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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 ⊢ ( 𝜑 → 𝐴 = ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) )
srapart.s ⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝑊 ) )
Assertion sraip ( 𝜑 → ( .r ‘ 𝑊 ) = ( ·𝑖 ‘ 𝐴 ) )

Proof

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