Theorem acopyeu 25725

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 762	. . . . 5 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> X e. P )
6	5	ad3antrrr 766	. . . 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 792	. . . 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 762	. . . . 5 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> Y e. P )
10	9	ad3antrrr 766	. . . 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 762	. . . . 5 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> G e. TarskiG )
13	12	ad3antrrr 766	. . . 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 762	. . . . 5 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> E e. P )
16	15	ad3antrrr 766	. . . 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 762	. . . . . . . 8 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> A e. P )
21	20	ad3antrrr 766	. . . . . . 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 762	. . . . . . . 8 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> B e. P )
24	23	ad3antrrr 766	. . . . . . 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 762	. . . . . . . 8 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> C e. P )
27	26	ad3antrrr 766	. . . . . . 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 792	. . . . . . . 8 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> d e. P )
29	28	ad3antrrr 766	. . . . . . 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 762	. . . . . . . 8 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> F e. P )
32	31	ad3antrrr 766	. . . . . . 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 762	. . . . . . . 8 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> -. ( A e. ( B L C ) \/ B = C ) )
35	34	ad3antrrr 766	. . . . . . 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 762	. . . . . . . . 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 762	. . . . . . . . 9 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> -. ( D e. ( E L F ) \/ E = F ) )
40		simprl 794	. . . . . . . . . 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 25502	. . . . . . . . 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 25500	. . . . . . . . 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 25541	. . . . . . . 8 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> -. ( d e. ( E L F ) \/ E = F ) )
44	43	ad3antrrr 766	. . . . . . 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 796	. . . . . . . . . 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 25375	. . . . . . . . 9 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> ( d .- E ) = ( A .- B ) )
47	46	eqcomd 2628	. . . . . . . 8 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> ( A .- B ) = ( d .- E ) )
48	47	ad3antrrr 766	. . . . . . 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 473	. . . . . . . . . . 11 |- ( ( u = a /\ v = b ) -> u = a )
50	49	eleq1d 2686	. . . . . . . . . 10 |- ( ( u = a /\ v = b ) -> ( u e. ( P \ ( d L E ) ) <-> a e. ( P \ ( d L E ) ) ) )
51		simpr 477	. . . . . . . . . . 11 |- ( ( u = a /\ v = b ) -> v = b )
52	51	eleq1d 2686	. . . . . . . . . 10 |- ( ( u = a /\ v = b ) -> ( v e. ( P \ ( d L E ) ) <-> b e. ( P \ ( d L E ) ) ) )
53	50, 52	anbi12d 747	. . . . . . . . 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 477	. . . . . . . . . . 11 |- ( ( ( u = a /\ v = b ) /\ w = t ) -> w = t )
55		simpll 790	. . . . . . . . . . . 12 |- ( ( ( u = a /\ v = b ) /\ w = t ) -> u = a )
56		simplr 792	. . . . . . . . . . . 12 |- ( ( ( u = a /\ v = b ) /\ w = t ) -> v = b )
57	55, 56	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ( u = a /\ v = b ) /\ w = t ) -> ( u I v ) = ( a I b ) )
58	54, 57	eleq12d 2695	. . . . . . . . . 10 |- ( ( ( u = a /\ v = b ) /\ w = t ) -> ( w e. ( u I v ) <-> t e. ( a I b ) ) )
59	58	cbvrexdva 3178	. . . . . . . . 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 747	. . . . . . . 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 4722	. . . . . . 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 799	. . . . . . 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 802	. . . . . . 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 804	. . . . . . 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 25520	. . . . . . . . . . . . 13 |- ( ph -> D =/= E )
66	1, 2, 18, 11, 36, 14, 65	tgelrnln 25525	. . . . . . . . . . . 12 |- ( ph -> ( D L E ) e. ran L )
67	66	ad2antrr 762	. . . . . . . . . . 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 25529	. . . . . . . . . . . 12 |- ( ph -> E e. ( D L E ) )
69	68	ad2antrr 762	. . . . . . . . . . 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 25531	. . . . . . . . . 10 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> ( D L E ) = ( d L E ) )
71	70, 67	eqeltrrd 2702	. . . . . . . . 9 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> ( d L E ) e. ran L )
72	71	ad3antrrr 766	. . . . . . . 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 2631	. . . . . . . 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 2703	. . . . . . . . . 10 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> E e. ( d L E ) )
