Metamath Proof Explorer

Theorem dvtanlem

Description: Lemma for dvtan - the domain of the tangent is open. (Contributed by Brendan Leahy, 8-Aug-2018) (Proof shortened by OpenAI, 3-Jul-2020)

Ref Expression
Assertion dvtanlem 
|- ( `' cos " ( CC \ { 0 } ) ) e. ( TopOpen ` CCfld )

Proof

Step Hyp Ref Expression
1 coscn 
 |-  cos e. ( CC -cn-> CC )
2 eqid 
 |-  ( TopOpen ` CCfld ) = ( TopOpen ` CCfld )
3 2 cncfcn1 
 |-  ( CC -cn-> CC ) = ( ( TopOpen ` CCfld ) Cn ( TopOpen ` CCfld ) )
4 1 3 eleqtri 
 |-  cos e. ( ( TopOpen ` CCfld ) Cn ( TopOpen ` CCfld ) )
5 2 cnfldhaus 
 |-  ( TopOpen ` CCfld ) e. Haus
6 0cn 
 |-  0 e. CC
7 2 cnfldtopon 
 |-  ( TopOpen ` CCfld ) e. ( TopOn ` CC )
8 7 toponunii 
 |-  CC = U. ( TopOpen ` CCfld )
9 8 sncld 
 |-  ( ( ( TopOpen ` CCfld ) e. Haus /\ 0 e. CC ) -> { 0 } e. ( Clsd ` ( TopOpen ` CCfld ) ) )
10 5 6 9 mp2an 
 |-  { 0 } e. ( Clsd ` ( TopOpen ` CCfld ) )
11 8 cldopn 
 |-  ( { 0 } e. ( Clsd ` ( TopOpen ` CCfld ) ) -> ( CC \ { 0 } ) e. ( TopOpen ` CCfld ) )
12 10 11 ax-mp 
 |-  ( CC \ { 0 } ) e. ( TopOpen ` CCfld )
13 cnima 
 |-  ( ( cos e. ( ( TopOpen ` CCfld ) Cn ( TopOpen ` CCfld ) ) /\ ( CC \ { 0 } ) e. ( TopOpen ` CCfld ) ) -> ( `' cos " ( CC \ { 0 } ) ) e. ( TopOpen ` CCfld ) )
14 4 12 13 mp2an 
 |-  ( `' cos " ( CC \ { 0 } ) ) e. ( TopOpen ` CCfld )