Theorem iseqlg 25747

Description: Property of a triangle being equilateral. (Contributed by Thierry Arnoux, 5-Oct-2020.)

Hypotheses

Ref	Expression
iseqlg.p	|- P = ( Base ` G )
iseqlg.m	|- .- = ( dist ` G )
iseqlg.i	|- I = (Itv ` G )
iseqlg.l	|- L = (LineG ` G )
iseqlg.g	|- ( ph -> G e. TarskiG )
iseqlg.a	|- ( ph -> A e. P )
iseqlg.b	|- ( ph -> B e. P )
iseqlg.c	|- ( ph -> C e. P )

Assertion

Ref	Expression
iseqlg	|- ( ph -> ( <" A B C "> e. (eqltrG ` G ) <-> <" A B C "> (cgrG ` G ) <" B C A "> ) )

Proof of Theorem iseqlg

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

Step	Hyp	Ref	Expression
1		iseqlg.g	. . . 4 |- ( ph -> G e. TarskiG )
2		elex 3212	. . . 4 |- ( G e. TarskiG -> G e. _V )
3		fveq2 6191	. . . . . . . 8 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		iseqlg.p	. . . . . . . 8 |- P = ( Base ` G )
5	3, 4	syl6eqr 2674	. . . . . . 7 |- ( g = G -> ( Base ` g ) = P )
6	5	oveq1d 6665	. . . . . 6 |- ( g = G -> ( ( Base ` g ) ^m ( 0..^ 3 ) ) = ( P ^m ( 0..^ 3 ) ) )
7		fveq2 6191	. . . . . . 7 |- ( g = G -> (cgrG ` g ) = (cgrG ` G ) )
8	7	breqd 4664	. . . . . 6 |- ( g = G -> ( x (cgrG ` g ) <" ( x ` 1 ) ( x ` 2 ) ( x ` 0 ) "> <-> x (cgrG ` G ) <" ( x ` 1 ) ( x ` 2 ) ( x ` 0 ) "> ) )
9	6, 8	rabeqbidv 3195	. . . . 5 |- ( g = G -> { x e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) | x (cgrG ` g ) <" ( x ` 1 ) ( x ` 2 ) ( x ` 0 ) "> } = { x e. ( P ^m ( 0..^ 3 ) ) | x (cgrG ` G ) <" ( x ` 1 ) ( x ` 2 ) ( x ` 0 ) "> } )
10		df-eqlg 25746	. . . . 5 |- eqltrG = ( g e. _V |-> { x e. ( ( Base ` g ) ^m ( 0..^ 3 ) ) | x (cgrG ` g ) <" ( x ` 1 ) ( x ` 2 ) ( x ` 0 ) "> } )
11		ovex 6678	. . . . . 6 |- ( P ^m ( 0..^ 3 ) ) e. _V
12	11	rabex 4813	. . . . 5 |- { x e. ( P ^m ( 0..^ 3 ) ) | x (cgrG ` G ) <" ( x ` 1 ) ( x ` 2 ) ( x ` 0 ) "> } e. _V
13	9, 10, 12	fvmpt 6282	. . . 4 |- ( G e. _V -> (eqltrG ` G ) = { x e. ( P ^m ( 0..^ 3 ) ) | x (cgrG ` G ) <" ( x ` 1 ) ( x ` 2 ) ( x ` 0 ) "> } )
14	1, 2, 13	3syl 18	. . 3 |- ( ph -> (eqltrG ` G ) = { x e. ( P ^m ( 0..^ 3 ) ) | x (cgrG ` G ) <" ( x ` 1 ) ( x ` 2 ) ( x ` 0 ) "> } )
15	14	eleq2d 2687	. 2 |- ( ph -> ( <" A B C "> e. (eqltrG ` G ) <-> <" A B C "> e. { x e. ( P ^m ( 0..^ 3 ) ) | x (cgrG ` G ) <" ( x ` 1 ) ( x ` 2 ) ( x ` 0 ) "> } ) )
16		id 22	. . . . 5 |- ( x = <" A B C "> -> x = <" A B C "> )
17		fveq1 6190	. . . . . 6 |- ( x = <" A B C "> -> ( x ` 1 ) = ( <" A B C "> ` 1 ) )
18		fveq1 6190	. . . . . 6 |- ( x = <" A B C "> -> ( x ` 2 ) = ( <" A B C "> ` 2 ) )
19		fveq1 6190	. . . . . 6 |- ( x = <" A B C "> -> ( x ` 0 ) = ( <" A B C "> ` 0 ) )
20	17, 18, 19	s3eqd 13609	. . . . 5 |- ( x = <" A B C "> -> <" ( x ` 1 ) ( x ` 2 ) ( x ` 0 ) "> = <" ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) ( <" A B C "> ` 0 ) "> )
21	16, 20	breq12d 4666	. . . 4 |- ( x = <" A B C "> -> ( x (cgrG ` G ) <" ( x ` 1 ) ( x ` 2 ) ( x ` 0 ) "> <-> <" A B C "> (cgrG ` G ) <" ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) ( <" A B C "> ` 0 ) "> ) )
22	21	elrab 3363	. . 3 |- ( <" A B C "> e. { x e. ( P ^m ( 0..^ 3 ) ) | x (cgrG ` G ) <" ( x ` 1 ) ( x ` 2 ) ( x ` 0 ) "> } <-> ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" A B C "> (cgrG ` G ) <" ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) ( <" A B C "> ` 0 ) "> ) )
