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 ( 𝜑𝐴 = ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) )
srapart.s ( 𝜑𝑆 ⊆ ( Base ‘ 𝑊 ) )
Assertion sraip ( 𝜑 → ( .r𝑊 ) = ( ·𝑖𝐴 ) )

Proof

Step Hyp Ref Expression
1 srapart.a ( 𝜑𝐴 = ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) )
2 srapart.s ( 𝜑𝑆 ⊆ ( Base ‘ 𝑊 ) )
3 ovex ( ( 𝑊 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑊s 𝑆 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r𝑊 ) ⟩ ) ∈ V
4 fvex ( .r𝑊 ) ∈ V
5 ipid ·𝑖 = Slot ( ·𝑖 ‘ ndx )
6 5 setsid ( ( ( ( 𝑊 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑊s 𝑆 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r𝑊 ) ⟩ ) ∈ V ∧ ( .r𝑊 ) ∈ V ) → ( .r𝑊 ) = ( ·𝑖 ‘ ( ( ( 𝑊 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑊s 𝑆 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r𝑊 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r𝑊 ) ⟩ ) ) )
7 3 4 6 mp2an ( .r𝑊 ) = ( ·𝑖 ‘ ( ( ( 𝑊 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑊s 𝑆 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r𝑊 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r𝑊 ) ⟩ ) )
8 1 adantl ( ( 𝑊 ∈ V ∧ 𝜑 ) → 𝐴 = ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) )
9 sraval ( ( 𝑊 ∈ V ∧ 𝑆 ⊆ ( Base ‘ 𝑊 ) ) → ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) = ( ( ( 𝑊 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑊s 𝑆 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r𝑊 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r𝑊 ) ⟩ ) )
10 2 9 sylan2 ( ( 𝑊 ∈ V ∧ 𝜑 ) → ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) = ( ( ( 𝑊 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑊s 𝑆 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r𝑊 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r𝑊 ) ⟩ ) )
11 8 10 eqtrd ( ( 𝑊 ∈ V ∧ 𝜑 ) → 𝐴 = ( ( ( 𝑊 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑊s 𝑆 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r𝑊 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r𝑊 ) ⟩ ) )
12 11 fveq2d ( ( 𝑊 ∈ V ∧ 𝜑 ) → ( ·𝑖𝐴 ) = ( ·𝑖 ‘ ( ( ( 𝑊 sSet ⟨ ( Scalar ‘ ndx ) , ( 𝑊s 𝑆 ) ⟩ ) sSet ⟨ ( ·𝑠 ‘ ndx ) , ( .r𝑊 ) ⟩ ) sSet ⟨ ( ·𝑖 ‘ ndx ) , ( .r𝑊 ) ⟩ ) ) )
13 7 12 eqtr4id ( ( 𝑊 ∈ V ∧ 𝜑 ) → ( .r𝑊 ) = ( ·𝑖𝐴 ) )
14 5 str0 ∅ = ( ·𝑖 ‘ ∅ )
15 fvprc ( ¬ 𝑊 ∈ V → ( .r𝑊 ) = ∅ )
16 15 adantr ( ( ¬ 𝑊 ∈ V ∧ 𝜑 ) → ( .r𝑊 ) = ∅ )
17 fv2prc ( ¬ 𝑊 ∈ V → ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) = ∅ )
18 1 17 sylan9eqr ( ( ¬ 𝑊 ∈ V ∧ 𝜑 ) → 𝐴 = ∅ )
19 18 fveq2d ( ( ¬ 𝑊 ∈ V ∧ 𝜑 ) → ( ·𝑖𝐴 ) = ( ·𝑖 ‘ ∅ ) )
20 14 16 19 3eqtr4a ( ( ¬ 𝑊 ∈ V ∧ 𝜑 ) → ( .r𝑊 ) = ( ·𝑖𝐴 ) )
21 13 20 pm2.61ian ( 𝜑 → ( .r𝑊 ) = ( ·𝑖𝐴 ) )