Theorem ecgrtg 25863

Description: The congruence relation used in the Tarski structure for the Euclidean geometry is the same as Cgr. (Contributed by Thierry Arnoux, 15-Mar-2019.)

Hypotheses

Ref	Expression
ecgrtg.1	|- ( ph -> N e. NN )
ecgrtg.2	|- P = ( Base ` (EEG ` N ) )
ecgrtg.3	|- .- = ( dist ` (EEG ` N ) )
ecgrtg.a	|- ( ph -> A e. P )
ecgrtg.b	|- ( ph -> B e. P )
ecgrtg.c	|- ( ph -> C e. P )
ecgrtg.d	|- ( ph -> D e. P )

Assertion

Ref	Expression
ecgrtg	|- ( ph -> ( <. A , B >.Cgr <. C , D >. <-> ( A .- B ) = ( C .- D ) ) )

Proof of Theorem ecgrtg

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

Step	Hyp	Ref	Expression
1		ecgrtg.a	. . . 4 |- ( ph -> A e. P )
2		ecgrtg.1	. . . . . 6 |- ( ph -> N e. NN )
3		eengbas 25861	. . . . . 6 |- ( N e. NN -> ( EE ` N ) = ( Base ` (EEG ` N ) ) )
4	2, 3	syl 17	. . . . 5 |- ( ph -> ( EE ` N ) = ( Base ` (EEG ` N ) ) )
5		ecgrtg.2	. . . . 5 |- P = ( Base ` (EEG ` N ) )
6	4, 5	syl6eqr 2674	. . . 4 |- ( ph -> ( EE ` N ) = P )
7	1, 6	eleqtrrd 2704	. . 3 |- ( ph -> A e. ( EE ` N ) )
8		ecgrtg.b	. . . 4 |- ( ph -> B e. P )
9	8, 6	eleqtrrd 2704	. . 3 |- ( ph -> B e. ( EE ` N ) )
10		ecgrtg.c	. . . 4 |- ( ph -> C e. P )
11	10, 6	eleqtrrd 2704	. . 3 |- ( ph -> C e. ( EE ` N ) )
12		ecgrtg.d	. . . 4 |- ( ph -> D e. P )
13	12, 6	eleqtrrd 2704	. . 3 |- ( ph -> D e. ( EE ` N ) )
14		brcgr 25780	. . 3 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. A , B >.Cgr <. C , D >. <-> sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) ) )
15	7, 9, 11, 13, 14	syl22anc 1327	. 2 |- ( ph -> ( <. A , B >.Cgr <. C , D >. <-> sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) ) )
16		dsid 16063	. . . . . . 7 |- dist = Slot ( dist ` ndx )
17		fvexd 6203	. . . . . . 7 |- ( ph -> (EEG ` N ) e. _V )
18		eengstr 25860	. . . . . . . . . 10 |- ( N e. NN -> (EEG ` N ) Struct <. 1 , ; 1 7 >. )
19	2, 18	syl 17	. . . . . . . . 9 |- ( ph -> (EEG ` N ) Struct <. 1 , ; 1 7 >. )
20		isstruct 15870	. . . . . . . . . 10 |- ( (EEG ` N ) Struct <. 1 , ; 1 7 >. <-> ( ( 1 e. NN /\ ; 1 7 e. NN /\ 1 <_ ; 1 7 ) /\ Fun ( (EEG ` N ) \ { (/) } ) /\ dom (EEG ` N ) C_ ( 1 ...; 1 7 ) ) )
21	20	simp2bi 1077	. . . . . . . . 9 |- ( (EEG ` N ) Struct <. 1 , ; 1 7 >. -> Fun ( (EEG ` N ) \ { (/) } ) )
22	19, 21	syl 17	. . . . . . . 8 |- ( ph -> Fun ( (EEG ` N ) \ { (/) } ) )
23		structcnvcnv 15871	. . . . . . . . . 10 |- ( (EEG ` N ) Struct <. 1 , ; 1 7 >. -> `' `' (EEG ` N ) = ( (EEG ` N ) \ { (/) } ) )
24	19, 23	syl 17	. . . . . . . . 9 |- ( ph -> `' `' (EEG ` N ) = ( (EEG ` N ) \ { (/) } ) )
25	24	funeqd 5910	. . . . . . . 8 |- ( ph -> ( Fun `' `' (EEG ` N ) <-> Fun ( (EEG ` N ) \ { (/) } ) ) )
26	22, 25	mpbird 247	. . . . . . 7 |- ( ph -> Fun `' `' (EEG ` N ) )
27		opex 4932	. . . . . . . . . 10 |- <. ( dist ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) >. e. _V
28	27	prid2 4298	. . . . . . . . 9 |- <. ( dist ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) >. e. { <. ( Base ` ndx ) , ( EE ` N ) >. , <. ( dist ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) >. }