23	22	a1i 11	. 2 |- ( ph -> ( <" A B C "> e. { x e. ( P ^m ( 0..^ 3 ) ) | x (cgrG ` G ) <" ( x ` 1 ) ( x ` 2 ) ( x ` 0 ) "> } <-> ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" A B C "> (cgrG ` G ) <" ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) ( <" A B C "> ` 0 ) "> ) ) )
24		iseqlg.a	. . . . . . 7 |- ( ph -> A e. P )
25		iseqlg.b	. . . . . . 7 |- ( ph -> B e. P )
26		iseqlg.c	. . . . . . 7 |- ( ph -> C e. P )
27	24, 25, 26	s3cld 13617	. . . . . 6 |- ( ph -> <" A B C "> e. Word P )
28		s3len 13639	. . . . . . 7 |- ( # ` <" A B C "> ) = 3
29	28	a1i 11	. . . . . 6 |- ( ph -> ( # ` <" A B C "> ) = 3 )
30	27, 29	jca 554	. . . . 5 |- ( ph -> ( <" A B C "> e. Word P /\ ( # ` <" A B C "> ) = 3 ) )
31		fvex 6201	. . . . . . 7 |- ( Base ` G ) e. _V
32	4, 31	eqeltri 2697	. . . . . 6 |- P e. _V
33		3nn0 11310	. . . . . 6 |- 3 e. NN0
34		wrdmap 13336	. . . . . 6 |- ( ( P e. _V /\ 3 e. NN0 ) -> ( ( <" A B C "> e. Word P /\ ( # ` <" A B C "> ) = 3 ) <-> <" A B C "> e. ( P ^m ( 0..^ 3 ) ) ) )
35	32, 33, 34	mp2an 708	. . . . 5 |- ( ( <" A B C "> e. Word P /\ ( # ` <" A B C "> ) = 3 ) <-> <" A B C "> e. ( P ^m ( 0..^ 3 ) ) )
36	30, 35	sylib 208	. . . 4 |- ( ph -> <" A B C "> e. ( P ^m ( 0..^ 3 ) ) )
37	36	biantrurd 529	. . 3 |- ( ph -> ( <" A B C "> (cgrG ` G ) <" ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) ( <" A B C "> ` 0 ) "> <-> ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" A B C "> (cgrG ` G ) <" ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) ( <" A B C "> ` 0 ) "> ) ) )
38		s3fv1 13637	. . . . . 6 |- ( B e. P -> ( <" A B C "> ` 1 ) = B )
39	25, 38	syl 17	. . . . 5 |- ( ph -> ( <" A B C "> ` 1 ) = B )
40		s3fv2 13638	. . . . . 6 |- ( C e. P -> ( <" A B C "> ` 2 ) = C )
41	26, 40	syl 17	. . . . 5 |- ( ph -> ( <" A B C "> ` 2 ) = C )
42		s3fv0 13636	. . . . . 6 |- ( A e. P -> ( <" A B C "> ` 0 ) = A )
43	24, 42	syl 17	. . . . 5 |- ( ph -> ( <" A B C "> ` 0 ) = A )
44	39, 41, 43	s3eqd 13609	. . . 4 |- ( ph -> <" ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) ( <" A B C "> ` 0 ) "> = <" B C A "> )
45	44	breq2d 4665	. . 3 |- ( ph -> ( <" A B C "> (cgrG ` G ) <" ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) ( <" A B C "> ` 0 ) "> <-> <" A B C "> (cgrG ` G ) <" B C A "> ) )
46	37, 45	bitr3d 270	. 2 |- ( ph -> ( ( <" A B C "> e. ( P ^m ( 0..^ 3 ) ) /\ <" A B C "> (cgrG ` G ) <" ( <" A B C "> ` 1 ) ( <" A B C "> ` 2 ) ( <" A B C "> ` 0 ) "> ) <-> <" A B C "> (cgrG ` G ) <" B C A "> ) )
47	15, 23, 46	3bitrd 294	1 |- ( ph -> ( <" A B C "> e. (eqltrG ` G ) <-> <" A B C "> (cgrG ` G ) <" B C A "> ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   0cc0 9936   1c1 9937   2c2 11070   3c3 11071   NN0cn0 11292  ..^cfzo 12465   #chash 13117  Word cword 13291   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  cgrGccgrg 25405  eqltrGceqlg 25745

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-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-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  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-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-eqlg 25746

This theorem is referenced by:  iseqlgd  25748