Theorem cgracgr 25710

Description: First direction of proposition 11.4 of [Schwabhauser] p. 95. Again, this is "half" of the proposition, i.e. only two additional points are used, while Schwabhauser has four. (Contributed by Thierry Arnoux, 31-Jul-2020.)

Hypotheses

Ref	Expression
iscgra.p	|- P = ( Base ` G )
iscgra.i	|- I = (Itv ` G )
iscgra.k	|- K = (hlG ` G )
iscgra.g	|- ( ph -> G e. TarskiG )
iscgra.a	|- ( ph -> A e. P )
iscgra.b	|- ( ph -> B e. P )
iscgra.c	|- ( ph -> C e. P )
iscgra.d	|- ( ph -> D e. P )
iscgra.e	|- ( ph -> E e. P )
iscgra.f	|- ( ph -> F e. P )
cgrahl1.2	|- ( ph -> <" A B C "> (cgrA ` G ) <" D E F "> )
cgrahl1.x	|- ( ph -> X e. P )
cgracgr.m	|- .- = ( dist ` G )
cgracgr.y	|- ( ph -> Y e. P )
cgracgr.1	|- ( ph -> X ( K ` B ) A )
cgracgr.2	|- ( ph -> Y ( K ` B ) C )
cgracgr.3	|- ( ph -> ( B .- X ) = ( E .- D ) )
cgracgr.4	|- ( ph -> ( B .- Y ) = ( E .- F ) )

Assertion

Ref	Expression
cgracgr	|- ( ph -> ( X .- Y ) = ( D .- F ) )

