Theorem cgrg3col4 25734

Description: Lemma 11.28 of [Schwabhauser] p. 102. Extend a congruence of three points with a fourth colinear point. (Contributed by Thierry Arnoux, 8-Oct-2020.)

Hypotheses

Ref	Expression
isleag.p	|- P = ( Base ` G )
isleag.g	|- ( ph -> G e. TarskiG )
isleag.a	|- ( ph -> A e. P )
isleag.b	|- ( ph -> B e. P )
isleag.c	|- ( ph -> C e. P )
isleag.d	|- ( ph -> D e. P )
isleag.e	|- ( ph -> E e. P )
isleag.f	|- ( ph -> F e. P )
cgrg3col4.l	|- L = (LineG ` G )
cgrg3col4.x	|- ( ph -> X e. P )
cgrg3col4.1	|- ( ph -> <" A B C "> (cgrG ` G ) <" D E F "> )
cgrg3col4.2	|- ( ph -> ( X e. ( A L C ) \/ A = C ) )

Assertion

Ref	Expression
cgrg3col4	|- ( ph -> E. y e. P <" A B C X "> (cgrG ` G ) <" D E F y "> )

Distinct variable groups:   y, A   y, B   y, C   y, D   y, E   y, F   y, G   y, L   y, P   y, X   ph, y

Proof of Theorem cgrg3col4

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isleag.p	. . . . 5 |- P = ( Base ` G )
2		cgrg3col4.l	. . . . 5 |- L = (LineG ` G )
3		eqid 2622	. . . . 5 |- (Itv ` G ) = (Itv ` G )
4		isleag.g	. . . . . 6 |- ( ph -> G e. TarskiG )
5	4	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) -> G e. TarskiG )
6		isleag.a	. . . . . 6 |- ( ph -> A e. P )
7	6	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) -> A e. P )
8		isleag.b	. . . . . 6 |- ( ph -> B e. P )
9	8	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) -> B e. P )
10		cgrg3col4.x	. . . . . 6 |- ( ph -> X e. P )
11	10	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) -> X e. P )
12		eqid 2622	. . . . 5 |- (cgrG ` G ) = (cgrG ` G )
13		isleag.d	. . . . . 6 |- ( ph -> D e. P )
14	13	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) -> D e. P )
15		isleag.e	. . . . . 6 |- ( ph -> E e. P )
16	15	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) -> E e. P )
17		eqid 2622	. . . . 5 |- ( dist ` G ) = ( dist ` G )
18		simpr 477	. . . . 5 |- ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) -> ( B e. ( A L X ) \/ A = X ) )
19		isleag.c	. . . . . . 7 |- ( ph -> C e. P )
20		isleag.f	. . . . . . 7 |- ( ph -> F e. P )
21		cgrg3col4.1	. . . . . . 7 |- ( ph -> <" A B C "> (cgrG ` G ) <" D E F "> )
22	1, 17, 3, 12, 4, 6, 8, 19, 13, 15, 20, 21	cgr3simp1 25415	. . . . . 6 |- ( ph -> ( A ( dist ` G ) B ) = ( D ( dist ` G ) E ) )
23	22	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) -> ( A ( dist ` G ) B ) = ( D ( dist ` G ) E ) )
24	1, 2, 3, 5, 7, 9, 11, 12, 14, 16, 17, 18, 23	lnext 25462	. . . 4 |- ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) -> E. y e. P <" A B X "> (cgrG ` G ) <" D E y "> )
25	21	ad4antr 768	. . . . . . . 8 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> <" A B C "> (cgrG ` G ) <" D E F "> )
26	5	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> G e. TarskiG )
27	11	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> X e. P )
28	7	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> A e. P )
29		simplr 792	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> y e. P )
30	14	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> D e. P )
31	9	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> B e. P )
32	16	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> E e. P )
33		simpr 477	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> <" A B X "> (cgrG ` G ) <" D E y "> )
34	1, 17, 3, 12, 26, 28, 31, 27, 30, 32, 29, 33	cgr3simp3 25417	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> ( X ( dist ` G ) A ) = ( y ( dist ` G ) D ) )
