Theorem trirn 15834

Description: Triangle inequality in R^n.

Assertion

Ref	Expression
trirn	|- ((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR) -> (sqr` sum_k e. (1...N)(((X` k) + (Y` k))^2)) <_ ((sqr` sum_k e. (1...N)((X` k)^2)) + (sqr` sum_k e. (1...N)((Y` k)^2))))

Distinct variable groups:   k,N   k,X   k,Y

Proof of Theorem trirn

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . 5 |- (N = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) -> (1...N) = (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1)))
2	1	sumeq1d 8250	. . . 4 |- (N = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) -> sum_k e. (1...N)(((X` k) + (Y` k))^2) = sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))(((X` k) + (Y` k))^2))
3	2	fveq2d 4685	. . 3 |- (N = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) -> (sqr` sum_k e. (1...N)(((X` k) + (Y` k))^2)) = (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))(((X` k) + (Y` k))^2)))
4	1	sumeq1d 8250	. . . . 5 |- (N = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) -> sum_k e. (1...N)((X` k)^2) = sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((X` k)^2))
5	4	fveq2d 4685	. . . 4 |- (N = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) -> (sqr` sum_k e. (1...N)((X` k)^2)) = (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((X` k)^2)))
6	1	sumeq1d 8250	. . . . 5 |- (N = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) -> sum_k e. (1...N)((Y` k)^2) = sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((Y` k)^2))
7	6	fveq2d 4685	. . . 4 |- (N = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) -> (sqr` sum_k e. (1...N)((Y` k)^2)) = (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((Y` k)^2)))
8	5, 7	opreq12d 4900	. . 3 |- (N = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) -> ((sqr` sum_k e. (1...N)((X` k)^2)) + (sqr` sum_k e. (1...N)((Y` k)^2))) = ((sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((X` k)^2)) + (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((Y` k)^2))))
9	3, 8	breq12d 3351	. 2 |- (N = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) -> ((sqr` sum_k e. (1...N)(((X` k) + (Y` k))^2)) <_ ((sqr` sum_k e. (1...N)((X` k)^2)) + (sqr` sum_k e. (1...N)((Y` k)^2))) <-> (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))(((X` k) + (Y` k))^2)) <_ ((sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((X` k)^2)) + (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((Y` k)^2)))))
10		fveq1 4680	. . . . . . 7 |- (X = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})) -> (X` k) = (if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k))
11	10	opreq1d 4897	. . . . . 6 |- (X = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})) -> ((X` k) + (Y` k)) = ((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k) + (Y` k)))
12	11	opreq1d 4897	. . . . 5 |- (X = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})) -> (((X` k) + (Y` k))^2) = (((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k) + (Y` k))^2))
13	12	sumeq2sdv 8253	. . . 4 |- (X = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})) -> sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))(((X` k) + (Y` k))^2) = sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))(((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k) + (Y` k))^2))
14	13	fveq2d 4685	. . 3 |- (X = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})) -> (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))(((X` k) + (Y` k))^2)) = (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))(((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k) + (Y` k))^2)))
15	10	opreq1d 4897	. . . . . 6 |- (X = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})) -> ((X` k)^2) = ((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k)^2))
16	15	sumeq2sdv 8253	. . . . 5 |- (X = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})) -> sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((X` k)^2) = sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k)^2))
17	16	fveq2d 4685	. . . 4 |- (X = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})) -> (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((X` k)^2)) = (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k)^2)))
18	17	opreq1d 4897	. . 3 |- (X = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})) -> ((sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((X` k)^2)) + (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((Y` k)^2))) = ((sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k)^2)) + (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((Y` k)^2))))
19	14, 18	breq12d 3351	. 2 |- (X = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})) -> ((sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))(((X` k) + (Y` k))^2)) <_ ((sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((X` k)^2)) + (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((Y` k)^2))) <-> (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))(((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k) + (Y` k))^2)) <_ ((sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k)^2)) + (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((Y` k)^2)))))
20		fveq1 4680	. . . . . . 7 |- (Y = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})) -> (Y` k) = (if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0}))` k))
21	20	opreq2d 4898	. . . . . 6 |- (Y = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})) -> ((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k) + (Y` k)) = ((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k) + (if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0}))` k)))
22	21	opreq1d 4897	. . . . 5 |- (Y = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})) -> (((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k) + (Y` k))^2) = (((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k) + (if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0}))` k))^2))
23	22	sumeq2sdv 8253	. . . 4 |- (Y = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})) -> sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))(((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k) + (Y` k))^2) = sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))(((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k) + (if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0}))` k))^2))
24	23	fveq2d 4685	. . 3 |- (Y = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})) -> (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))(((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k) + (Y` k))^2)) = (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))(((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k) + (if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0}))` k))^2)))
25	20	opreq1d 4897	. . . . . 6 |- (Y = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})) -> ((Y` k)^2) = ((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0}))` k)^2))
26	25	sumeq2sdv 8253	. . . . 5 |- (Y = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})) -> sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((Y` k)^2) = sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0}))` k)^2))
27	26	fveq2d 4685	. . . 4 |- (Y = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})) -> (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((Y` k)^2)) = (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0}))` k)^2)))
28	27	opreq2d 4898	. . 3 |- (Y = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})) -> ((sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k)^2)) + (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((Y` k)^2))) = ((sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k)^2)) + (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0}))` k)^2))))
