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Metamath Proof Explorer

Theorem resvplusg

Description: +g 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 )
resvplusg.2 ⊢ + = ( +g ‘ 𝐺 )
Assertion resvplusg ( 𝐴 ∈ 𝑉 → + = ( +g ‘ 𝐻 ) )

Proof

Step Hyp Ref Expression
1 resvbas.1 ⊢ 𝐻 = ( 𝐺 ↾v 𝐴 )
2 resvplusg.2 ⊢ + = ( +g ‘ 𝐺 )
3 plusgid ⊢ +g = Slot ( +g ‘ ndx )
4 scandxnplusgndx ⊢ ( Scalar ‘ ndx ) ≠ ( +g ‘ ndx )
5 4 necomi ⊢ ( +g ‘ ndx ) ≠ ( Scalar ‘ ndx )
6 1 2 3 5 resvlem ⊢ ( 𝐴 ∈ 𝑉 → + = ( +g ‘ 𝐻 ) )