Theorem iscgrg 25407

Description: The congruence property for sequences of points. (Contributed by Thierry Arnoux, 3-Apr-2019.)

Hypotheses

Ref	Expression
iscgrg.p	|- P = ( Base ` G )
iscgrg.m	|- .- = ( dist ` G )
iscgrg.e	|- .~ = (cgrG ` G )

Assertion

Ref	Expression
iscgrg	|- ( G e. V -> ( A .~ B <-> ( ( A e. ( P ^pm RR ) /\ B e. ( P ^pm RR ) ) /\ ( dom A = dom B /\ A. i e. dom A A. j e. dom A ( ( A ` i ) .- ( A ` j ) ) = ( ( B ` i ) .- ( B ` j ) ) ) ) ) )

Distinct variable groups:   i, j, G   A, i, j   B, i, j

Allowed substitution hints:   P( i, j)   .~ ( i, j)   .- ( i, j)   V( i, j)

Proof of Theorem iscgrg

Dummy variables a b g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscgrg.e	. . . 4 |- .~ = (cgrG ` G )
2		elex 3212	. . . . 5 |- ( G e. V -> G e. _V )
3		fveq2 6191	. . . . . . . . . . . 12 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
4		iscgrg.p	. . . . . . . . . . . 12 |- P = ( Base ` G )
5	3, 4	syl6eqr 2674	. . . . . . . . . . 11 |- ( g = G -> ( Base ` g ) = P )
6	5	oveq1d 6665	. . . . . . . . . 10 |- ( g = G -> ( ( Base ` g ) ^pm RR ) = ( P ^pm RR ) )
7	6	eleq2d 2687	. . . . . . . . 9 |- ( g = G -> ( a e. ( ( Base ` g ) ^pm RR ) <-> a e. ( P ^pm RR ) ) )
8	6	eleq2d 2687	. . . . . . . . 9 |- ( g = G -> ( b e. ( ( Base ` g ) ^pm RR ) <-> b e. ( P ^pm RR ) ) )
9	7, 8	anbi12d 747	. . . . . . . 8 |- ( g = G -> ( ( a e. ( ( Base ` g ) ^pm RR ) /\ b e. ( ( Base ` g ) ^pm RR ) ) <-> ( a e. ( P ^pm RR ) /\ b e. ( P ^pm RR ) ) ) )
10		fveq2 6191	. . . . . . . . . . . . 13 |- ( g = G -> ( dist ` g ) = ( dist ` G ) )
11		iscgrg.m	. . . . . . . . . . . . 13 |- .- = ( dist ` G )
12	10, 11	syl6eqr 2674	. . . . . . . . . . . 12 |- ( g = G -> ( dist ` g ) = .- )
13	12	oveqd 6667	. . . . . . . . . . 11 |- ( g = G -> ( ( a ` i ) ( dist ` g ) ( a ` j ) ) = ( ( a ` i ) .- ( a ` j ) ) )
14	12	oveqd 6667	. . . . . . . . . . 11 |- ( g = G -> ( ( b ` i ) ( dist ` g ) ( b ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) )
15	13, 14	eqeq12d 2637	. . . . . . . . . 10 |- ( g = G -> ( ( ( a ` i ) ( dist ` g ) ( a ` j ) ) = ( ( b ` i ) ( dist ` g ) ( b ` j ) ) <-> ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) )
16	15	2ralbidv 2989	. . . . . . . . 9 |- ( g = G -> ( A. i e. dom a A. j e. dom a ( ( a ` i ) ( dist ` g ) ( a ` j ) ) = ( ( b ` i ) ( dist ` g ) ( b ` j ) ) <-> A. i e. dom a A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) )
17	16	anbi2d 740	. . . . . . . 8 |- ( g = G -> ( ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) ( dist ` g ) ( a ` j ) ) = ( ( b ` i ) ( dist ` g ) ( b ` j ) ) ) <-> ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) ) )
18	9, 17	anbi12d 747	. . . . . . 7 |- ( g = G -> ( ( ( a e. ( ( Base ` g ) ^pm RR ) /\ b e. ( ( Base ` g ) ^pm RR ) ) /\ ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) ( dist ` g ) ( a ` j ) ) = ( ( b ` i ) ( dist ` g ) ( b ` j ) ) ) ) <-> ( ( a e. ( P ^pm RR ) /\ b e. ( P ^pm RR ) ) /\ ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) ) ) )
