L E ) e. ran L )
73	61	eqcomi 2470	. . . . . . . 8 |- { <. a , b >. | ( ( a e. ( P \ ( d L E ) ) /\ b e. ( P \ ( d L E ) ) ) /\ E. t e. ( d L E ) t e. ( a I b ) ) } = { <. u , v >. | ( ( u e. ( P \ ( d L E ) ) /\ v e. ( P \ ( d L E ) ) ) /\ E. w e. ( d L E ) w e. ( u I v ) ) }
74	69, 70	eleqtrd 2541	. . . . . . . . . 10 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> E e. ( d L E ) )
75	74	ad3antrrr 741	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> E e. ( d L E ) )
76	37	ad3antrrr 741	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> D e. P )
77		acopyeu.1	. . . . . . . . . . . . . . . 16 |- ( ph -> <" A B C "> (cgrA ` G ) <" D E X "> )
78	1, 18, 2, 11, 22, 25, 19, 33	ncolrot2 24656	. . . . . . . . . . . . . . . 16 |- ( ph -> -. ( C e. ( A L B ) \/ A = B ) )
79	1, 2, 17, 11, 19, 22, 25, 36, 14, 4, 77, 18, 78	cgrancol 24918	. . . . . . . . . . . . . . 15 |- ( ph -> -. ( X e. ( D L E ) \/ D = E ) )
80	1, 18, 2, 11, 36, 14, 4, 79	ncolcom 24654	. . . . . . . . . . . . . 14 |- ( ph -> -. ( X e. ( E L D ) \/ E = D ) )
81	80	ad5antr 745	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> -. ( X e. ( E L D ) \/ E = D ) )
82		simprlr 778	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> x ( K ` E ) X )
83	1, 2, 3, 62, 6, 16, 13, 18, 82	hlln 24700	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> x e. ( X L E ) )
84	1, 2, 3, 62, 6, 16, 13, 82	hlne1 24698	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> x =/= E )
85	1, 2, 18, 13, 6, 16, 76, 62, 81, 83, 84	ncolncol 24739	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> -. ( x e. ( E L D ) \/ E = D ) )
86	1, 18, 2, 13, 16, 76, 62, 85	ncolcom 24654	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> -. ( x e. ( D L E ) \/ D = E ) )
87		pm2.45 403	. . . . . . . . . . 11 |- ( -. ( x e. ( D L E ) \/ D = E ) -> -. x e. ( D L E ) )
88	86, 87	syl 17	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> -. x e. ( D L E ) )
89	70	ad3antrrr 741	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> ( D L E ) = ( d L E ) )
90	89	eleq2d 2524	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> ( x e. ( D L E ) <-> x e. ( d L E ) ) )
91	88, 90	mtbid 306	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> -. x e. ( d L E ) )
92	1, 2, 18, 13, 72, 16, 61, 3, 75, 62, 6, 91, 82	hphl 24861	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> x ( (hpG ` G ) ` ( d L E ) ) X )
93		acopyeu.3	. . . . . . . . . 10 |- ( ph -> X ( (hpG ` G ) ` ( D L E ) ) F )
94	93	ad5antr 745	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> X ( (hpG ` G ) ` ( D L E ) ) F )
95	70	fveq2d 5891	. . . . . . . . . . 11 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> ( (hpG ` G ) ` ( D L E ) ) = ( (hpG ` G ) ` ( d L E ) ) )
96	95	ad3antrrr 741	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> ( (hpG ` G ) ` ( D L E ) ) = ( (hpG ` G ) ` ( d L E ) ) )
97	96	breqd 4426	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> ( X ( (hpG ` G ) ` ( D L E ) ) F <-> X ( (hpG ` G ) ` ( d L E ) ) F ) )
98	94, 97	mpbid 215	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> X ( (hpG ` G ) ` ( d L E ) ) F )
99	1, 2, 18, 13, 72, 62, 73, 6, 92, 32, 98	hpgtr 24858	. . . . . . 7 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> x ( (hpG ` G ) ` ( d L E ) ) F )
100		acopyeu.2	. . . . . . . . . . . . . . . 16 |- ( ph -> <" A B C "> (cgrA ` G ) <" D E Y "> )
101	1, 2, 17, 11, 19, 22, 25, 36, 14, 8, 100, 18, 78	cgrancol 24918	. . . . . . . . . . . . . . 15 |- ( ph -> -. ( Y e. ( D L E ) \/ D = E ) )
102	1, 18, 2, 11, 36, 14, 8, 101	ncolcom 24654	. . . . . . . . . . . . . 14 |- ( ph -> -. ( Y e. ( E L D ) \/ E = D ) )
103	102	ad5antr 745	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> -. ( Y e. ( E L D ) \/ E = D ) )
104		simprrr 780	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> y ( K ` E ) Y )
105	1, 2, 3, 7, 10, 16, 13, 18, 104	hlln 24700	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> y e. ( Y L E ) )
106	1, 2, 3, 7, 10, 16, 13, 104	hlne1 24698	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> y =/= E )
107	1, 2, 18, 13, 10, 16, 76, 7, 103, 105, 106	ncolncol 24739	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> -. ( y e. ( E L D ) \/ E = D ) )
108	1, 18, 2, 13, 16, 76, 7, 107	ncolcom 24654	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> -. ( y e. ( D L E ) \/ D = E ) )
109		pm2.45 403	. . . . . . . . . . 11 |- ( -. ( y e. ( D L E ) \/ D = E ) -> -. y e. ( D L E ) )
110	108, 109	syl 17	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> -. y e. ( D L E ) )
111	89	eleq2d 2524	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> ( y e. ( D L E ) <-> y e. ( d L E ) ) )
112	110, 111	mtbid 306	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> -. y e. ( d L E ) )
