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Metamath Proof Explorer

Theorem rhmresel

Description: An element of the unital ring homomorphisms restricted to a subset of unital rings is a unital ring homomorphism. (Contributed by AV, 10-Mar-2020)

Ref Expression
Hypothesis rhmresel.h ⊢ ( 𝜑 → 𝐻 = ( RingHom ↾ ( 𝐵 × 𝐵 ) ) )
Assertion rhmresel ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ) → 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) )

Proof

Step Hyp Ref Expression
1 rhmresel.h ⊢ ( 𝜑 → 𝐻 = ( RingHom ↾ ( 𝐵 × 𝐵 ) ) )
2 1 adantr ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐻 = ( RingHom ↾ ( 𝐵 × 𝐵 ) ) )
3 2 oveqd ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 ( RingHom ↾ ( 𝐵 × 𝐵 ) ) 𝑌 ) )
4 ovres ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ( RingHom ↾ ( 𝐵 × 𝐵 ) ) 𝑌 ) = ( 𝑋 RingHom 𝑌 ) )
5 4 adantl ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( RingHom ↾ ( 𝐵 × 𝐵 ) ) 𝑌 ) = ( 𝑋 RingHom 𝑌 ) )
6 3 5 eqtrd ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 RingHom 𝑌 ) )
7 6 eleq2d ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ↔ 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) ) )
8 7 biimp3a ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ) → 𝐹 ∈ ( 𝑋 RingHom 𝑌 ) )