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Metamath Proof Explorer

Theorem rnasclassa

Description: The scalar multiples of the unit vector form a subalgebra of the vectors. (Contributed by SN, 16-Nov-2023)

Ref Expression
Hypotheses rnasclassa.a 
|- A = ( algSc ` W )
rnasclassa.u 
|- U = ( W |`s ran A )
rnasclassa.w 
|- ( ph -> W e. AssAlg )
Assertion rnasclassa 
|- ( ph -> U e. AssAlg )

Proof

Step Hyp Ref Expression
1 rnasclassa.a 
 |-  A = ( algSc ` W )
2 rnasclassa.u 
 |-  U = ( W |`s ran A )
3 rnasclassa.w 
 |-  ( ph -> W e. AssAlg )
4 ssidd 
 |-  ( ph -> ran A C_ ran A )
5 1 3 rnasclsubrg 
 |-  ( ph -> ran A e. ( SubRing ` W ) )
6 eqid 
 |-  ( LSubSp ` W ) = ( LSubSp ` W )
7 1 6 issubassa2 
 |-  ( ( W e. AssAlg /\ ran A e. ( SubRing ` W ) ) -> ( ran A e. ( LSubSp ` W ) <-> ran A C_ ran A ) )
8 2 6 issubassa3 
 |-  ( ( W e. AssAlg /\ ( ran A e. ( SubRing ` W ) /\ ran A e. ( LSubSp ` W ) ) ) -> U e. AssAlg )
9 8 expr 
 |-  ( ( W e. AssAlg /\ ran A e. ( SubRing ` W ) ) -> ( ran A e. ( LSubSp ` W ) -> U e. AssAlg ) )
10 7 9 sylbird 
 |-  ( ( W e. AssAlg /\ ran A e. ( SubRing ` W ) ) -> ( ran A C_ ran A -> U e. AssAlg ) )
11 3 5 10 syl2anc 
 |-  ( ph -> ( ran A C_ ran A -> U e. AssAlg ) )
12 4 11 mpd 
 |-  ( ph -> U e. AssAlg )