Metamath Proof Explorer


Theorem ehl2eudisval0

Description: The Euclidean distance of a point to the origin in a real Euclidean space of dimension 2. (Contributed by AV, 26-Feb-2023)

Ref Expression
Hypotheses ehl2eudisval0.e E = 𝔼 hil 2
ehl2eudisval0.x X = 1 2
ehl2eudisval0.d D = dist E
ehl2eudisval0.0 0 ˙ = 1 2 × 0
Assertion ehl2eudisval0 F X F D 0 ˙ = F 1 2 + F 2 2

Proof

Step Hyp Ref Expression
1 ehl2eudisval0.e E = 𝔼 hil 2
2 ehl2eudisval0.x X = 1 2
3 ehl2eudisval0.d D = dist E
4 ehl2eudisval0.0 0 ˙ = 1 2 × 0
5 prex 1 2 V
6 4 2 rrx0el 1 2 V 0 ˙ X
7 5 6 mp1i F X 0 ˙ X
8 1 2 3 ehl2eudisval F X 0 ˙ X F D 0 ˙ = F 1 0 ˙ 1 2 + F 2 0 ˙ 2 2
9 7 8 mpdan F X F D 0 ˙ = F 1 0 ˙ 1 2 + F 2 0 ˙ 2 2
10 1ex 1 V
11 2ex 2 V
12 c0ex 0 V
13 xpprsng 1 V 2 V 0 V 1 2 × 0 = 1 0 2 0
14 10 11 12 13 mp3an 1 2 × 0 = 1 0 2 0
15 4 14 eqtri 0 ˙ = 1 0 2 0
16 15 fveq1i 0 ˙ 1 = 1 0 2 0 1
17 1ne2 1 2
18 10 12 fvpr1 1 2 1 0 2 0 1 = 0
19 17 18 ax-mp 1 0 2 0 1 = 0
20 16 19 eqtri 0 ˙ 1 = 0
21 20 a1i F X 0 ˙ 1 = 0
22 21 oveq2d F X F 1 0 ˙ 1 = F 1 0
23 eqid 1 2 = 1 2
24 23 2 rrx2pxel F X F 1
25 24 recnd F X F 1
26 25 subid1d F X F 1 0 = F 1
27 22 26 eqtrd F X F 1 0 ˙ 1 = F 1
28 27 oveq1d F X F 1 0 ˙ 1 2 = F 1 2
29 15 fveq1i 0 ˙ 2 = 1 0 2 0 2
30 11 12 fvpr2 1 2 1 0 2 0 2 = 0
31 17 30 mp1i F X 1 0 2 0 2 = 0
32 29 31 syl5eq F X 0 ˙ 2 = 0
33 32 oveq2d F X F 2 0 ˙ 2 = F 2 0
34 23 2 rrx2pyel F X F 2
35 34 recnd F X F 2
36 35 subid1d F X F 2 0 = F 2
37 33 36 eqtrd F X F 2 0 ˙ 2 = F 2
38 37 oveq1d F X F 2 0 ˙ 2 2 = F 2 2
39 28 38 oveq12d F X F 1 0 ˙ 1 2 + F 2 0 ˙ 2 2 = F 1 2 + F 2 2
40 39 fveq2d F X F 1 0 ˙ 1 2 + F 2 0 ˙ 2 2 = F 1 2 + F 2 2
41 9 40 eqtrd F X F D 0 ˙ = F 1 2 + F 2 2