29		elun1 3780	. . . . . . . . 9 |- ( <. ( dist ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) >. e. { <. ( Base ` ndx ) , ( EE ` N ) >. , <. ( dist ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) >. } -> <. ( dist ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) >. e. ( { <. ( Base ` ndx ) , ( EE ` N ) >. , <. ( dist ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) >. } u. { <. (Itv ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> { z e. ( EE ` N ) | z Btwn <. x , y >. } ) >. , <. (LineG ` ndx ) , ( x e. ( EE ` N ) , y e. ( ( EE ` N ) \ { x } ) |-> { z e. ( EE ` N ) | ( z Btwn <. x , y >. \/ x Btwn <. z , y >. \/ y Btwn <. x , z >. ) } ) >. } ) )
30	28, 29	ax-mp 5	. . . . . . . 8 |- <. ( dist ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) >. e. ( { <. ( Base ` ndx ) , ( EE ` N ) >. , <. ( dist ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) >. } u. { <. (Itv ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> { z e. ( EE ` N ) | z Btwn <. x , y >. } ) >. , <. (LineG ` ndx ) , ( x e. ( EE ` N ) , y e. ( ( EE ` N ) \ { x } ) |-> { z e. ( EE ` N ) | ( z Btwn <. x , y >. \/ x Btwn <. z , y >. \/ y Btwn <. x , z >. ) } ) >. } )
31		eengv 25859	. . . . . . . . 9 |- ( N e. NN -> (EEG ` N ) = ( { <. ( Base ` ndx ) , ( EE ` N ) >. , <. ( dist ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) >. } u. { <. (Itv ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> { z e. ( EE ` N ) | z Btwn <. x , y >. } ) >. , <. (LineG ` ndx ) , ( x e. ( EE ` N ) , y e. ( ( EE ` N ) \ { x } ) |-> { z e. ( EE ` N ) | ( z Btwn <. x , y >. \/ x Btwn <. z , y >. \/ y Btwn <. x , z >. ) } ) >. } ) )
32	2, 31	syl 17	. . . . . . . 8 |- ( ph -> (EEG ` N ) = ( { <. ( Base ` ndx ) , ( EE ` N ) >. , <. ( dist ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) >. } u. { <. (Itv ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> { z e. ( EE ` N ) | z Btwn <. x , y >. } ) >. , <. (LineG ` ndx ) , ( x e. ( EE ` N ) , y e. ( ( EE ` N ) \ { x } ) |-> { z e. ( EE ` N ) | ( z Btwn <. x , y >. \/ x Btwn <. z , y >. \/ y Btwn <. x , z >. ) } ) >. } ) )
33	30, 32	syl5eleqr 2708	. . . . . . 7 |- ( ph -> <. ( dist ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) >. e. (EEG ` N ) )
34		fvex 6201	. . . . . . . . 9 |- ( EE ` N ) e. _V
35	34, 34	mpt2ex 7247	. . . . . . . 8 |- ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) e. _V
36	35	a1i 11	. . . . . . 7 |- ( ph -> ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) e. _V )
37	16, 17, 26, 33, 36	strfv2d 15905	. . . . . 6 |- ( ph -> ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) = ( dist ` (EEG ` N ) ) )
38		ecgrtg.3	. . . . . 6 |- .- = ( dist ` (EEG ` N ) )
39	37, 38	syl6reqr 2675	. . . . 5 |- ( ph -> .- = ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) )
40		simplrl 800	. . . . . . . . 9 |- ( ( ( ph /\ ( x = A /\ y = B ) ) /\ i e. ( 1 ... N ) ) -> x = A )
41	40	fveq1d 6193	. . . . . . . 8 |- ( ( ( ph /\ ( x = A /\ y = B ) ) /\ i e. ( 1 ... N ) ) -> ( x ` i ) = ( A ` i ) )
42		simplrr 801	. . . . . . . . 9 |- ( ( ( ph /\ ( x = A /\ y = B ) ) /\ i e. ( 1 ... N ) ) -> y = B )
43	42	fveq1d 6193	. . . . . . . 8 |- ( ( ( ph /\ ( x = A /\ y = B ) ) /\ i e. ( 1 ... N ) ) -> ( y ` i ) = ( B ` i ) )
44	41, 43	oveq12d 6668	. . . . . . 7 |- ( ( ( ph /\ ( x = A /\ y = B ) ) /\ i e. ( 1 ... N ) ) -> ( ( x ` i ) - ( y ` i ) ) = ( ( A ` i ) - ( B ` i ) ) )
