Theorem brcgr 24026

Description: The binary relationship form of the congruence predicate. The statement <. A , B >.Cgr <. C , D >. should be read informally as "the N dimensional point A is as far from B as C is from D, or "the line segment A B is congruent to the line segment C D. This particular definition is encapsulated by Tarski's axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)

Assertion

Ref	Expression
brcgr	|- ( ( ( 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 ) ) )

Distinct variable groups:   i, N   A, i   B, i   C, i   D, i

Proof of Theorem brcgr

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

Step	Hyp	Ref	Expression
1		opex 4717	. . 3 |- <. A , B >. e. _V
2		opex 4717	. . 3 |- <. C , D >. e. _V
3		eleq1 2539	. . . . . 6 |- ( x = <. A , B >. -> ( x e. ( ( EE ` n ) X. ( EE ` n ) ) <-> <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) )
4	3	anbi1d 704	. . . . 5 |- ( x = <. A , B >. -> ( ( x e. ( ( EE ` n ) X. ( EE ` n ) ) /\ y e. ( ( EE ` n ) X. ( EE ` n ) ) ) <-> ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ y e. ( ( EE ` n ) X. ( EE ` n ) ) ) ) )
5		fveq2 5872	. . . . . . . . . 10 |- ( x = <. A , B >. -> ( 1st ` x ) = ( 1st ` <. A , B >. ) )
6	5	fveq1d 5874	. . . . . . . . 9 |- ( x = <. A , B >. -> ( ( 1st ` x ) ` i ) = ( ( 1st ` <. A , B >. ) ` i ) )
7		fveq2 5872	. . . . . . . . . 10 |- ( x = <. A , B >. -> ( 2nd ` x ) = ( 2nd ` <. A , B >. ) )
8	7	fveq1d 5874	. . . . . . . . 9 |- ( x = <. A , B >. -> ( ( 2nd ` x ) ` i ) = ( ( 2nd ` <. A , B >. ) ` i ) )
9	6, 8	oveq12d 6313	. . . . . . . 8 |- ( x = <. A , B >. -> ( ( ( 1st ` x ) ` i ) - ( ( 2nd ` x ) ` i ) ) = ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) )
10	9	oveq1d 6310	. . . . . . 7 |- ( x = <. A , B >. -> ( ( ( ( 1st ` x ) ` i ) - ( ( 2nd ` x ) ` i ) ) ^ 2 ) = ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) )
11	10	sumeq2sdv 13506	. . . . . 6 |- ( x = <. A , B >. -> sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` x ) ` i ) - ( ( 2nd ` x ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) )
12	11	eqeq1d 2469	. . . . 5 |- ( x = <. A , B >. -> ( sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` x ) ` i ) - ( ( 2nd ` x ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` y ) ` i ) - ( ( 2nd ` y ) ` i ) ) ^ 2 ) <-> sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` y ) ` i ) - ( ( 2nd ` y ) ` i ) ) ^ 2 ) ) )
13	4, 12	anbi12d 710	. . . 4 |- ( x = <. A , B >. -> ( ( ( x e. ( ( EE ` n ) X. ( EE ` n ) ) /\ y e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` x ) ` i ) - ( ( 2nd ` x ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` y ) ` i ) - ( ( 2nd ` y ) ` i ) ) ^ 2 ) ) <-> ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ y e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` y ) ` i ) - ( ( 2nd ` y ) ` i ) ) ^ 2 ) ) ) )
14	13	rexbidv 2978	. . 3 |- ( x = <. A , B >. -> ( E. n e. NN ( ( x e. ( ( EE ` n ) X. ( EE ` n ) ) /\ y e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` x ) ` i ) - ( ( 2nd ` x ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` y ) ` i ) - ( ( 2nd ` y ) ` i ) ) ^ 2 ) ) <-> E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ y e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` y ) ` i ) - ( ( 2nd ` y ) ` i ) ) ^ 2 ) ) ) )
15		eleq1 2539	. . . . . 6 |- ( y = <. C , D >. -> ( y e. ( ( EE ` n ) X. ( EE ` n ) ) <-> <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) )
16	15	anbi2d 703	. . . . 5 |- ( y = <. C , D >. -> ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ y e. ( ( EE ` n ) X. ( EE ` n ) ) ) <-> ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) ) )
17		fveq2 5872	. . . . . . . . . 10 |- ( y = <. C , D >. -> ( 1st ` y ) = ( 1st ` <. C , D >. ) )
18	17	fveq1d 5874	. . . . . . . . 9 |- ( y = <. C , D >. -> ( ( 1st ` y ) ` i ) = ( ( 1st ` <. C , D >. ) ` i ) )
19		fveq2 5872	. . . . . . . . . 10 |- ( y = <. C , D >. -> ( 2nd ` y ) = ( 2nd ` <. C , D >. ) )
20	19	fveq1d 5874	. . . . . . . . 9 |- ( y = <. C , D >. -> ( ( 2nd ` y ) ` i ) = ( ( 2nd ` <. C , D >. ) ` i ) )
21	18, 20	oveq12d 6313	. . . . . . . 8 |- ( y = <. C , D >. -> ( ( ( 1st ` y ) ` i ) - ( ( 2nd ` y ) ` i ) ) = ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) )
22	21	oveq1d 6310	. . . . . . 7 |- ( y = <. C , D >. -> ( ( ( ( 1st ` y ) ` i ) - ( ( 2nd ` y ) ` i ) ) ^ 2 ) = ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) )
23	22	sumeq2sdv 13506	. . . . . 6 |- ( y = <. C , D >. -> sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` y ) ` i ) - ( ( 2nd ` y ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) )
24	23	eqeq2d 2481	. . . . 5 |- ( y = <. C , D >. -> ( sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` y ) ` i ) - ( ( 2nd ` y ) ` i ) ) ^ 2 ) <-> sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) ) )
