Theorem istvc 21995

Description: A topological vector space is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypothesis

Ref	Expression
tlmtrg.f	|- F = (Scalar ` W )

Assertion

Ref	Expression
istvc	|- ( W e. TopVec <-> ( W e. TopMod /\ F e. TopDRing ) )

Proof of Theorem istvc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( x = W -> (Scalar ` x ) = (Scalar ` W ) )
2		tlmtrg.f	. . . 4 |- F = (Scalar ` W )
3	1, 2	syl6eqr 2674	. . 3 |- ( x = W -> (Scalar ` x ) = F )
4	3	eleq1d 2686	. 2 |- ( x = W -> ( (Scalar ` x ) e. TopDRing <-> F e. TopDRing ) )
5		df-tvc 21966	. 2 |- TopVec = { x e. TopMod | (Scalar ` x ) e. TopDRing }
6	4, 5	elrab2 3366	1 |- ( W e. TopVec <-> ( W e. TopMod /\ F e. TopDRing ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  Scalarcsca 15944  TopDRingctdrg 21960  TopModctlm 21961   TopVecctvc 21962

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  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-tvc 21966

This theorem is referenced by:  tvctdrg  21996  tvctlm  22000  nvctvc  22504