35	1, 17, 3, 26, 27, 28, 29, 30, 34	tgcgrcomlr 25375	. . . . . . . . 9 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> ( A ( dist ` G ) X ) = ( D ( dist ` G ) y ) )
36	1, 17, 3, 12, 26, 28, 31, 27, 30, 32, 29, 33	cgr3simp2 25416	. . . . . . . . 9 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> ( B ( dist ` G ) X ) = ( E ( dist ` G ) y ) )
37	19	ad4antr 768	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> C e. P )
38	20	ad4antr 768	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> F e. P )
39		simpr 477	. . . . . . . . . . . . 13 |- ( ( ph /\ A = C ) -> A = C )
40	39	ad3antrrr 766	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> A = C )
41	40	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> ( X ( dist ` G ) A ) = ( X ( dist ` G ) C ) )
42	4	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ A = C ) -> G e. TarskiG )
43	6	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ A = C ) -> A e. P )
44	19	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ A = C ) -> C e. P )
45	13	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ A = C ) -> D e. P )
46	20	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ A = C ) -> F e. P )
47	1, 17, 3, 12, 4, 6, 8, 19, 13, 15, 20, 21	cgr3simp3 25417	. . . . . . . . . . . . . . . 16 |- ( ph -> ( C ( dist ` G ) A ) = ( F ( dist ` G ) D ) )
48	1, 17, 3, 4, 19, 6, 20, 13, 47	tgcgrcomlr 25375	. . . . . . . . . . . . . . 15 |- ( ph -> ( A ( dist ` G ) C ) = ( D ( dist ` G ) F ) )
49	48	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ A = C ) -> ( A ( dist ` G ) C ) = ( D ( dist ` G ) F ) )
50	1, 17, 3, 42, 43, 44, 45, 46, 49, 39	tgcgreq 25377	. . . . . . . . . . . . 13 |- ( ( ph /\ A = C ) -> D = F )
51	50	ad3antrrr 766	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> D = F )
52	51	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> ( y ( dist ` G ) D ) = ( y ( dist ` G ) F ) )
53	34, 41, 52	3eqtr3d 2664	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> ( X ( dist ` G ) C ) = ( y ( dist ` G ) F ) )
54	1, 17, 3, 26, 27, 37, 29, 38, 53	tgcgrcomlr 25375	. . . . . . . . 9 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> ( C ( dist ` G ) X ) = ( F ( dist ` G ) y ) )
55	35, 36, 54	3jca 1242	. . . . . . . 8 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> ( ( A ( dist ` G ) X ) = ( D ( dist ` G ) y ) /\ ( B ( dist ` G ) X ) = ( E ( dist ` G ) y ) /\ ( C ( dist ` G ) X ) = ( F ( dist ` G ) y ) ) )
56	25, 55	jca 554	. . . . . . 7 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> ( <" A B C "> (cgrG ` G ) <" D E F "> /\ ( ( A ( dist ` G ) X ) = ( D ( dist ` G ) y ) /\ ( B ( dist ` G ) X ) = ( E ( dist ` G ) y ) /\ ( C ( dist ` G ) X ) = ( F ( dist ` G ) y ) ) ) )
57	1, 17, 3, 12, 26, 28, 31, 37, 27, 30, 32, 38, 29	tgcgr4 25426	. . . . . . 7 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> ( <" A B C X "> (cgrG ` G ) <" D E F y "> <-> ( <" A B C "> (cgrG ` G ) <" D E F "> /\ ( ( A ( dist ` G ) X ) = ( D ( dist ` G ) y ) /\ ( B ( dist ` G ) X ) = ( E ( dist ` G ) y ) /\ ( C ( dist ` G ) X ) = ( F ( dist ` G ) y ) ) ) ) )
58	56, 57	mpbird 247	. . . . . 6 |- ( ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) /\ <" A B X "> (cgrG ` G ) <" D E y "> ) -> <" A B C X "> (cgrG ` G ) <" D E F y "> )
