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Metamath Proof Explorer

Theorem resvmulr

Description: .r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018) (Revised by AV, 31-Oct-2024)

Ref Expression
Hypotheses resvbas.1 ⊢ 𝐻 = ( 𝐺 ↾v 𝐴 )
resvmulr.2 ⊢ · = ( .r ‘ 𝐺 )
Assertion resvmulr ( 𝐴 ∈ 𝑉 → · = ( .r ‘ 𝐻 ) )

Proof

Step Hyp Ref Expression
1 resvbas.1 ⊢ 𝐻 = ( 𝐺 ↾v 𝐴 )
2 resvmulr.2 ⊢ · = ( .r ‘ 𝐺 )
3 mulridx ⊢ .r = Slot ( .r ‘ ndx )
4 scandxnmulrndx ⊢ ( Scalar ‘ ndx ) ≠ ( .r ‘ ndx )
5 4 necomi ⊢ ( .r ‘ ndx ) ≠ ( Scalar ‘ ndx )
6 1 2 3 5 resvlem ⊢ ( 𝐴 ∈ 𝑉 → · = ( .r ‘ 𝐻 ) )