19	18	opabbidv 4716	. . . . . 6 |- ( g = G -> { <. a , b >. | ( ( a e. ( ( Base ` g ) ^pm RR ) /\ b e. ( ( Base ` g ) ^pm RR ) ) /\ ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) ( dist ` g ) ( a ` j ) ) = ( ( b ` i ) ( dist ` g ) ( b ` j ) ) ) ) } = { <. a , b >. | ( ( a e. ( P ^pm RR ) /\ b e. ( P ^pm RR ) ) /\ ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) ) } )
20		df-cgrg 25406	. . . . . 6 |- cgrG = ( g e. _V |-> { <. a , b >. | ( ( a e. ( ( Base ` g ) ^pm RR ) /\ b e. ( ( Base ` g ) ^pm RR ) ) /\ ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) ( dist ` g ) ( a ` j ) ) = ( ( b ` i ) ( dist ` g ) ( b ` j ) ) ) ) } )
21		df-xp 5120	. . . . . . . 8 |- ( ( P ^pm RR ) X. ( P ^pm RR ) ) = { <. a , b >. | ( a e. ( P ^pm RR ) /\ b e. ( P ^pm RR ) ) }
22		ovex 6678	. . . . . . . . 9 |- ( P ^pm RR ) e. _V
23	22, 22	xpex 6962	. . . . . . . 8 |- ( ( P ^pm RR ) X. ( P ^pm RR ) ) e. _V
24	21, 23	eqeltrri 2698	. . . . . . 7 |- { <. a , b >. | ( a e. ( P ^pm RR ) /\ b e. ( P ^pm RR ) ) } e. _V
25		simpl 473	. . . . . . . 8 |- ( ( ( a e. ( P ^pm RR ) /\ b e. ( P ^pm RR ) ) /\ ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) ) -> ( a e. ( P ^pm RR ) /\ b e. ( P ^pm RR ) ) )
26	25	ssopab2i 5003	. . . . . . 7 |- { <. a , b >. | ( ( a e. ( P ^pm RR ) /\ b e. ( P ^pm RR ) ) /\ ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) ) } C_ { <. a , b >. | ( a e. ( P ^pm RR ) /\ b e. ( P ^pm RR ) ) }
27	24, 26	ssexi 4803	. . . . . 6 |- { <. a , b >. | ( ( a e. ( P ^pm RR ) /\ b e. ( P ^pm RR ) ) /\ ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) ) } e. _V
28	19, 20, 27	fvmpt 6282	. . . . 5 |- ( G e. _V -> (cgrG ` G ) = { <. a , b >. | ( ( a e. ( P ^pm RR ) /\ b e. ( P ^pm RR ) ) /\ ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) ) } )
29	2, 28	syl 17	. . . 4 |- ( G e. V -> (cgrG ` G ) = { <. a , b >. | ( ( a e. ( P ^pm RR ) /\ b e. ( P ^pm RR ) ) /\ ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) ) } )
30	1, 29	syl5eq 2668	. . 3 |- ( G e. V -> .~ = { <. a , b >. | ( ( a e. ( P ^pm RR ) /\ b e. ( P ^pm RR ) ) /\ ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) ) } )
31	30	breqd 4664	. 2 |- ( G e. V -> ( A .~ B <-> A { <. a , b >. | ( ( a e. ( P ^pm RR ) /\ b e. ( P ^pm RR ) ) /\ ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) ) } B ) )
32		dmeq 5324	. . . . . 6 |- ( a = A -> dom a = dom A )
33	32	eqeq1d 2624	. . . . 5 |- ( a = A -> ( dom a = dom b <-> dom A = dom b ) )
34	32	adantr 481	. . . . . . 7 |- ( ( a = A /\ i e. dom a ) -> dom a = dom A )
35		simpll 790	. . . . . . . . . 10 |- ( ( ( a = A /\ i e. dom a ) /\ j e. dom a ) -> a = A )
36	35	fveq1d 6193	. . . . . . . . 9 |- ( ( ( a = A /\ i e. dom a ) /\ j e. dom a ) -> ( a ` i ) = ( A ` i ) )
37	35	fveq1d 6193	. . . . . . . . 9 |- ( ( ( a = A /\ i e. dom a ) /\ j e. dom a ) -> ( a ` j ) = ( A ` j ) )
38	36, 37	oveq12d 6668	. . . . . . . 8 |- ( ( ( a = A /\ i e. dom a ) /\ j e. dom a ) -> ( ( a ` i ) .- ( a ` j ) ) = ( ( A ` i ) .- ( A ` j ) ) )
39	38	eqeq1d 2624	. . . . . . 7 |- ( ( ( a = A /\ i e. dom a ) /\ j e. dom a ) -> ( ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) <-> ( ( A ` i ) .- ( A ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) )