59	58	ex 450	. . . . 5 |- ( ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) /\ y e. P ) -> ( <" A B X "> (cgrG ` G ) <" D E y "> -> <" A B C X "> (cgrG ` G ) <" D E F y "> ) )
60	59	reximdva 3017	. . . 4 |- ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) -> ( E. y e. P <" A B X "> (cgrG ` G ) <" D E y "> -> E. y e. P <" A B C X "> (cgrG ` G ) <" D E F y "> ) )
61	24, 60	mpd 15	. . 3 |- ( ( ( ph /\ A = C ) /\ ( B e. ( A L X ) \/ A = X ) ) -> E. y e. P <" A B C X "> (cgrG ` G ) <" D E F y "> )
62		eqid 2622	. . . . . 6 |- (hlG ` G ) = (hlG ` G )
63	42	adantr 481	. . . . . . 7 |- ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) -> G e. TarskiG )
64	63	ad2antrr 762	. . . . . 6 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> G e. TarskiG )
65	8	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) -> B e. P )
66	65	ad2antrr 762	. . . . . 6 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> B e. P )
67	43	adantr 481	. . . . . . 7 |- ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) -> A e. P )
68	67	ad2antrr 762	. . . . . 6 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> A e. P )
69	10	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) -> X e. P )
70	69	ad2antrr 762	. . . . . 6 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> X e. P )
71	15	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) -> E e. P )
72	71	ad2antrr 762	. . . . . 6 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> E e. P )
73	45	adantr 481	. . . . . . 7 |- ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) -> D e. P )
74	73	ad2antrr 762	. . . . . 6 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> D e. P )
75		simplr 792	. . . . . 6 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> x e. P )
76		simpr 477	. . . . . . 7 |- ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) -> -. ( B e. ( A L X ) \/ A = X ) )
77	76	ad2antrr 762	. . . . . 6 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> -. ( B e. ( A L X ) \/ A = X ) )
78		simpr 477	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> -. x e. ( D L E ) )
79	22	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) -> ( A ( dist ` G ) B ) = ( D ( dist ` G ) E ) )
80	1, 3, 2, 63, 65, 67, 69, 76	ncolne1 25520	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) -> B =/= A )
81	80	necomd 2849	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) -> A =/= B )
82	1, 17, 3, 63, 67, 65, 73, 71, 79, 81	tgcgrneq 25378	. . . . . . . . . . . 12 |- ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) -> D =/= E )
83	82	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> D =/= E )
84	83	neneqd 2799	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> -. D = E )
85	78, 84	jca 554	. . . . . . . . 9 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> ( -. x e. ( D L E ) /\ -. D = E ) )
86		ioran 511	. . . . . . . . 9 |- ( -. ( x e. ( D L E ) \/ D = E ) <-> ( -. x e. ( D L E ) /\ -. D = E ) )
87	85, 86	sylibr 224	. . . . . . . 8 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> -. ( x e. ( D L E ) \/ D = E ) )
88	1, 2, 3, 64, 74, 72, 75, 87	ncolcom 25456	. . . . . . 7 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> -. ( x e. ( E L D ) \/ E = D ) )
89	1, 2, 3, 64, 72, 74, 75, 88	ncolrot1 25457	. . . . . 6 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> -. ( E e. ( D L x ) \/ D = x ) )
90	1, 17, 3, 4, 6, 8, 13, 15, 22	tgcgrcomlr 25375	. . . . . . 7 |- ( ph -> ( B ( dist ` G ) A ) = ( E ( dist ` G ) D ) )
