Metamath Proof Explorer

Theorem isghm3

Description: Property of a group homomorphism, similar to ismhm . (Contributed by Mario Carneiro, 7-Mar-2015)

Ref Expression
Hypotheses isghm.w âŠĒ 𝑋 = ( Base ‘ 𝑆 )
isghm.x âŠĒ 𝑌 = ( Base ‘ 𝑇 )
isghm.a âŠĒ + = ( +g ‘ 𝑆 )
isghm.b âŠĒ âĻĢ = ( +g ‘ 𝑇 )
Assertion isghm3 ( ( 𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ) → ( ðđ ∈ ( 𝑆 GrpHom 𝑇 ) ↔ ( ðđ : 𝑋 âŸķ 𝑌 ∧ ∀ ð‘Ē ∈ 𝑋 ∀ ð‘Ģ ∈ 𝑋 ( ðđ ‘ ( ð‘Ē + ð‘Ģ ) ) = ( ( ðđ ‘ ð‘Ē ) âĻĢ ( ðđ ‘ ð‘Ģ ) ) ) ) )

Proof

Step Hyp Ref Expression
1 isghm.w âŠĒ 𝑋 = ( Base ‘ 𝑆 )
2 isghm.x âŠĒ 𝑌 = ( Base ‘ 𝑇 )
3 isghm.a âŠĒ + = ( +g ‘ 𝑆 )
4 isghm.b âŠĒ âĻĢ = ( +g ‘ 𝑇 )
5 1 2 3 4 isghm âŠĒ ( ðđ ∈ ( 𝑆 GrpHom 𝑇 ) ↔ ( ( 𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ) ∧ ( ðđ : 𝑋 âŸķ 𝑌 ∧ ∀ ð‘Ē ∈ 𝑋 ∀ ð‘Ģ ∈ 𝑋 ( ðđ ‘ ( ð‘Ē + ð‘Ģ ) ) = ( ( ðđ ‘ ð‘Ē ) âĻĢ ( ðđ ‘ ð‘Ģ ) ) ) ) )
6 5 baib âŠĒ ( ( 𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ) → ( ðđ ∈ ( 𝑆 GrpHom 𝑇 ) ↔ ( ðđ : 𝑋 âŸķ 𝑌 ∧ ∀ ð‘Ē ∈ 𝑋 ∀ ð‘Ģ ∈ 𝑋 ( ðđ ‘ ( ð‘Ē + ð‘Ģ ) ) = ( ( ðđ ‘ ð‘Ē ) âĻĢ ( ðđ ‘ ð‘Ģ ) ) ) ) )