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Metamath Proof Explorer

Theorem sratset

Description: Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015) (Revised by Thierry Arnoux, 16-Jun-2019) (Revised by AV, 29-Oct-2024)

Ref Expression
Hypotheses srapart.a 
|- ( ph -> A = ( ( subringAlg ` W ) ` S ) )
srapart.s 
|- ( ph -> S C_ ( Base ` W ) )
Assertion sratset 
|- ( ph -> ( TopSet ` W ) = ( TopSet ` A ) )

Proof

Step Hyp Ref Expression
1 srapart.a 
 |-  ( ph -> A = ( ( subringAlg ` W ) ` S ) )
2 srapart.s 
 |-  ( ph -> S C_ ( Base ` W ) )
3 tsetid 
 |-  TopSet = Slot ( TopSet ` ndx )
4 slotstnscsi 
 |-  ( ( TopSet ` ndx ) =/= ( Scalar ` ndx ) /\ ( TopSet ` ndx ) =/= ( .s ` ndx ) /\ ( TopSet ` ndx ) =/= ( .i ` ndx ) )
5 4 simp1i 
 |-  ( TopSet ` ndx ) =/= ( Scalar ` ndx )
6 5 necomi 
 |-  ( Scalar ` ndx ) =/= ( TopSet ` ndx )
7 4 simp2i 
 |-  ( TopSet ` ndx ) =/= ( .s ` ndx )
8 7 necomi 
 |-  ( .s ` ndx ) =/= ( TopSet ` ndx )
9 4 simp3i 
 |-  ( TopSet ` ndx ) =/= ( .i ` ndx )
10 9 necomi 
 |-  ( .i ` ndx ) =/= ( TopSet ` ndx )
11 1 2 3 6 8 10 sralem 
 |-  ( ph -> ( TopSet ` W ) = ( TopSet ` A ) )