91	90	ad4antr 768	. . . . . 6 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> ( B ( dist ` G ) A ) = ( E ( dist ` G ) D ) )
92	1, 17, 3, 2, 62, 64, 66, 68, 70, 72, 74, 75, 77, 89, 91	trgcopy 25696	. . . . 5 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> E. y e. P ( <" B A X "> (cgrG ` G ) <" E D y "> /\ y ( (hpG ` G ) ` ( E L D ) ) x ) )
93	21	ad6antr 772	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> <" A B C "> (cgrG ` G ) <" D E F "> )
94	64	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> G e. TarskiG )
95	66	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> B e. P )
96	68	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> A e. P )
97	70	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> X e. P )
98	72	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> E e. P )
99	74	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> D e. P )
100		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> y e. P )
101		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> <" B A X "> (cgrG ` G ) <" E D y "> )
102	1, 17, 3, 12, 94, 95, 96, 97, 98, 99, 100, 101	cgr3simp2 25416	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> ( A ( dist ` G ) X ) = ( D ( dist ` G ) y ) )
103	1, 17, 3, 12, 94, 95, 96, 97, 98, 99, 100, 101	cgr3simp3 25417	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> ( X ( dist ` G ) B ) = ( y ( dist ` G ) E ) )
104	1, 17, 3, 94, 97, 95, 100, 98, 103	tgcgrcomlr 25375	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> ( B ( dist ` G ) X ) = ( E ( dist ` G ) y ) )
105	44	ad5antr 770	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> C e. P )
106	46	ad5antr 770	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> F e. P )
107	1, 17, 3, 94, 96, 97, 99, 100, 102	tgcgrcomlr 25375	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> ( X ( dist ` G ) A ) = ( y ( dist ` G ) D ) )
108		simp-6r 811	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> A = C )
109	108	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> ( X ( dist ` G ) A ) = ( X ( dist ` G ) C ) )
110	50	ad5antr 770	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> D = F )
111	110	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> ( y ( dist ` G ) D ) = ( y ( dist ` G ) F ) )
112	107, 109, 111	3eqtr3d 2664	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> ( X ( dist ` G ) C ) = ( y ( dist ` G ) F ) )
113	1, 17, 3, 94, 97, 105, 100, 106, 112	tgcgrcomlr 25375	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> ( C ( dist ` G ) X ) = ( F ( dist ` G ) y ) )
114	102, 104, 113	3jca 1242	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> ( ( A ( dist ` G ) X ) = ( D ( dist ` G ) y ) /\ ( B ( dist ` G ) X ) = ( E ( dist ` G ) y ) /\ ( C ( dist ` G ) X ) = ( F ( dist ` G ) y ) ) )
115	93, 114	jca 554	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> ( <" A B C "> (cgrG ` G ) <" D E F "> /\ ( ( A ( dist ` G ) X ) = ( D ( dist ` G ) y ) /\ ( B ( dist ` G ) X ) = ( E ( dist ` G ) y ) /\ ( C ( dist ` G ) X ) = ( F ( dist ` G ) y ) ) ) )
116	1, 17, 3, 12, 94, 96, 95, 105, 97, 99, 98, 106, 100	tgcgr4 25426	. . . . . . . . 9 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> ( <" A B C X "> (cgrG ` G ) <" D E F y "> <-> ( <" A B C "> (cgrG ` G ) <" D E F "> /\ ( ( A ( dist ` G ) X ) = ( D ( dist ` G ) y ) /\ ( B ( dist ` G ) X ) = ( E ( dist ` G ) y ) /\ ( C ( dist ` G ) X ) = ( F ( dist ` G ) y ) ) ) ) )
