Theorem tmdcn 21887

Description: In a topological monoid, the operation F representing the functionalization of the operator slot +g is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)

Hypotheses

Ref	Expression
tgpcn.j	|- J = ( TopOpen ` G )
tgpcn.1	|- F = ( +f ` G )

Assertion

Ref	Expression
tmdcn	|- ( G e. TopMnd -> F e. ( ( J tX J ) Cn J ) )

Proof of Theorem tmdcn

Step	Hyp	Ref	Expression
1		tgpcn.1	. . 3 |- F = ( +f ` G )
2		tgpcn.j	. . 3 |- J = ( TopOpen ` G )
3	1, 2	istmd 21878	. 2 |- ( G e. TopMnd <-> ( G e. Mnd /\ G e. TopSp /\ F e. ( ( J tX J ) Cn J ) ) )
4	3	simp3bi 1078	1 |- ( G e. TopMnd -> F e. ( ( J tX J ) Cn J ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   TopOpenctopn 16082   +fcplusf 17239   Mndcmnd 17294   TopSpctps 20736   Cn ccn 21028   tX ctx 21363  TopMndctmd 21874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-tmd 21876

This theorem is referenced by:  tgpcn  21888  cnmpt1plusg  21891  cnmpt2plusg  21892  tmdcn2  21893  submtmd  21908  tsmsadd  21950  mulrcn  21982  mhmhmeotmd  29973  xrge0pluscn  29986