Theorem rnghmrcl

Description: Reverse closure of a non-unital ring homomorphism. (Contributed by AV, 22-Feb-2020)

	Ref	Expression
Assertion	rnghmrcl	⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) )

Proof

Step	Hyp	Ref	Expression
1		df-rnghomo	⊢ RngHomo = ( 𝑟 ∈ Rng , 𝑠 ∈ Rng ↦ ⦋ ( Base ‘ 𝑟 ) / 𝑣 ⦌ ⦋ ( Base ‘ 𝑠 ) / 𝑤 ⦌ { 𝑓 ∈ ( 𝑤 ↑m 𝑣 ) ∣ ∀ 𝑥 ∈ 𝑣 ∀ 𝑦 ∈ 𝑣 ( ( 𝑓 ‘ ( 𝑥 ( +g ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ∧ ( 𝑓 ‘ ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ 𝑠 ) ( 𝑓 ‘ 𝑦 ) ) ) } )
2	1	elmpocl	⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → ( 𝑅 ∈ Rng ∧ 𝑆 ∈ Rng ) )