117	115, 116	mpbird 247	. . . . . . . 8 |- ( ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) /\ <" B A X "> (cgrG ` G ) <" E D y "> ) -> <" A B C X "> (cgrG ` G ) <" D E F y "> )
118	117	ex 450	. . . . . . 7 |- ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) -> ( <" B A X "> (cgrG ` G ) <" E D y "> -> <" A B C X "> (cgrG ` G ) <" D E F y "> ) )
119	118	adantrd 484	. . . . . 6 |- ( ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) /\ y e. P ) -> ( ( <" B A X "> (cgrG ` G ) <" E D y "> /\ y ( (hpG ` G ) ` ( E L D ) ) x ) -> <" A B C X "> (cgrG ` G ) <" D E F y "> ) )
120	119	reximdva 3017	. . . . 5 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> ( E. y e. P ( <" B A X "> (cgrG ` G ) <" E D y "> /\ y ( (hpG ` G ) ` ( E L D ) ) x ) -> E. y e. P <" A B C X "> (cgrG ` G ) <" D E F y "> ) )
121	92, 120	mpd 15	. . . 4 |- ( ( ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) /\ x e. P ) /\ -. x e. ( D L E ) ) -> E. y e. P <" A B C X "> (cgrG ` G ) <" D E F y "> )
122	1, 2, 3, 63, 67, 69, 65, 76	ncoltgdim2 25460	. . . . 5 |- ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) -> GDimTarskiG≥ 2 )
123	1, 3, 2, 63, 122, 73, 71, 82	tglowdim2ln 25546	. . . 4 |- ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) -> E. x e. P -. x e. ( D L E ) )
124	121, 123	r19.29a 3078	. . 3 |- ( ( ( ph /\ A = C ) /\ -. ( B e. ( A L X ) \/ A = X ) ) -> E. y e. P <" A B C X "> (cgrG ` G ) <" D E F y "> )
125	61, 124	pm2.61dan 832	. 2 |- ( ( ph /\ A = C ) -> E. y e. P <" A B C X "> (cgrG ` G ) <" D E F y "> )
126		cgrg3col4.2	. . . . . . 7 |- ( ph -> ( X e. ( A L C ) \/ A = C ) )
127	1, 2, 3, 4, 6, 19, 10, 126	colcom 25453	. . . . . 6 |- ( ph -> ( X e. ( C L A ) \/ C = A ) )
128	1, 2, 3, 4, 19, 6, 10, 127	colrot1 25454	. . . . 5 |- ( ph -> ( C e. ( A L X ) \/ A = X ) )
129	1, 2, 3, 4, 6, 19, 10, 12, 13, 20, 17, 128, 48	lnext 25462	. . . 4 |- ( ph -> E. y e. P <" A C X "> (cgrG ` G ) <" D F y "> )
130	129	adantr 481	. . 3 |- ( ( ph /\ A =/= C ) -> E. y e. P <" A C X "> (cgrG ` G ) <" D F y "> )
131	21	ad3antrrr 766	. . . . . . 7 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> <" A B C "> (cgrG ` G ) <" D E F "> )
132	4	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> G e. TarskiG )
133	10	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> X e. P )
134	6	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> A e. P )
135		simplr 792	. . . . . . . . 9 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> y e. P )
136	13	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> D e. P )
137	19	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> C e. P )
138	20	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> F e. P )
139		simpr 477	. . . . . . . . . 10 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> <" A C X "> (cgrG ` G ) <" D F y "> )
140	1, 17, 3, 12, 132, 134, 137, 133, 136, 138, 135, 139	cgr3simp3 25417	. . . . . . . . 9 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> ( X ( dist ` G ) A ) = ( y ( dist ` G ) D ) )
141	1, 17, 3, 132, 133, 134, 135, 136, 140	tgcgrcomlr 25375	. . . . . . . 8 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> ( A ( dist ` G ) X ) = ( D ( dist ` G ) y ) )
142	8	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> B e. P )
143	15	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> E e. P )
144	128	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> ( C e. ( A L X ) \/ A = X ) )