40	34, 39	raleqbidva 3154	. . . . . 6 |- ( ( a = A /\ i e. dom a ) -> ( A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) <-> A. j e. dom A ( ( A ` i ) .- ( A ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) )
41	32, 40	raleqbidva 3154	. . . . 5 |- ( a = A -> ( A. i e. dom a A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) <-> A. i e. dom A A. j e. dom A ( ( A ` i ) .- ( A ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) )
42	33, 41	anbi12d 747	. . . 4 |- ( a = A -> ( ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) <-> ( dom A = dom b /\ A. i e. dom A A. j e. dom A ( ( A ` i ) .- ( A ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) ) )
43		dmeq 5324	. . . . . 6 |- ( b = B -> dom b = dom B )
44	43	eqeq2d 2632	. . . . 5 |- ( b = B -> ( dom A = dom b <-> dom A = dom B ) )
45		fveq1 6190	. . . . . . . 8 |- ( b = B -> ( b ` i ) = ( B ` i ) )
46		fveq1 6190	. . . . . . . 8 |- ( b = B -> ( b ` j ) = ( B ` j ) )
47	45, 46	oveq12d 6668	. . . . . . 7 |- ( b = B -> ( ( b ` i ) .- ( b ` j ) ) = ( ( B ` i ) .- ( B ` j ) ) )
48	47	eqeq2d 2632	. . . . . 6 |- ( b = B -> ( ( ( A ` i ) .- ( A ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) <-> ( ( A ` i ) .- ( A ` j ) ) = ( ( B ` i ) .- ( B ` j ) ) ) )
49	48	2ralbidv 2989	. . . . 5 |- ( b = B -> ( A. i e. dom A A. j e. dom A ( ( A ` i ) .- ( A ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) <-> A. i e. dom A A. j e. dom A ( ( A ` i ) .- ( A ` j ) ) = ( ( B ` i ) .- ( B ` j ) ) ) )
50	44, 49	anbi12d 747	. . . 4 |- ( b = B -> ( ( dom A = dom b /\ A. i e. dom A A. j e. dom A ( ( A ` i ) .- ( A ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) <-> ( dom A = dom B /\ A. i e. dom A A. j e. dom A ( ( A ` i ) .- ( A ` j ) ) = ( ( B ` i ) .- ( B ` j ) ) ) ) )
51	42, 50	sylan9bb 736	. . 3 |- ( ( a = A /\ b = B ) -> ( ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) <-> ( dom A = dom B /\ A. i e. dom A A. j e. dom A ( ( A ` i ) .- ( A ` j ) ) = ( ( B ` i ) .- ( B ` j ) ) ) ) )
52		eqid 2622	. . 3 |- { <. a , b >. | ( ( a e. ( P ^pm RR ) /\ b e. ( P ^pm RR ) ) /\ ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) ) } = { <. a , b >. | ( ( a e. ( P ^pm RR ) /\ b e. ( P ^pm RR ) ) /\ ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) ) }
53	51, 52	brab2a 5194	. 2 |- ( A { <. a , b >. | ( ( a e. ( P ^pm RR ) /\ b e. ( P ^pm RR ) ) /\ ( dom a = dom b /\ A. i e. dom a A. j e. dom a ( ( a ` i ) .- ( a ` j ) ) = ( ( b ` i ) .- ( b ` j ) ) ) ) } B <-> ( ( A e. ( P ^pm RR ) /\ B e. ( P ^pm RR ) ) /\ ( dom A = dom B /\ A. i e. dom A A. j e. dom A ( ( A ` i ) .- ( A ` j ) ) = ( ( B ` i ) .- ( B ` j ) ) ) ) )
54	31, 53	syl6bb 276	1 |- ( G e. V -> ( A .~ B <-> ( ( A e. ( P ^pm RR ) /\ B e. ( P ^pm RR ) ) /\ ( dom A = dom B /\ A. i e. dom A A. j e. dom A ( ( A ` i ) .- ( A ` j ) ) = ( ( B ` i ) .- ( B ` j ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   class class class wbr 4653   {copab 4712   X. cxp 5112   dom cdm 5114   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   RRcr 9935   Basecbs 15857   distcds 15950  cgrGccgrg 25405

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-cgrg 25406

This theorem is referenced by:  iscgrgd  25408  ercgrg  25412