Metamath Proof Explorer


Theorem cnfldtgp

Description: The complex numbers form a topological group under addition, with the standard topology induced by the absolute value metric. (Contributed by Mario Carneiro, 2-Sep-2015)

Ref Expression
Assertion cnfldtgp fld TopGrp

Proof

Step Hyp Ref Expression
1 cnring fld Ring
2 ringgrp fld Ring fld Grp
3 1 2 ax-mp fld Grp
4 cnfldtps fld TopSp
5 eqid TopOpen fld = TopOpen fld
6 5 subcn TopOpen fld × t TopOpen fld Cn TopOpen fld
7 cnfldsub = - fld
8 5 7 istgp2 fld TopGrp fld Grp fld TopSp TopOpen fld × t TopOpen fld Cn TopOpen fld
9 3 4 6 8 mpbir3an fld TopGrp