Theorem acopy 25724

Description: Angle construction. Theorem 11.15 of [Schwabhauser] p. 98. This is Hilbert's axiom III.4 for geometry. (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 ) )

Assertion

Ref	Expression
acopy	|- ( ph -> E. f e. P ( <" A B C "> (cgrA ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) )

Distinct variable groups:   .- , f   A, f   B, f   C, f   D, f   f, E   f, F   f, G   f, I   P, f   f, L   ph, f

Proof of Theorem acopy

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfcgra2.p	. . . 4 |- P = ( Base ` G )
2		dfcgra2.m	. . . 4 |- .- = ( dist ` G )
3		dfcgra2.i	. . . 4 |- I = (Itv ` G )
4		acopy.l	. . . 4 |- L = (LineG ` G )
5		eqid 2622	. . . 4 |- (hlG ` G ) = (hlG ` G )
6		dfcgra2.g	. . . . 5 |- ( ph -> G e. TarskiG )
7	6	ad2antrr 762	. . . 4 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> G e. TarskiG )
8		dfcgra2.a	. . . . 5 |- ( ph -> A e. P )
9	8	ad2antrr 762	. . . 4 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> A e. P )
10		dfcgra2.b	. . . . 5 |- ( ph -> B e. P )
11	10	ad2antrr 762	. . . 4 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> B e. P )
12		dfcgra2.c	. . . . 5 |- ( ph -> C e. P )
13	12	ad2antrr 762	. . . 4 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> C e. P )
14		simplr 792	. . . 4 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> d e. P )
15		dfcgra2.e	. . . . 5 |- ( ph -> E e. P )
16	15	ad2antrr 762	. . . 4 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> E e. P )
17		dfcgra2.f	. . . . 5 |- ( ph -> F e. P )
18	17	ad2antrr 762	. . . 4 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> F e. P )
19		acopy.1	. . . . 5 |- ( ph -> -. ( A e. ( B L C ) \/ B = C ) )
20	19	ad2antrr 762	. . . 4 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> -. ( A e. ( B L C ) \/ B = C ) )
21		dfcgra2.d	. . . . . 6 |- ( ph -> D e. P )
22	21	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> D e. P )
23		acopy.2	. . . . . 6 |- ( ph -> -. ( D e. ( E L F ) \/ E = F ) )
24	23	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> -. ( D e. ( E L F ) \/ E = F ) )
25		simprl 794	. . . . . 6 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> d ( (hlG ` G ) ` E ) D )
26	1, 3, 5, 14, 22, 16, 7, 4, 25	hlln 25502	. . . . 5 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> d e. ( D L E ) )
27	1, 3, 5, 14, 22, 16, 7, 25	hlne1 25500	. . . . 5 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> d =/= E )
28	1, 3, 4, 7, 22, 16, 18, 14, 24, 26, 27	ncolncol 25541	. . . 4 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> -. ( d e. ( E L F ) \/ E = F ) )
29		simprr 796	. . . . . 6 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> ( E .- d ) = ( B .- A ) )
30	29	eqcomd 2628	. . . . 5 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> ( B .- A ) = ( E .- d ) )
31	1, 2, 3, 7, 11, 9, 16, 14, 30	tgcgrcomlr 25375	. . . 4 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> ( A .- B ) = ( d .- E ) )
32	1, 2, 3, 4, 5, 7, 9, 11, 13, 14, 16, 18, 20, 28, 31	trgcopy 25696	. . 3 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> E. f e. P ( <" A B C "> (cgrG ` G ) <" d E f "> /\ f ( (hpG ` G ) ` ( d L E ) ) F ) )
33	7	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ <" A B C "> (cgrG ` G ) <" d E f "> ) -> G e. TarskiG )
34	9	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ <" A B C "> (cgrG ` G ) <" d E f "> ) -> A e. P )
35	11	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ <" A B C "> (cgrG ` G ) <" d E f "> ) -> B e. P )
36	13	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ <" A B C "> (cgrG ` G ) <" d E f "> ) -> C e. P )
37	14	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ <" A B C "> (cgrG ` G ) <" d E f "> ) -> d e. P )
38	16	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ <" A B C "> (cgrG ` G ) <" d E f "> ) -> E e. P )
39		simplr 792	. . . . . . 7 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ <" A B C "> (cgrG ` G ) <" d E f "> ) -> f e. P )
40	1, 3, 4, 6, 8, 10, 12, 19	ncolne1 25520	. . . . . . . . 9 |- ( ph -> A =/= B )
41	40	ad4antr 768	. . . . . . . 8 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ <" A B C "> (cgrG ` G ) <" d E f "> ) -> A =/= B )
42	1, 4, 3, 6, 10, 12, 8, 19	ncolrot1 25457	. . . . . . . . . 10 |- ( ph -> -. ( B e. ( C L A ) \/ C = A ) )
43	1, 3, 4, 6, 10, 12, 8, 42	ncolne1 25520	. . . . . . . . 9 |- ( ph -> B =/= C )
44	43	ad4antr 768	. . . . . . . 8 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ <" A B C "> (cgrG ` G ) <" d E f "> ) -> B =/= C )
