Theorem iscgrgd 25408

Description: The property for two sequences A and B of points to be congruent. (Contributed by Thierry Arnoux, 3-Apr-2019.)

Hypotheses

Ref	Expression
iscgrg.p	|- P = ( Base ` G )
iscgrg.m	|- .- = ( dist ` G )
iscgrg.e	|- .~ = (cgrG ` G )
iscgrgd.g	|- ( ph -> G e. V )
iscgrgd.d	|- ( ph -> D C_ RR )
iscgrgd.a	|- ( ph -> A : D --> P )
iscgrgd.b	|- ( ph -> B : D --> P )

Assertion

Ref	Expression
iscgrgd	|- ( ph -> ( A .~ 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:   ph( i, j)   D( i, j)   P( i, j)   .~ ( i, j)   .- ( i, j)   V( i, j)

Proof of Theorem iscgrgd

Step	Hyp	Ref	Expression
1		iscgrgd.a	. . . . 5 |- ( ph -> A : D --> P )
2		iscgrgd.d	. . . . 5 |- ( ph -> D C_ RR )
3		iscgrg.p	. . . . . . 7 |- P = ( Base ` G )
4		fvex 6201	. . . . . . 7 |- ( Base ` G ) e. _V
5	3, 4	eqeltri 2697	. . . . . 6 |- P e. _V
6		reex 10027	. . . . . 6 |- RR e. _V
7		elpm2r 7875	. . . . . 6 |- ( ( ( P e. _V /\ RR e. _V ) /\ ( A : D --> P /\ D C_ RR ) ) -> A e. ( P ^pm RR ) )
8	5, 6, 7	mpanl12 718	. . . . 5 |- ( ( A : D --> P /\ D C_ RR ) -> A e. ( P ^pm RR ) )
9	1, 2, 8	syl2anc 693	. . . 4 |- ( ph -> A e. ( P ^pm RR ) )
10		iscgrgd.b	. . . . 5 |- ( ph -> B : D --> P )
11		elpm2r 7875	. . . . . 6 |- ( ( ( P e. _V /\ RR e. _V ) /\ ( B : D --> P /\ D C_ RR ) ) -> B e. ( P ^pm RR ) )
12	5, 6, 11	mpanl12 718	. . . . 5 |- ( ( B : D --> P /\ D C_ RR ) -> B e. ( P ^pm RR ) )
13	10, 2, 12	syl2anc 693	. . . 4 |- ( ph -> B e. ( P ^pm RR ) )
14	9, 13	jca 554	. . 3 |- ( ph -> ( A e. ( P ^pm RR ) /\ B e. ( P ^pm RR ) ) )
15	14	biantrurd 529	. 2 |- ( ph -> ( ( 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 ) ) /\ ( dom A = dom B /\ A. i e. dom A A. j e. dom A ( ( A ` i ) .- ( A ` j ) ) = ( ( B ` i ) .- ( B ` j ) ) ) ) ) )
16		fdm 6051	. . . . 5 |- ( A : D --> P -> dom A = D )
17	1, 16	syl 17	. . . 4 |- ( ph -> dom A = D )
18		fdm 6051	. . . . 5 |- ( B : D --> P -> dom B = D )
19	10, 18	syl 17	. . . 4 |- ( ph -> dom B = D )
20	17, 19	eqtr4d 2659	. . 3 |- ( ph -> dom A = dom B )
21	20	biantrurd 529	. 2 |- ( ph -> ( 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 ) ) ) ) )
22		iscgrgd.g	. . 3 |- ( ph -> G e. V )
23		iscgrg.m	. . . 4 |- .- = ( dist ` G )
24		iscgrg.e	. . . 4 |- .~ = (cgrG ` G )
25	3, 23, 24	iscgrg 25407	. . 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 ) ) ) ) ) )
26	22, 25	syl 17	. 2 |- ( ph -> ( 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 ) ) ) ) ) )
27	15, 21, 26	3bitr4rd 301	1 |- ( ph -> ( A .~ 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   C_ wss 3574   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` 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  ax-cnex 9992  ax-resscn 9993

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-ne 2795  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-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-pm 7860  df-cgrg 25406

This theorem is referenced by:  iscgrglt  25409  trgcgrg  25410  motcgrg  25439