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Metamath Proof Explorer

Theorem imasgrpf1

Description: The image of a group under an injection is a group. (Contributed by Mario Carneiro, 20-Aug-2015)

Ref Expression
Hypotheses imasgrpf1.u ⊢ 𝑈 = ( 𝐹 “s 𝑅 )
imasgrpf1.v ⊢ 𝑉 = ( Base ‘ 𝑅 )
Assertion imasgrpf1 ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Grp ) → 𝑈 ∈ Grp )

Proof

Step Hyp Ref Expression
1 imasgrpf1.u ⊢ 𝑈 = ( 𝐹 “s 𝑅 )
2 imasgrpf1.v ⊢ 𝑉 = ( Base ‘ 𝑅 )
3 1 a1i ⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Grp ) → 𝑈 = ( 𝐹 “s 𝑅 ) )
4 2 a1i ⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Grp ) → 𝑉 = ( Base ‘ 𝑅 ) )
5 eqidd ⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Grp ) → ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) )
6 f1f1orn ⊢ ( 𝐹 : 𝑉 –1-1→ 𝐵 → 𝐹 : 𝑉 –1-1-onto→ ran 𝐹 )
7 6 adantr ⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Grp ) → 𝐹 : 𝑉 –1-1-onto→ ran 𝐹 )
8 f1ofo ⊢ ( 𝐹 : 𝑉 –1-1-onto→ ran 𝐹 → 𝐹 : 𝑉 –onto→ ran 𝐹 )
9 7 8 syl ⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Grp ) → 𝐹 : 𝑉 –onto→ ran 𝐹 )
10 7 f1ocpbl ⊢ ( ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Grp ) ∧ ( 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 ) ) ) )
11 simpr ⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Grp ) → 𝑅 ∈ Grp )
12 eqid ⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
13 3 4 5 9 10 11 12 imasgrp ⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Grp ) → ( 𝑈 ∈ Grp ∧ ( 𝐹 ‘ ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑈 ) ) )
14 13 simpld ⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝐵 ∧ 𝑅 ∈ Grp ) → 𝑈 ∈ Grp )