Metamath Proof Explorer

Theorem isghmd

Description: Deduction for a group homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015)

Ref Expression
Hypotheses isghmd.x âŠĒ 𝑋 = ( Base ‘ 𝑆 )
isghmd.y âŠĒ 𝑌 = ( Base ‘ 𝑇 )
isghmd.a âŠĒ + = ( +g ‘ 𝑆 )
isghmd.b âŠĒ âĻĢ = ( +g ‘ 𝑇 )
isghmd.s âŠĒ ( 𝜑 → 𝑆 ∈ Grp )
isghmd.t âŠĒ ( 𝜑 → 𝑇 ∈ Grp )
isghmd.f âŠĒ ( 𝜑 → ðđ : 𝑋 âŸķ 𝑌 )
isghmd.l âŠĒ ( ( 𝜑 ∧ ( ð‘Ĩ ∈ 𝑋 ∧ ð‘Ķ ∈ 𝑋 ) ) → ( ðđ ‘ ( ð‘Ĩ + ð‘Ķ ) ) = ( ( ðđ ‘ ð‘Ĩ ) âĻĢ ( ðđ ‘ ð‘Ķ ) ) )
Assertion isghmd ( 𝜑 → ðđ ∈ ( 𝑆 GrpHom 𝑇 ) )

Proof

Step Hyp Ref Expression
1 isghmd.x âŠĒ 𝑋 = ( Base ‘ 𝑆 )
2 isghmd.y âŠĒ 𝑌 = ( Base ‘ 𝑇 )
3 isghmd.a âŠĒ + = ( +g ‘ 𝑆 )
4 isghmd.b âŠĒ âĻĢ = ( +g ‘ 𝑇 )
5 isghmd.s âŠĒ ( 𝜑 → 𝑆 ∈ Grp )
6 isghmd.t âŠĒ ( 𝜑 → 𝑇 ∈ Grp )
7 isghmd.f âŠĒ ( 𝜑 → ðđ : 𝑋 âŸķ 𝑌 )
8 isghmd.l âŠĒ ( ( 𝜑 ∧ ( ð‘Ĩ ∈ 𝑋 ∧ ð‘Ķ ∈ 𝑋 ) ) → ( ðđ ‘ ( ð‘Ĩ + ð‘Ķ ) ) = ( ( ðđ ‘ ð‘Ĩ ) âĻĢ ( ðđ ‘ ð‘Ķ ) ) )
9 8 ralrimivva âŠĒ ( 𝜑 → ∀ ð‘Ĩ ∈ 𝑋 ∀ ð‘Ķ ∈ 𝑋 ( ðđ ‘ ( ð‘Ĩ + ð‘Ķ ) ) = ( ( ðđ ‘ ð‘Ĩ ) âĻĢ ( ðđ ‘ ð‘Ķ ) ) )
10 7 9 jca âŠĒ ( 𝜑 → ( ðđ : 𝑋 âŸķ 𝑌 ∧ ∀ ð‘Ĩ ∈ 𝑋 ∀ ð‘Ķ ∈ 𝑋 ( ðđ ‘ ( ð‘Ĩ + ð‘Ķ ) ) = ( ( ðđ ‘ ð‘Ĩ ) âĻĢ ( ðđ ‘ ð‘Ķ ) ) ) )
11 1 2 3 4 isghm âŠĒ ( ðđ ∈ ( 𝑆 GrpHom 𝑇 ) ↔ ( ( 𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ) ∧ ( ðđ : 𝑋 âŸķ 𝑌 ∧ ∀ ð‘Ĩ ∈ 𝑋 ∀ ð‘Ķ ∈ 𝑋 ( ðđ ‘ ( ð‘Ĩ + ð‘Ķ ) ) = ( ( ðđ ‘ ð‘Ĩ ) âĻĢ ( ðđ ‘ ð‘Ķ ) ) ) ) )
12 5 6 10 11 syl21anbrc âŠĒ ( 𝜑 → ðđ ∈ ( 𝑆 GrpHom 𝑇 ) )