145	22	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> ( A ( dist ` G ) B ) = ( D ( dist ` G ) E ) )
146	1, 17, 3, 12, 4, 6, 8, 19, 13, 15, 20, 21	cgr3simp2 25416	. . . . . . . . . . . 12 |- ( ph -> ( B ( dist ` G ) C ) = ( E ( dist ` G ) F ) )
147	1, 17, 3, 4, 8, 19, 15, 20, 146	tgcgrcomlr 25375	. . . . . . . . . . 11 |- ( ph -> ( C ( dist ` G ) B ) = ( F ( dist ` G ) E ) )
148	147	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> ( C ( dist ` G ) B ) = ( F ( dist ` G ) E ) )
149		simpllr 799	. . . . . . . . . 10 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> A =/= C )
150	1, 2, 3, 132, 134, 137, 133, 12, 136, 138, 17, 142, 135, 143, 144, 139, 145, 148, 149	tgfscgr 25463	. . . . . . . . 9 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> ( X ( dist ` G ) B ) = ( y ( dist ` G ) E ) )
151	1, 17, 3, 132, 133, 142, 135, 143, 150	tgcgrcomlr 25375	. . . . . . . 8 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> ( B ( dist ` G ) X ) = ( E ( dist ` G ) y ) )
152	1, 17, 3, 12, 132, 134, 137, 133, 136, 138, 135, 139	cgr3simp2 25416	. . . . . . . 8 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> ( C ( dist ` G ) X ) = ( F ( dist ` G ) y ) )
153	141, 151, 152	3jca 1242	. . . . . . 7 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> ( ( A ( dist ` G ) X ) = ( D ( dist ` G ) y ) /\ ( B ( dist ` G ) X ) = ( E ( dist ` G ) y ) /\ ( C ( dist ` G ) X ) = ( F ( dist ` G ) y ) ) )
154	131, 153	jca 554	. . . . . 6 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> ( <" A B C "> (cgrG ` G ) <" D E F "> /\ ( ( A ( dist ` G ) X ) = ( D ( dist ` G ) y ) /\ ( B ( dist ` G ) X ) = ( E ( dist ` G ) y ) /\ ( C ( dist ` G ) X ) = ( F ( dist ` G ) y ) ) ) )
155	1, 17, 3, 12, 132, 134, 142, 137, 133, 136, 143, 138, 135	tgcgr4 25426	. . . . . 6 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> ( <" A B C X "> (cgrG ` G ) <" D E F y "> <-> ( <" A B C "> (cgrG ` G ) <" D E F "> /\ ( ( A ( dist ` G ) X ) = ( D ( dist ` G ) y ) /\ ( B ( dist ` G ) X ) = ( E ( dist ` G ) y ) /\ ( C ( dist ` G ) X ) = ( F ( dist ` G ) y ) ) ) ) )
156	154, 155	mpbird 247	. . . . 5 |- ( ( ( ( ph /\ A =/= C ) /\ y e. P ) /\ <" A C X "> (cgrG ` G ) <" D F y "> ) -> <" A B C X "> (cgrG ` G ) <" D E F y "> )
157	156	ex 450	. . . 4 |- ( ( ( ph /\ A =/= C ) /\ y e. P ) -> ( <" A C X "> (cgrG ` G ) <" D F y "> -> <" A B C X "> (cgrG ` G ) <" D E F y "> ) )
158	157	reximdva 3017	. . 3 |- ( ( ph /\ A =/= C ) -> ( E. y e. P <" A C X "> (cgrG ` G ) <" D F y "> -> E. y e. P <" A B C X "> (cgrG ` G ) <" D E F y "> ) )
159	130, 158	mpd 15	. 2 |- ( ( ph /\ A =/= C ) -> E. y e. P <" A B C X "> (cgrG ` G ) <" D E F y "> )
160	125, 159	pm2.61dane 2881	1 |- ( ph -> E. y e. P <" A B C X "> (cgrG ` G ) <" D E F y "> )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ 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   <"cs4 13588   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  cgrGccgrg 25405  hlGchlg 25495  hpGchpg 25649

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-fal 1489  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-4 11081  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-s4 13595  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkgld 25351  df-trkg 25352  df-cgrg 25406  df-ismt 25428  df-leg 25478  df-hlg 25496  df-mir 25548  df-rag 25589  df-perpg 25591  df-hpg 25650  df-mid 25666  df-lmi 25667

This theorem is referenced by: (None)