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Metamath Proof Explorer

Theorem sraip

Description: The inner product operation of a subring algebra. (Contributed by Thierry Arnoux, 16-Jun-2019)

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

Proof

Step Hyp Ref Expression
1 srapart.a 
 |-  ( ph -> A = ( ( subringAlg ` W ) ` S ) )
2 srapart.s 
 |-  ( ph -> S C_ ( Base ` W ) )
3 ovex 
 |-  ( ( W sSet <. ( Scalar ` ndx ) , ( W |`s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V
4 fvex 
 |-  ( .r ` W ) e. _V
5 ipid 
 |-  .i = Slot ( .i ` ndx )
6 5 setsid 
 |-  ( ( ( ( W sSet <. ( Scalar ` ndx ) , ( W |`s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) e. _V /\ ( .r ` W ) e. _V ) -> ( .r ` W ) = ( .i ` ( ( ( W sSet <. ( Scalar ` ndx ) , ( W |`s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
7 3 4 6 mp2an 
 |-  ( .r ` W ) = ( .i ` ( ( ( W sSet <. ( Scalar ` ndx ) , ( W |`s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
8 1 adantl 
 |-  ( ( W e. _V /\ ph ) -> A = ( ( subringAlg ` W ) ` S ) )
9 sraval 
 |-  ( ( W e. _V /\ S C_ ( Base ` W ) ) -> ( ( subringAlg ` W ) ` S ) = ( ( ( W sSet <. ( Scalar ` ndx ) , ( W |`s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
10 2 9 sylan2 
 |-  ( ( W e. _V /\ ph ) -> ( ( subringAlg ` W ) ` S ) = ( ( ( W sSet <. ( Scalar ` ndx ) , ( W |`s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
11 8 10 eqtrd 
 |-  ( ( W e. _V /\ ph ) -> A = ( ( ( W sSet <. ( Scalar ` ndx ) , ( W |`s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
12 11 fveq2d 
 |-  ( ( W e. _V /\ ph ) -> ( .i ` A ) = ( .i ` ( ( ( W sSet <. ( Scalar ` ndx ) , ( W |`s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
13 7 12 eqtr4id 
 |-  ( ( W e. _V /\ ph ) -> ( .r ` W ) = ( .i ` A ) )
14 5 str0 
 |-  (/) = ( .i ` (/) )
15 fvprc 
 |-  ( -. W e. _V -> ( .r ` W ) = (/) )
16 15 adantr 
 |-  ( ( -. W e. _V /\ ph ) -> ( .r ` W ) = (/) )
17 fv2prc 
 |-  ( -. W e. _V -> ( ( subringAlg ` W ) ` S ) = (/) )
18 1 17 sylan9eqr 
 |-  ( ( -. W e. _V /\ ph ) -> A = (/) )
19 18 fveq2d 
 |-  ( ( -. W e. _V /\ ph ) -> ( .i ` A ) = ( .i ` (/) ) )
20 14 16 19 3eqtr4a 
 |-  ( ( -. W e. _V /\ ph ) -> ( .r ` W ) = ( .i ` A ) )
21 13 20 pm2.61ian 
 |-  ( ph -> ( .r ` W ) = ( .i ` A ) )