45	44	oveq1d 6665	. . . . . 6 |- ( ( ( ph /\ ( x = A /\ y = B ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) = ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) )
46	45	sumeq2dv 14433	. . . . 5 |- ( ( ph /\ ( x = A /\ y = B ) ) -> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) )
47		sumex 14418	. . . . . 6 |- sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) e. _V
48	47	a1i 11	. . . . 5 |- ( ph -> sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) e. _V )
49	39, 46, 7, 9, 48	ovmpt2d 6788	. . . 4 |- ( ph -> ( A .- B ) = sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) )
50	49	eqcomd 2628	. . 3 |- ( ph -> sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) = ( A .- B ) )
51		simplrl 800	. . . . . . . . 9 |- ( ( ( ph /\ ( x = C /\ y = D ) ) /\ i e. ( 1 ... N ) ) -> x = C )
52	51	fveq1d 6193	. . . . . . . 8 |- ( ( ( ph /\ ( x = C /\ y = D ) ) /\ i e. ( 1 ... N ) ) -> ( x ` i ) = ( C ` i ) )
53		simplrr 801	. . . . . . . . 9 |- ( ( ( ph /\ ( x = C /\ y = D ) ) /\ i e. ( 1 ... N ) ) -> y = D )
54	53	fveq1d 6193	. . . . . . . 8 |- ( ( ( ph /\ ( x = C /\ y = D ) ) /\ i e. ( 1 ... N ) ) -> ( y ` i ) = ( D ` i ) )
55	52, 54	oveq12d 6668	. . . . . . 7 |- ( ( ( ph /\ ( x = C /\ y = D ) ) /\ i e. ( 1 ... N ) ) -> ( ( x ` i ) - ( y ` i ) ) = ( ( C ` i ) - ( D ` i ) ) )
56	55	oveq1d 6665	. . . . . 6 |- ( ( ( ph /\ ( x = C /\ y = D ) ) /\ i e. ( 1 ... N ) ) -> ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) = ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) )
57	56	sumeq2dv 14433	. . . . 5 |- ( ( ph /\ ( x = C /\ y = D ) ) -> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) )
58		sumex 14418	. . . . . 6 |- sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) e. _V
59	58	a1i 11	. . . . 5 |- ( ph -> sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) e. _V )
60	39, 57, 11, 13, 59	ovmpt2d 6788	. . . 4 |- ( ph -> ( C .- D ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) )
61	60	eqcomd 2628	. . 3 |- ( ph -> sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) = ( C .- D ) )
62	50, 61	eqeq12d 2637	. 2 |- ( ph -> ( sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) <-> ( A .- B ) = ( C .- D ) ) )
63	15, 62	bitrd 268	1 |- ( ph -> ( <. A , B >.Cgr <. C , D >. <-> ( A .- B ) = ( C .- D ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   <.cop 4183   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1c1 9937   <_ cle 10075   - cmin 10266   NNcn 11020   2c2 11070   7c7 11075  ;cdc 11493   ...cfz 12326   ^cexp 12860   sum_csu 14416   Struct cstr 15853   ndxcnx 15854   Basecbs 15857   distcds 15950  Itvcitv 25335  LineGclng 25336   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770  EEGceeng 25857

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-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-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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-seq 12802  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-ds 15964  df-itv 25337  df-lng 25338  df-ee 25771  df-cgr 25773  df-eeng 25858

This theorem is referenced by:  eengtrkg  25865