113	1, 2, 18, 13, 72, 16, 61, 3, 75, 7, 10, 112, 104	hphl 24861	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> y ( (hpG ` G ) ` ( d L E ) ) Y )
114		acopyeu.4	. . . . . . . . . 10 |- ( ph -> Y ( (hpG ` G ) ` ( D L E ) ) F )
115	114	ad5antr 745	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> Y ( (hpG ` G ) ` ( D L E ) ) F )
116	96	breqd 4426	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> ( Y ( (hpG ` G ) ` ( D L E ) ) F <-> Y ( (hpG ` G ) ` ( d L E ) ) F ) )
117	115, 116	mpbid 215	. . . . . . . 8 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> Y ( (hpG ` G ) ` ( d L E ) ) F )
118	1, 2, 18, 13, 72, 7, 73, 10, 113, 32, 117	hpgtr 24858	. . . . . . 7 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> y ( (hpG ` G ) ` ( d L E ) ) F )
119	1, 17, 2, 18, 3, 13, 21, 24, 27, 29, 16, 32, 35, 44, 48, 61, 62, 7, 63, 64, 99, 118	trgcopyeulem 24895	. . . . . 6 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> x = y )
120	119, 82	eqbrtrrd 4438	. . . . 5 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> y ( K ` E ) X )
121	1, 2, 3, 7, 6, 16, 13, 120	hlcomd 24697	. . . 4 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> X ( K ` E ) y )
122	1, 2, 3, 6, 7, 10, 13, 16, 121, 104	hltr 24703	. . 3 |- ( ( ( ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ x e. P ) /\ y e. P ) /\ ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) ) -> X ( K ` E ) Y )
123	77	ad2antrr 737	. . . . . 6 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> <" A B C "> (cgrA ` G ) <" D E X "> )
124	1, 2, 3, 12, 20, 23, 26, 37, 15, 5, 123, 28, 40	cgrahl1 24906	. . . . 5 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> <" A B C "> (cgrA ` G ) <" d E X "> )
125	1, 2, 18, 11, 19, 22, 25, 33	ncolne1 24718	. . . . . . 7 |- ( ph -> A =/= B )
126	125	ad2antrr 737	. . . . . 6 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> A =/= B )
127	1, 2, 3, 12, 20, 23, 26, 28, 15, 5, 17, 126, 47	iscgra1 24900	. . . . 5 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> ( <" A B C "> (cgrA ` G ) <" d E X "> <-> E. x e. P ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) ) )
128	124, 127	mpbid 215	. . . 4 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> E. x e. P ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) )
129	100	ad2antrr 737	. . . . . 6 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> <" A B C "> (cgrA ` G ) <" D E Y "> )
130	1, 2, 3, 12, 20, 23, 26, 37, 15, 9, 129, 28, 40	cgrahl1 24906	. . . . 5 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> <" A B C "> (cgrA ` G ) <" d E Y "> )
131	1, 2, 3, 12, 20, 23, 26, 28, 15, 9, 17, 126, 47	iscgra1 24900	. . . . 5 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> ( <" A B C "> (cgrA ` G ) <" d E Y "> <-> E. y e. P ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) )
132	130, 131	mpbid 215	. . . 4 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> E. y e. P ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) )
133		reeanv 2969	. . . 4 |- ( E. x e. P E. y e. P ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) <-> ( E. x e. P ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ E. y e. P ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) )
134	128, 132, 133	sylanbrc 675	. . 3 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> E. x e. P E. y e. P ( ( <" A B C "> (cgrG ` G ) <" d E x "> /\ x ( K ` E ) X ) /\ ( <" A B C "> (cgrG ` G ) <" d E y "> /\ y ( K ` E ) Y ) ) )
135	122, 134	r19.29vva 2945	. 2 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> X ( K ` E ) Y )
136	125	necomd 2690	. . 3 |- ( ph -> B =/= A )
137	1, 2, 3, 14, 22, 19, 11, 36, 17, 65, 136	hlcgrex 24709	. 2 |- ( ph -> E. d e. P ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) )
138	135, 137	r19.29a 2943	1 |- ( ph -> X ( K ` E ) Y )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 374   /\ wa 375   = wceq 1454   e. wcel 1897   =/= wne 2632   E.wrex 2749   \ cdif 3412   class class class wbr 4415   {copab 4473   ran crn 4853   ` cfv 5600  (class class class)co 6314   <"cs3 12974   Basecbs 15169   distcds 15247  TarskiGcstrkg 24526  Itvcitv 24532  LineGclng 24533  cgrGccgrg 24603  hlGchlg 24693  hpGchpg 24847  cgrAccgra 24897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-map 7499  df-pm 7500  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-card 8398  df-cda 8623  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-3 10696  df-n0 10898  df-z 10966  df-uz 11188  df-fz 11813  df-fzo 11946  df-hash 12547  df-word 12696  df-concat 12698  df-s1 12699  df-s2 12980  df-s3 12981  df-trkgc 24544  df-trkgb 24545  df-trkgcb 24546  df-trkgld 24548  df-trkg 24549  df-cgrg 24604  df-leg 24676  df-hlg 24694  df-mir 24746  df-rag 24787  df-perpg 24789  df-hpg 24848  df-mid 24864  df-lmi 24865  df-cgra 24898

This theorem is referenced by:  tgasa1  24937