Metamath Proof Explorer


Theorem rrxtopn

Description: The topology of the generalized real Euclidean space. (Contributed by Glauco Siliprandi, 24-Dec-2020)

Ref Expression
Hypothesis rrxtopn.1 φ I V
Assertion rrxtopn φ TopOpen I = MetOpen f Base I , g Base I fld x I f x g x 2

Proof

Step Hyp Ref Expression
1 rrxtopn.1 φ I V
2 eqid I = I
3 2 rrxval I V I = toCPreHil fld freeLMod I
4 1 3 syl φ I = toCPreHil fld freeLMod I
5 4 fveq2d φ TopOpen I = TopOpen toCPreHil fld freeLMod I
6 ovex fld freeLMod I V
7 eqid toCPreHil fld freeLMod I = toCPreHil fld freeLMod I
8 eqid dist toCPreHil fld freeLMod I = dist toCPreHil fld freeLMod I
9 eqid TopOpen toCPreHil fld freeLMod I = TopOpen toCPreHil fld freeLMod I
10 7 8 9 tcphtopn fld freeLMod I V TopOpen toCPreHil fld freeLMod I = MetOpen dist toCPreHil fld freeLMod I
11 6 10 ax-mp TopOpen toCPreHil fld freeLMod I = MetOpen dist toCPreHil fld freeLMod I
12 11 a1i φ TopOpen toCPreHil fld freeLMod I = MetOpen dist toCPreHil fld freeLMod I
13 4 eqcomd φ toCPreHil fld freeLMod I = I
14 13 fveq2d φ dist toCPreHil fld freeLMod I = dist I
15 14 fveq2d φ MetOpen dist toCPreHil fld freeLMod I = MetOpen dist I
16 5 12 15 3eqtrd φ TopOpen I = MetOpen dist I
17 eqid Base I = Base I
18 2 17 rrxds I V f Base I , g Base I fld x I f x g x 2 = dist I
19 1 18 syl φ f Base I , g Base I fld x I f x g x 2 = dist I
20 19 eqcomd φ dist I = f Base I , g Base I fld x I f x g x 2
21 20 fveq2d φ MetOpen dist I = MetOpen f Base I , g Base I fld x I f x g x 2
22 16 21 eqtrd φ TopOpen I = MetOpen f Base I , g Base I fld x I f x g x 2