Theorem imasringf1

Description: The image of a ring under an injection is a ring ( imasmndf1 analog). (Contributed by AV, 27-Feb-2025)

	Ref	Expression
Hypotheses	imasringf1.u	⊢ 𝑈 = ( 𝐹 “s 𝑅 )
	imasringf1.v	⊢ 𝑉 = ( Base ‘ 𝑅 )
Assertion	imasringf1	⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Ring ) → 𝑈 ∈ Ring )

Proof

Step	Hyp	Ref	Expression
1		imasringf1.u	⊢ 𝑈 = ( 𝐹 “s 𝑅 )
2		imasringf1.v	⊢ 𝑉 = ( Base ‘ 𝑅 )
3	1	a1i	⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Ring ) → 𝑈 = ( 𝐹 “s 𝑅 ) )
4	2	a1i	⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Ring ) → 𝑉 = ( Base ‘ 𝑅 ) )
5		eqid	⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
6		eqid	⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
7		eqid	⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
8		f1f1orn	⊢ ( 𝐹 : 𝑉 –1-1→ 𝐵 → 𝐹 : 𝑉 –1-1-onto→ ran 𝐹 )
9	8	adantr	⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Ring ) → 𝐹 : 𝑉 –1-1-onto→ ran 𝐹 )
10		f1ofo	⊢ ( 𝐹 : 𝑉 –1-1-onto→ ran 𝐹 → 𝐹 : 𝑉 –onto→ ran 𝐹 )
11	9 10	syl	⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Ring ) → 𝐹 : 𝑉 –onto→ ran 𝐹 )
12	9	f1ocpbl	⊢ ( ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Ring ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉 ) ) → ( ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑝 ) ∧ ( 𝐹 ‘ 𝑏 ) = ( 𝐹 ‘ 𝑞 ) ) → ( 𝐹 ‘ ( 𝑎 ( +g ‘ 𝑅 ) 𝑏 ) ) = ( 𝐹 ‘ ( 𝑝 ( +g ‘ 𝑅 ) 𝑞 ) ) ) )
13	9	f1ocpbl	⊢ ( ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Ring ) ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ ( 𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉 ) ) → ( ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑝 ) ∧ ( 𝐹 ‘ 𝑏 ) = ( 𝐹 ‘ 𝑞 ) ) → ( 𝐹 ‘ ( 𝑎 ( .r ‘ 𝑅 ) 𝑏 ) ) = ( 𝐹 ‘ ( 𝑝 ( .r ‘ 𝑅 ) 𝑞 ) ) ) )
14		simpr	⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Ring ) → 𝑅 ∈ Ring )
15	3 4 5 6 7 11 12 13 14	imasring	⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Ring ) → ( 𝑈 ∈ Ring ∧ ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑈 ) ) )
16	15	simpld	⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Ring ) → 𝑈 ∈ Ring )