Proof of Theorem cgracgr

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

Step	Hyp	Ref	Expression
1		iscgra.p	. . 3 |- P = ( Base ` G )
2		eqid 2622	. . 3 |- (LineG ` G ) = (LineG ` G )
3		iscgra.i	. . 3 |- I = (Itv ` G )
4		iscgra.g	. . . 4 |- ( ph -> G e. TarskiG )
5	4	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> G e. TarskiG )
6		iscgra.a	. . . 4 |- ( ph -> A e. P )
7	6	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> A e. P )
8		iscgra.b	. . . 4 |- ( ph -> B e. P )
9	8	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> B e. P )
10		cgrahl1.x	. . . 4 |- ( ph -> X e. P )
11	10	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> X e. P )
12		eqid 2622	. . 3 |- (cgrG ` G ) = (cgrG ` G )
13		simpllr 799	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> x e. P )
14		iscgra.e	. . . 4 |- ( ph -> E e. P )
15	14	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> E e. P )
16		cgracgr.m	. . 3 |- .- = ( dist ` G )
17		cgracgr.y	. . . 4 |- ( ph -> Y e. P )
18	17	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> Y e. P )
19		iscgra.d	. . . 4 |- ( ph -> D e. P )
20	19	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> D e. P )
21		iscgra.f	. . . 4 |- ( ph -> F e. P )
22	21	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> F e. P )
23		iscgra.k	. . . . . . . . 9 |- K = (hlG ` G )
24		iscgra.c	. . . . . . . . 9 |- ( ph -> C e. P )
25		cgrahl1.2	. . . . . . . . 9 |- ( ph -> <" A B C "> (cgrA ` G ) <" D E F "> )
26	1, 3, 23, 4, 6, 8, 24, 19, 14, 21, 25	cgrane1 25704	. . . . . . . 8 |- ( ph -> A =/= B )
27	26	necomd 2849	. . . . . . 7 |- ( ph -> B =/= A )
28		cgracgr.1	. . . . . . . 8 |- ( ph -> X ( K ` B ) A )
29	1, 3, 23, 10, 6, 8, 4, 2, 28	hlln 25502	. . . . . . 7 |- ( ph -> X e. ( A (LineG ` G ) B ) )
30	1, 3, 2, 4, 8, 6, 10, 27, 29	lncom 25517	. . . . . 6 |- ( ph -> X e. ( B (LineG ` G ) A ) )
31	30	orcd 407	. . . . 5 |- ( ph -> ( X e. ( B (LineG ` G ) A ) \/ B = A ) )
32	1, 2, 3, 4, 8, 6, 10, 31	colrot1 25454	. . . 4 |- ( ph -> ( B e. ( A (LineG ` G ) X ) \/ A = X ) )
33	32	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( B e. ( A (LineG ` G ) X ) \/ A = X ) )
34	24	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> C e. P )
35		simplr 792	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> y e. P )
36		simpr1 1067	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> <" A B C "> (cgrG ` G ) <" x E y "> )
37	1, 16, 3, 12, 5, 7, 9, 34, 13, 15, 35, 36	cgr3simp1 25415	. . . 4 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( A .- B ) = ( x .- E ) )
38		cgracgr.3	. . . . 5 |- ( ph -> ( B .- X ) = ( E .- D ) )
39	38	ad3antrrr 766	. . . 4 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( B .- X ) = ( E .- D ) )
40		eqid 2622	. . . . . . 7 |- (≤G ` G ) = (≤G ` G )
41		simpr2 1068	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> x ( K ` E ) D )
42	1, 3, 23, 13, 20, 15, 5	ishlg 25497	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( x ( K ` E ) D <-> ( x =/= E /\ D =/= E /\ ( x e. ( E I D ) \/ D e. ( E I x ) ) ) ) )
43	41, 42	mpbid 222	. . . . . . . 8 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( x =/= E /\ D =/= E /\ ( x e. ( E I D ) \/ D e. ( E I x ) ) ) )
44	43	simp3d 1075	. . . . . . 7 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( x e. ( E I D ) \/ D e. ( E I x ) ) )
45	1, 3, 23, 10, 6, 8, 4	ishlg 25497	. . . . . . . . . . 11 |- ( ph -> ( X ( K ` B ) A <-> ( X =/= B /\ A =/= B /\ ( X e. ( B I A ) \/ A e. ( B I X ) ) ) ) )
46	28, 45	mpbid 222	. . . . . . . . . 10 |- ( ph -> ( X =/= B /\ A =/= B /\ ( X e. ( B I A ) \/ A e. ( B I X ) ) ) )
47	46	simp3d 1075	. . . . . . . . 9 |- ( ph -> ( X e. ( B I A ) \/ A e. ( B I X ) ) )
48	47	orcomd 403	. . . . . . . 8 |- ( ph -> ( A e. ( B I X ) \/ X e. ( B I A ) ) )
49	48	ad3antrrr 766	. . . . . . 7 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( A e. ( B I X ) \/ X e. ( B I A ) ) )
50	37	eqcomd 2628	. . . . . . . 8 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( x .- E ) = ( A .- B ) )
51	1, 16, 3, 5, 13, 15, 7, 9, 50	tgcgrcomlr 25375	. . . . . . 7 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( E .- x ) = ( B .- A ) )
52	39	eqcomd 2628	. . . . . . 7 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( E .- D ) = ( B .- X ) )
53	1, 16, 3, 40, 5, 15, 13, 20, 9, 9, 7, 11, 44, 49, 51, 52	tgcgrsub2 25490	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( x .- D ) = ( A .- X ) )
54	53	eqcomd 2628	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( A .- X ) = ( x .- D ) )