29	24, 28	breq12d 3351	. 2 |- (Y = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})) -> ((sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))(((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k) + (Y` k))^2)) <_ ((sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k)^2)) + (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((Y` k)^2))) <-> (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))(((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k) + (if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0}))` k))^2)) <_ ((sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k)^2)) + (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0}))` k)^2)))))
30		eleq1 1957	. . . . . 6 |- (N = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) -> (N e. NN <-> if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) e. NN))
31	1	feq2d 4557	. . . . . 6 |- (N = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) -> (X:(1...N)-->RR <-> X:(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR))
32	1	feq2d 4557	. . . . . 6 |- (N = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) -> (Y:(1...N)-->RR <-> Y:(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR))
33	30, 31, 32	3anbi123d 1168	. . . . 5 |- (N = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) -> ((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR) <-> (if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) e. NN /\ X:(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR /\ Y:(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR)))
34		feq1 4551	. . . . . 6 |- (X = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})) -> (X:(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR <-> if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR))
35	34	3anbi2d 1173	. . . . 5 |- (X = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})) -> ((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) e. NN /\ X:(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR /\ Y:(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR) <-> (if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) e. NN /\ if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR /\ Y:(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR)))
36		feq1 4551	. . . . . 6 |- (Y = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})) -> (Y:(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR <-> if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR))
37	36	3anbi3d 1174	. . . . 5 |- (Y = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})) -> ((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) e. NN /\ if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR /\ Y:(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR) <-> (if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) e. NN /\ if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR /\ if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR)))
38		eleq1 1957	. . . . . 6 |- (1 = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) -> (1 e. NN <-> if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) e. NN))
39		opreq2 4890	. . . . . . 7 |- (1 = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) -> (1...1) = (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1)))
40	39	feq2d 4557	. . . . . 6 |- (1 = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) -> (((1...1) X. {0}):(1...1)-->RR <-> ((1...1) X. {0}):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR))
41	38, 40, 40	3anbi123d 1168	. . . . 5 |- (1 = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) -> ((1 e. NN /\ ((1...1) X. {0}):(1...1)-->RR /\ ((1...1) X. {0}):(1...1)-->RR) <-> (if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) e. NN /\ ((1...1) X. {0}):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR /\ ((1...1) X. {0}):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR)))
42		feq1 4551	. . . . . 6 |- (((1...1) X. {0}) = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})) -> (((1...1) X. {0}):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR <-> if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR))
43	42	3anbi2d 1173	. . . . 5 |- (((1...1) X. {0}) = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})) -> ((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) e. NN /\ ((1...1) X. {0}):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR /\ ((1...1) X. {0}):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR) <-> (if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) e. NN /\ if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR /\ ((1...1) X. {0}):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR)))
44		feq1 4551	. . . . . 6 |- (((1...1) X. {0}) = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})) -> (((1...1) X. {0}):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR <-> if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR))
45	44	3anbi3d 1174	. . . . 5 |- (((1...1) X. {0}) = if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})) -> ((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) e. NN /\ if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR /\ ((1...1) X. {0}):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR) <-> (if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) e. NN /\ if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR /\ if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR)))
46		1nn 7117	. . . . . 6 |- 1 e. NN
47		0re 6603	. . . . . . . . 9 |- 0 e. RR
48	47	elisseti 2301	. . . . . . . 8 |- 0 e. _V
49	48	fconst 4602	. . . . . . 7 |- ((1...1) X. {0}):(1...1)-->{0}
50	48	snss 3122	. . . . . . . 8 |- (0 e. RR <-> {0} C_ RR)
51	47, 50	mpbi 206	. . . . . . 7 |- {0} C_ RR
52		fss 4571	. . . . . . 7 |- ((((1...1) X. {0}):(1...1)-->{0} /\ {0} C_ RR) -> ((1...1) X. {0}):(1...1)-->RR)
53	49, 51, 52	mp2an 761	. . . . . 6 |- ((1...1) X. {0}):(1...1)-->RR
54	46, 53, 53	3pm3.2i 1048	. . . . 5 |- (1 e. NN /\ ((1...1) X. {0}):(1...1)-->RR /\ ((1...1) X. {0}):(1...1)-->RR)
55	33, 35, 37, 41, 43, 45, 54	elimhyp3v 3023	. . . 4 |- (if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) e. NN /\ if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR /\ if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR)
56	55	simp1i 885	. . 3 |- if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1) e. NN
57	55	simp2i 886	. . 3 |- if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0})):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR
58	55	simp3i 887	. . 3 |- if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0})):(1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))-->RR
59	56, 57, 58	trirni 15833	. 2 |- (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))(((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k) + (if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0}))` k))^2)) <_ ((sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), X, ((1...1) X. {0}))` k)^2)) + (sqr` sum_k e. (1...if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), N, 1))((if((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR), Y, ((1...1) X. {0}))` k)^2)))
60	9, 19, 29, 59	dedth3v 3019	1 |- ((N e. NN /\ X:(1...N)-->RR /\ Y:(1...N)-->RR) -> (sqr` sum_k e. (1...N)(((X` k) + (Y` k))^2)) <_ ((sqr` sum_k e. (1...N)((X` k)^2)) + (sqr` sum_k e. (1...N)((Y` k)^2))))

Syntax hints:   -> wi 3   /\ w3a 858   = wceq 1298   e. wcel 1300   C_ wss 2593  ifcif 2982  {csn 3044   class class class wbr 3338   X. cxp 3984  -->wf 3994  ` cfv 3998  (class class class)co 4884  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   <_ cle 6448  NNcn 6449  2c2 7145  ...cfz 7637  ^cexp 7811  sqrcsqr 7919  sum_csu 8239

This theorem is referenced by:  rrnmet 16016

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-sum 8240

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