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Metamath Proof Explorer

Theorem tgpmulg2

Description: In a topological monoid, the group multiple function is jointly continuous (although this is not saying much as one of the factors is discrete). Use zdis to write the left topology as a subset of the complex numbers. (Contributed by Mario Carneiro, 19-Sep-2015)

Ref Expression
Hypotheses tgpmulg.j 
|- J = ( TopOpen ` G )
tgpmulg.t 
|- .x. = ( .g ` G )
Assertion tgpmulg2 
|- ( G e. TopGrp -> .x. e. ( ( ~P ZZ tX J ) Cn J ) )

Proof

Step Hyp Ref Expression
1 tgpmulg.j 
 |-  J = ( TopOpen ` G )
2 tgpmulg.t 
 |-  .x. = ( .g ` G )
3 zex 
 |-  ZZ e. _V
4 3 a1i 
 |-  ( G e. TopGrp -> ZZ e. _V )
5 eqid 
 |-  ( Base ` G ) = ( Base ` G )
6 1 5 tgptopon 
 |-  ( G e. TopGrp -> J e. ( TopOn ` ( Base ` G ) ) )
7 topontop 
 |-  ( J e. ( TopOn ` ( Base ` G ) ) -> J e. Top )
8 6 7 syl 
 |-  ( G e. TopGrp -> J e. Top )
9 5 2 mulgfn 
 |-  .x. Fn ( ZZ X. ( Base ` G ) )
10 9 a1i 
 |-  ( G e. TopGrp -> .x. Fn ( ZZ X. ( Base ` G ) ) )
11 1 2 5 tgpmulg 
 |-  ( ( G e. TopGrp /\ n e. ZZ ) -> ( x e. ( Base ` G ) |-> ( n .x. x ) ) e. ( J Cn J ) )
12 4 6 8 10 11 txdis1cn 
 |-  ( G e. TopGrp -> .x. e. ( ( ~P ZZ tX J ) Cn J ) )