55	1, 16, 3, 5, 7, 11, 13, 20, 54	tgcgrcomlr 25375	. . . 4 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( X .- A ) = ( D .- x ) )
56	1, 16, 12, 5, 7, 9, 11, 13, 15, 20, 37, 39, 55	trgcgr 25411	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> <" A B X "> (cgrG ` G ) <" x E D "> )
57		cgracgr.2	. . . . . . . . 9 |- ( ph -> Y ( K ` B ) C )
58	1, 3, 23, 17, 24, 8, 4, 2, 57	hlln 25502	. . . . . . . 8 |- ( ph -> Y e. ( C (LineG ` G ) B ) )
59	58	orcd 407	. . . . . . 7 |- ( ph -> ( Y e. ( C (LineG ` G ) B ) \/ C = B ) )
60	1, 2, 3, 4, 24, 8, 17, 59	colrot1 25454	. . . . . 6 |- ( ph -> ( C e. ( B (LineG ` G ) Y ) \/ B = Y ) )
61	60	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( C e. ( B (LineG ` G ) Y ) \/ B = Y ) )
62	1, 16, 3, 12, 5, 7, 9, 34, 13, 15, 35, 36	cgr3simp2 25416	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( B .- C ) = ( E .- y ) )
63	1, 3, 23, 17, 24, 8, 4	ishlg 25497	. . . . . . . . . . 11 |- ( ph -> ( Y ( K ` B ) C <-> ( Y =/= B /\ C =/= B /\ ( Y e. ( B I C ) \/ C e. ( B I Y ) ) ) ) )
64	57, 63	mpbid 222	. . . . . . . . . 10 |- ( ph -> ( Y =/= B /\ C =/= B /\ ( Y e. ( B I C ) \/ C e. ( B I Y ) ) ) )
65	64	simp3d 1075	. . . . . . . . 9 |- ( ph -> ( Y e. ( B I C ) \/ C e. ( B I Y ) ) )
66	65	orcomd 403	. . . . . . . 8 |- ( ph -> ( C e. ( B I Y ) \/ Y e. ( B I C ) ) )
67	66	ad3antrrr 766	. . . . . . 7 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( C e. ( B I Y ) \/ Y e. ( B I C ) ) )
68		simpr3 1069	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> y ( K ` E ) F )
69	1, 3, 23, 35, 22, 15, 5	ishlg 25497	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( y ( K ` E ) F <-> ( y =/= E /\ F =/= E /\ ( y e. ( E I F ) \/ F e. ( E I y ) ) ) ) )
70	68, 69	mpbid 222	. . . . . . . 8 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( y =/= E /\ F =/= E /\ ( y e. ( E I F ) \/ F e. ( E I y ) ) ) )
71	70	simp3d 1075	. . . . . . 7 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( y e. ( E I F ) \/ F e. ( E I y ) ) )
72		cgracgr.4	. . . . . . . 8 |- ( ph -> ( B .- Y ) = ( E .- F ) )
73	72	ad3antrrr 766	. . . . . . 7 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( B .- Y ) = ( E .- F ) )
74	1, 16, 3, 40, 5, 9, 34, 18, 15, 15, 35, 22, 67, 71, 62, 73	tgcgrsub2 25490	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( C .- Y ) = ( y .- F ) )
75	1, 16, 3, 5, 9, 18, 15, 22, 73	tgcgrcomlr 25375	. . . . . 6 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( Y .- B ) = ( F .- E ) )
76	1, 16, 12, 5, 9, 34, 18, 15, 35, 22, 62, 74, 75	trgcgr 25411	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> <" B C Y "> (cgrG ` G ) <" E y F "> )
77	51	eqcomd 2628	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( B .- A ) = ( E .- x ) )
78	1, 16, 3, 12, 5, 7, 9, 34, 13, 15, 35, 36	cgr3simp3 25417	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( C .- A ) = ( y .- x ) )
79	1, 3, 23, 4, 6, 8, 24, 19, 14, 21, 25	cgrane2 25705	. . . . . 6 |- ( ph -> B =/= C )
80	79	ad3antrrr 766	. . . . 5 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> B =/= C )
81	1, 2, 3, 5, 9, 34, 18, 12, 15, 35, 16, 7, 22, 13, 61, 76, 77, 78, 80	tgfscgr 25463	. . . 4 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( Y .- A ) = ( F .- x ) )
82	1, 16, 3, 5, 18, 7, 22, 13, 81	tgcgrcomlr 25375	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( A .- Y ) = ( x .- F ) )
83	26	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> A =/= B )
84	1, 2, 3, 5, 7, 9, 11, 12, 13, 15, 16, 18, 20, 22, 33, 56, 82, 73, 83	tgfscgr 25463	. 2 |- ( ( ( ( ph /\ x e. P ) /\ y e. P ) /\ ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) -> ( X .- Y ) = ( D .- F ) )
85	1, 3, 23, 4, 6, 8, 24, 19, 14, 21	iscgra 25701	. . 3 |- ( ph -> ( <" A B C "> (cgrA ` G ) <" D E F "> <-> E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) ) )
86	25, 85	mpbid 222	. 2 |- ( ph -> E. x e. P E. y e. P ( <" A B C "> (cgrG ` G ) <" x E y "> /\ x ( K ` E ) D /\ y ( K ` E ) F ) )
87	84, 86	r19.29vva 3081	1 |- ( ph -> ( X .- Y ) = ( D .- F ) )

Syntax hints:   -> 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   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  cgrGccgrg 25405  ≤Gcleg 25477  hlGchlg 25495  cgrAccgra 25699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-cgrg 25406  df-leg 25478  df-hlg 25496  df-cgra 25700

This theorem is referenced by:  cgracom  25714  cgratr  25715  dfcgra2  25721  tgsas1  25735  tgasa1  25739