25	16, 24	anbi12d 710	. . . 4 |- ( y = <. C , D >. -> ( ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ y e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` y ) ` i ) - ( ( 2nd ` y ) ` i ) ) ^ 2 ) ) <-> ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) ) ) )
26	25	rexbidv 2978	. . 3 |- ( y = <. C , D >. -> ( E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ y e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` y ) ` i ) - ( ( 2nd ` y ) ` i ) ) ^ 2 ) ) <-> E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) ) ) )
27		df-cgr 24019	. . 3 |- Cgr = { <. x , y >. | E. n e. NN ( ( x e. ( ( EE ` n ) X. ( EE ` n ) ) /\ y e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` x ) ` i ) - ( ( 2nd ` x ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` y ) ` i ) - ( ( 2nd ` y ) ` i ) ) ^ 2 ) ) }
28	1, 2, 14, 26, 27	brab 4776	. 2 |- ( <. A , B >.Cgr <. C , D >. <-> E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) ) )
29		opelxp2 5039	. . . . . . . . . . 11 |- ( <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) -> D e. ( EE ` n ) )
30	29	ad2antll 728	. . . . . . . . . 10 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) ) -> D e. ( EE ` n ) )
31		simplrr 760	. . . . . . . . . 10 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) ) -> D e. ( EE ` N ) )
32		eedimeq 24024	. . . . . . . . . 10 |- ( ( D e. ( EE ` n ) /\ D e. ( EE ` N ) ) -> n = N )
33	30, 31, 32	syl2anc 661	. . . . . . . . 9 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) ) -> n = N )
34	33	adantlr 714	. . . . . . . 8 |- ( ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ n e. NN ) /\ ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) ) -> n = N )
35		oveq2 6303	. . . . . . . . . 10 |- ( n = N -> ( 1 ... n ) = ( 1 ... N ) )
36	35	sumeq1d 13503	. . . . . . . . 9 |- ( n = N -> sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) )
37	35	sumeq1d 13503	. . . . . . . . 9 |- ( n = N -> sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) )
38	36, 37	eqeq12d 2489	. . . . . . . 8 |- ( n = N -> ( sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) <-> sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) ) )
39	34, 38	syl 16	. . . . . . 7 |- ( ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ n e. NN ) /\ ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) ) -> ( sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) <-> sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) ) )
40		op1stg 6807	. . . . . . . . . . . . 13 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( 1st ` <. A , B >. ) = A )
41	40	fveq1d 5874	. . . . . . . . . . . 12 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( ( 1st ` <. A , B >. ) ` i ) = ( A ` i ) )
42		op2ndg 6808	. . . . . . . . . . . . 13 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( 2nd ` <. A , B >. ) = B )
43	42	fveq1d 5874	. . . . . . . . . . . 12 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( ( 2nd ` <. A , B >. ) ` i ) = ( B ` i ) )
44	41, 43	oveq12d 6313	. . . . . . . . . . 11 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) = ( ( A ` i ) - ( B ` i ) ) )
45	44	oveq1d 6310	. . . . . . . . . 10 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) )
46	45	sumeq2sdv 13506	. . . . . . . . 9 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) )
47		op1stg 6807	. . . . . . . . . . . . 13 |- ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> ( 1st ` <. C , D >. ) = C )
48	47	fveq1d 5874	. . . . . . . . . . . 12 |- ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> ( ( 1st ` <. C , D >. ) ` i ) = ( C ` i ) )
49		op2ndg 6808	. . . . . . . . . . . . 13 |- ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> ( 2nd ` <. C , D >. ) = D )
50	49	fveq1d 5874	. . . . . . . . . . . 12 |- ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> ( ( 2nd ` <. C , D >. ) ` i ) = ( D ` i ) )
51	48, 50	oveq12d 6313	. . . . . . . . . . 11 |- ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) = ( ( C ` i ) - ( D ` i ) ) )
52	51	oveq1d 6310	. . . . . . . . . 10 |- ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) = ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) )
53	52	sumeq2sdv 13506	. . . . . . . . 9 |- ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) )
54	46, 53	eqeqan12d 2490	. . . . . . . 8 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) <-> sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) ) )
55	54	ad2antrr 725	. . . . . . 7 |- ( ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ n e. NN ) /\ ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) ) -> ( sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) <-> sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) ) )
56	39, 55	bitrd 253	. . . . . 6 |- ( ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ n e. NN ) /\ ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) ) -> ( sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) <-> sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) ) )