75	74	ad3antrrr 766	. . . . . . . . 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 766	. . . . . . . . . . . 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 25458	. . . . . . . . . . . . . . . 16 |- ( ph -> -. ( C e. ( A L B ) \/ A = B ) )
79	1, 2, 17, 11, 19, 22, 25, 36, 14, 4, 77, 18, 78	cgrancol 25720	. . . . . . . . . . . . . . 15 |- ( ph -> -. ( X e. ( D L E ) \/ D = E ) )
80	1, 18, 2, 11, 36, 14, 4, 79	ncolcom 25456	. . . . . . . . . . . . . 14 |- ( ph -> -. ( X e. ( E L D ) \/ E = D ) )
81	80	ad5antr 770	. . . . . . . . . . . . 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 803	. . . . . . . . . . . . . 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 25502	. . . . . . . . . . . . 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 25500	. . . . . . . . . . . . 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 25541	. . . . . . . . . . . 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 25456	. . . . . . . . . . 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		pm2.45 412	. . . . . . . . . . 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 766	. . . . . . . . . . 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 2687	. . . . . . . . . 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 314	. . . . . . . . 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 25663	. . . . . . . 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 770	. . . . . . . . 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 6195	. . . . . . . . . . 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 766	. . . . . . . . . 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 4664	. . . . . . . . 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 222	. . . . . . . 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 25660	. . . . . . 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 25720	. . . . . . . . . . . . . . 15 |- ( ph -> -. ( Y e. ( D L E ) \/ D = E ) )
102	1, 18, 2, 11, 36, 14, 8, 101	ncolcom 25456	. . . . . . . . . . . . . 14 |- ( ph -> -. ( Y e. ( E L D ) \/ E = D ) )
103	102	ad5antr 770	. . . . . . . . . . . . 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 805	. . . . . . . . . . . . . 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 25502	. . . . . . . . . . . . 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 25500	. . . . . . . . . . . . 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 25541	. . . . . . . . . . . 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 25456	. . . . . . . . . . 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		pm2.45 412	. . . . . . . . . . 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 2687	. . . . . . . . . 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 314	. . . . . . . . 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 25663	. . . . . . . 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 770	. . . . . . . . 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 4664	. . . . . . . . 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 222	. . . . . . . 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 25660	. . . . . . 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 25697	. . . . . 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 4677	. . . . 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 25499	. . . 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 25505	. . 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 762	. . . . . 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 25708	. . . . 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 25520	. . . . . . 7 |- ( ph -> A =/= B )
126	125	ad2antrr 762	. . . . . 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 25702	. . . . 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 222	. . . 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 762	. . . . . 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 25708	. . . . 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 25702	. . . . 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 222	. . . 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 3107	. . . 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 698	. . 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 3081	. 2 |- ( ( ( ph /\ d e. P ) /\ ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> X ( K ` E ) Y )
136	125	necomd 2849	. . 3 |- ( ph -> B =/= A )
137	1, 2, 3, 14, 22, 19, 11, 36, 17, 65, 136	hlcgrex 25511	. 2 |- ( ph -> E. d e. P ( d ( K ` E ) D /\ ( E .- d ) = ( B .- A ) ) )
138	135, 137	r19.29a 3078	1 |- ( ph -> X ( K ` E ) Y )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   class class class wbr 4653   {copab 4712   ran crn 5115   ` cfv 5888  (class class class)co 6650   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  cgrGccgrg 25405  hlGchlg 25495  hpGchpg 25649  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-trkgld 25351  df-trkg 25352  df-cgrg 25406  df-leg 25478  df-hlg 25496  df-mir 25548  df-rag 25589  df-perpg 25591  df-hpg 25650  df-mid 25666  df-lmi 25667  df-cgra 25700

This theorem is referenced by:  tgasa1  25739