# Metamath Proof Explorer

## Theorem cnfldtset

Description: The topology component of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015) (Revised by Mario Carneiro, 6-Oct-2015) (Revised by Thierry Arnoux, 17-Dec-2017)

Ref Expression
Assertion cnfldtset ${⊢}\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)=\mathrm{TopSet}\left({ℂ}_{\mathrm{fld}}\right)$

### Proof

Step Hyp Ref Expression
1 fvex ${⊢}\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)\in \mathrm{V}$
2 cnfldstr ${⊢}{ℂ}_{\mathrm{fld}}\mathrm{Struct}⟨1,13⟩$
3 tsetid ${⊢}\mathrm{TopSet}=\mathrm{Slot}\mathrm{TopSet}\left(\mathrm{ndx}\right)$
4 snsstp1 ${⊢}\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)⟩\right\}\subseteq \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)⟩,⟨{\le }_{\mathrm{ndx}},\le ⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{abs}\circ -⟩\right\}$
5 ssun1 ${⊢}\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)⟩,⟨{\le }_{\mathrm{ndx}},\le ⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{abs}\circ -⟩\right\}\subseteq \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)⟩,⟨{\le }_{\mathrm{ndx}},\le ⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{abs}\circ -⟩\right\}\cup \left\{⟨\mathrm{UnifSet}\left(\mathrm{ndx}\right),\mathrm{metUnif}\left(\mathrm{abs}\circ -\right)⟩\right\}$
6 ssun2 ${⊢}\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)⟩,⟨{\le }_{\mathrm{ndx}},\le ⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{abs}\circ -⟩\right\}\cup \left\{⟨\mathrm{UnifSet}\left(\mathrm{ndx}\right),\mathrm{metUnif}\left(\mathrm{abs}\circ -\right)⟩\right\}\subseteq \left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},ℂ⟩,⟨{+}_{\mathrm{ndx}},+⟩,⟨{\cdot }_{\mathrm{ndx}},×⟩\right\}\cup \left\{⟨{*}_{\mathrm{ndx}},*⟩\right\}\right)\cup \left(\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)⟩,⟨{\le }_{\mathrm{ndx}},\le ⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{abs}\circ -⟩\right\}\cup \left\{⟨\mathrm{UnifSet}\left(\mathrm{ndx}\right),\mathrm{metUnif}\left(\mathrm{abs}\circ -\right)⟩\right\}\right)$
7 df-cnfld ${⊢}{ℂ}_{\mathrm{fld}}=\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},ℂ⟩,⟨{+}_{\mathrm{ndx}},+⟩,⟨{\cdot }_{\mathrm{ndx}},×⟩\right\}\cup \left\{⟨{*}_{\mathrm{ndx}},*⟩\right\}\right)\cup \left(\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)⟩,⟨{\le }_{\mathrm{ndx}},\le ⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{abs}\circ -⟩\right\}\cup \left\{⟨\mathrm{UnifSet}\left(\mathrm{ndx}\right),\mathrm{metUnif}\left(\mathrm{abs}\circ -\right)⟩\right\}\right)$
8 6 7 sseqtrri ${⊢}\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)⟩,⟨{\le }_{\mathrm{ndx}},\le ⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{abs}\circ -⟩\right\}\cup \left\{⟨\mathrm{UnifSet}\left(\mathrm{ndx}\right),\mathrm{metUnif}\left(\mathrm{abs}\circ -\right)⟩\right\}\subseteq {ℂ}_{\mathrm{fld}}$
9 5 8 sstri ${⊢}\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)⟩,⟨{\le }_{\mathrm{ndx}},\le ⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\mathrm{abs}\circ -⟩\right\}\subseteq {ℂ}_{\mathrm{fld}}$
10 4 9 sstri ${⊢}\left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)⟩\right\}\subseteq {ℂ}_{\mathrm{fld}}$
11 2 3 10 strfv ${⊢}\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)\in \mathrm{V}\to \mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)=\mathrm{TopSet}\left({ℂ}_{\mathrm{fld}}\right)$
12 1 11 ax-mp ${⊢}\mathrm{MetOpen}\left(\mathrm{abs}\circ -\right)=\mathrm{TopSet}\left({ℂ}_{\mathrm{fld}}\right)$