57	56	biimpd 207	. . . . 5 |- ( ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ n e. NN ) /\ ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) ) -> ( sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) -> sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) ) )
58	57	expimpd 603	. . . 4 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ n e. NN ) -> ( ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) ) -> sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) ) )
59	58	rexlimdva 2959	. . 3 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) ) -> sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) ) )
60		eleenn 24022	. . . . 5 |- ( D e. ( EE ` N ) -> N e. NN )
61	60	ad2antll 728	. . . 4 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> N e. NN )
62		opelxpi 5037	. . . . . . . . 9 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) )
63		opelxpi 5037	. . . . . . . . 9 |- ( ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) -> <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) )
64	62, 63	anim12i 566	. . . . . . . 8 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) /\ <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) ) )
65	64	adantr 465	. . . . . . 7 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) ) -> ( <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) /\ <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) ) )
66	54	biimpar 485	. . . . . . 7 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) ) -> sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) )
67	65, 66	jca 532	. . . . . 6 |- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) ) -> ( ( <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) /\ <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) ) /\ sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) ) )
68		fveq2 5872	. . . . . . . . . . 11 |- ( n = N -> ( EE ` n ) = ( EE ` N ) )
69	68, 68	xpeq12d 5030	. . . . . . . . . 10 |- ( n = N -> ( ( EE ` n ) X. ( EE ` n ) ) = ( ( EE ` N ) X. ( EE ` N ) ) )
70	69	eleq2d 2537	. . . . . . . . 9 |- ( n = N -> ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) <-> <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) ) )
71	69	eleq2d 2537	. . . . . . . . 9 |- ( n = N -> ( <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) <-> <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) ) )
72	70, 71	anbi12d 710	. . . . . . . 8 |- ( n = N -> ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) <-> ( <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) /\ <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) ) ) )
73	72, 38	anbi12d 710	. . . . . . 7 |- ( n = N -> ( ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) ) <-> ( ( <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) /\ <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) ) /\ sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) ) ) )
74	73	rspcev 3219	. . . . . 6 |- ( ( N e. NN /\ ( ( <. A , B >. e. ( ( EE ` N ) X. ( EE ` N ) ) /\ <. C , D >. e. ( ( EE ` N ) X. ( EE ` N ) ) ) /\ sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) ) ) -> E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) ) )
75	67, 74	sylan2 474	. . . . 5 |- ( ( N e. NN /\ ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) /\ sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) ) ) -> E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) ) )
76	75	exp32 605	. . . 4 |- ( N e. NN -> ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) -> E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) ) ) ) )
77	61, 76	mpcom 36	. . 3 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) -> E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) ) ) )
78	59, 77	impbid 191	. 2 |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( E. n e. NN ( ( <. A , B >. e. ( ( EE ` n ) X. ( EE ` n ) ) /\ <. C , D >. e. ( ( EE ` n ) X. ( EE ` n ) ) ) /\ sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. A , B >. ) ` i ) - ( ( 2nd ` <. A , B >. ) ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... n ) ( ( ( ( 1st ` <. C , D >. ) ` i ) - ( ( 2nd ` <. C , D >. ) ` i ) ) ^ 2 ) ) <-> sum_ i e. ( 1 ... N ) ( ( ( A ` i ) - ( B ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) ) )
79	28, 78	syl5bb 257	1 |- ( ( ( 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 ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   E.wrex 2818   <.cop 4039   class class class wbr 4453   X. cxp 5003   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   1c1 9505   - cmin 9817   NNcn 10548   2c2 10597   ...cfz 11684   ^cexp 12146   sum_csu 13488   EEcee 24014  Cgrccgr 24016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-seq 12088  df-sum 13489  df-ee 24017  df-cgr 24019

This theorem is referenced by:  axcgrrflx  24040  axcgrtr  24041  axcgrid  24042  axsegcon  24053  ax5seglem3  24057  ax5seglem6  24060  ax5seg  24064  axlowdimlem17  24084  ecgrtg  24109