45		simpr 477	. . . . . . . 8 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ <" A B C "> (cgrG ` G ) <" d E f "> ) -> <" A B C "> (cgrG ` G ) <" d E f "> )
46	1, 3, 33, 5, 34, 35, 36, 37, 38, 39, 41, 44, 45	cgrcgra 25713	. . . . . . 7 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ <" A B C "> (cgrG ` G ) <" d E f "> ) -> <" A B C "> (cgrA ` G ) <" d E f "> )
47	22	ad2antrr 762	. . . . . . 7 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ <" A B C "> (cgrG ` G ) <" d E f "> ) -> D e. P )
48	25	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ <" A B C "> (cgrG ` G ) <" d E f "> ) -> d ( (hlG ` G ) ` E ) D )
49	1, 3, 5, 37, 47, 38, 33, 48	hlcomd 25499	. . . . . . 7 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ <" A B C "> (cgrG ` G ) <" d E f "> ) -> D ( (hlG ` G ) ` E ) d )
50	1, 3, 5, 33, 34, 35, 36, 37, 38, 39, 46, 47, 49	cgrahl1 25708	. . . . . 6 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ <" A B C "> (cgrG ` G ) <" d E f "> ) -> <" A B C "> (cgrA ` G ) <" D E f "> )
51	50	ex 450	. . . . 5 |- ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) -> ( <" A B C "> (cgrG ` G ) <" d E f "> -> <" A B C "> (cgrA ` G ) <" D E f "> ) )
52		simpr 477	. . . . . . 7 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ f ( (hpG ` G ) ` ( d L E ) ) F ) -> f ( (hpG ` G ) ` ( d L E ) ) F )
53	7	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ f ( (hpG ` G ) ` ( d L E ) ) F ) -> G e. TarskiG )
54	14	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ f ( (hpG ` G ) ` ( d L E ) ) F ) -> d e. P )
55	16	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ f ( (hpG ` G ) ` ( d L E ) ) F ) -> E e. P )
56	27	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ f ( (hpG ` G ) ` ( d L E ) ) F ) -> d =/= E )
57	1, 3, 4, 6, 21, 15, 17, 23	ncolne1 25520	. . . . . . . . . . . 12 |- ( ph -> D =/= E )
58	1, 3, 4, 6, 21, 15, 57	tgelrnln 25525	. . . . . . . . . . 11 |- ( ph -> ( D L E ) e. ran L )
59	58	ad4antr 768	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ f ( (hpG ` G ) ` ( d L E ) ) F ) -> ( D L E ) e. ran L )
60	26	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ f ( (hpG ` G ) ` ( d L E ) ) F ) -> d e. ( D L E ) )
61	1, 3, 4, 6, 21, 15, 57	tglinerflx2 25529	. . . . . . . . . . 11 |- ( ph -> E e. ( D L E ) )
62	61	ad4antr 768	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ f ( (hpG ` G ) ` ( d L E ) ) F ) -> E e. ( D L E ) )
63	1, 3, 4, 53, 54, 55, 56, 56, 59, 60, 62	tglinethru 25531	. . . . . . . . 9 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ f ( (hpG ` G ) ` ( d L E ) ) F ) -> ( D L E ) = ( d L E ) )
64	63	fveq2d 6195	. . . . . . . 8 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ f ( (hpG ` G ) ` ( d L E ) ) F ) -> ( (hpG ` G ) ` ( D L E ) ) = ( (hpG ` G ) ` ( d L E ) ) )
65	64	breqd 4664	. . . . . . 7 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ f ( (hpG ` G ) ` ( d L E ) ) F ) -> ( f ( (hpG ` G ) ` ( D L E ) ) F <-> f ( (hpG ` G ) ` ( d L E ) ) F ) )
66	52, 65	mpbird 247	. . . . . 6 |- ( ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) /\ f ( (hpG ` G ) ` ( d L E ) ) F ) -> f ( (hpG ` G ) ` ( D L E ) ) F )
67	66	ex 450	. . . . 5 |- ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) -> ( f ( (hpG ` G ) ` ( d L E ) ) F -> f ( (hpG ` G ) ` ( D L E ) ) F ) )
68	51, 67	anim12d 586	. . . 4 |- ( ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) /\ f e. P ) -> ( ( <" A B C "> (cgrG ` G ) <" d E f "> /\ f ( (hpG ` G ) ` ( d L E ) ) F ) -> ( <" A B C "> (cgrA ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) )
69	68	reximdva 3017	. . 3 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> ( E. f e. P ( <" A B C "> (cgrG ` G ) <" d E f "> /\ f ( (hpG ` G ) ` ( d L E ) ) F ) -> E. f e. P ( <" A B C "> (cgrA ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) ) )
70	32, 69	mpd 15	. 2 |- ( ( ( ph /\ d e. P ) /\ ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) ) -> E. f e. P ( <" A B C "> (cgrA ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) )
71	40	necomd 2849	. . 3 |- ( ph -> B =/= A )
72	1, 3, 5, 15, 10, 8, 6, 21, 2, 57, 71	hlcgrex 25511	. 2 |- ( ph -> E. d e. P ( d ( (hlG ` G ) ` E ) D /\ ( E .- d ) = ( B .- A ) ) )
73	70, 72	r19.29a 3078	1 |- ( ph -> E. f e. P ( <" A B C "> (cgrA ` G ) <" D E f "> /\ f ( (hpG ` G ) ` ( D L E ) ) F ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   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-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-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-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  df-cgra 25700

This theorem is referenced by: (None)