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Metamath Proof Explorer

Theorem imasrngf1

Description: The image of a non-unital ring under an injection is a non-unital ring ( imasmndf1 analog). (Contributed by AV, 22-Feb-2025)

Ref Expression
Hypotheses imasrngf1.u ⊢ 𝑈 = ( 𝐹 “s 𝑅 )
imasrngf1.v ⊢ 𝑉 = ( Base ‘ 𝑅 )
Assertion imasrngf1 ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Rng ) → 𝑈 ∈ Rng )

Proof

Step Hyp Ref Expression
1 imasrngf1.u ⊢ 𝑈 = ( 𝐹 “s 𝑅 )
2 imasrngf1.v ⊢ 𝑉 = ( Base ‘ 𝑅 )
3 1 a1i ⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Rng ) → 𝑈 = ( 𝐹 “s 𝑅 ) )
4 2 a1i ⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Rng ) → 𝑉 = ( Base ‘ 𝑅 ) )
5 eqid ⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
6 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
7 f1f1orn ⊢ ( 𝐹 : 𝑉 –1-1→ 𝐵 → 𝐹 : 𝑉 –1-1-onto→ ran 𝐹 )
8 7 adantr ⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Rng ) → 𝐹 : 𝑉 –1-1-onto→ ran 𝐹 )
9 f1ofo ⊢ ( 𝐹 : 𝑉 –1-1-onto→ ran 𝐹 → 𝐹 : 𝑉 –onto→ ran 𝐹 )
10 8 9 syl ⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Rng ) → 𝐹 : 𝑉 –onto→ ran 𝐹 )
11 8 f1ocpbl ⊢ ( ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Rng ) ∧ ( 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 ) ) ) )
12 8 f1ocpbl ⊢ ( ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Rng ) ∧ ( 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 ) ) ) )
13 simpr ⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Rng ) → 𝑅 ∈ Rng )
14 3 4 5 6 10 11 12 13 imasrng ⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Rng ) → 𝑈 ∈ Rng )