Metamath Proof Explorer


Theorem ismhmd

Description: Deduction version of ismhm . (Contributed by SN, 27-Jul-2024)

Ref Expression
Hypotheses ismhmd.b 𝐵 = ( Base ‘ 𝑆 )
ismhmd.c 𝐶 = ( Base ‘ 𝑇 )
ismhmd.p + = ( +g𝑆 )
ismhmd.q = ( +g𝑇 )
ismhmd.0 0 = ( 0g𝑆 )
ismhmd.z 𝑍 = ( 0g𝑇 )
ismhmd.s ( 𝜑𝑆 ∈ Mnd )
ismhmd.t ( 𝜑𝑇 ∈ Mnd )
ismhmd.f ( 𝜑𝐹 : 𝐵𝐶 )
ismhmd.a ( ( 𝜑 ∧ ( 𝑥𝐵𝑦𝐵 ) ) → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹𝑥 ) ( 𝐹𝑦 ) ) )
ismhmd.h ( 𝜑 → ( 𝐹0 ) = 𝑍 )
Assertion ismhmd ( 𝜑𝐹 ∈ ( 𝑆 MndHom 𝑇 ) )

Proof

Step Hyp Ref Expression
1 ismhmd.b 𝐵 = ( Base ‘ 𝑆 )
2 ismhmd.c 𝐶 = ( Base ‘ 𝑇 )
3 ismhmd.p + = ( +g𝑆 )
4 ismhmd.q = ( +g𝑇 )
5 ismhmd.0 0 = ( 0g𝑆 )
6 ismhmd.z 𝑍 = ( 0g𝑇 )
7 ismhmd.s ( 𝜑𝑆 ∈ Mnd )
8 ismhmd.t ( 𝜑𝑇 ∈ Mnd )
9 ismhmd.f ( 𝜑𝐹 : 𝐵𝐶 )
10 ismhmd.a ( ( 𝜑 ∧ ( 𝑥𝐵𝑦𝐵 ) ) → ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹𝑥 ) ( 𝐹𝑦 ) ) )
11 ismhmd.h ( 𝜑 → ( 𝐹0 ) = 𝑍 )
12 10 ralrimivva ( 𝜑 → ∀ 𝑥𝐵𝑦𝐵 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹𝑥 ) ( 𝐹𝑦 ) ) )
13 9 12 11 3jca ( 𝜑 → ( 𝐹 : 𝐵𝐶 ∧ ∀ 𝑥𝐵𝑦𝐵 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹𝑥 ) ( 𝐹𝑦 ) ) ∧ ( 𝐹0 ) = 𝑍 ) )
14 1 2 3 4 5 6 ismhm ( 𝐹 ∈ ( 𝑆 MndHom 𝑇 ) ↔ ( ( 𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ) ∧ ( 𝐹 : 𝐵𝐶 ∧ ∀ 𝑥𝐵𝑦𝐵 ( 𝐹 ‘ ( 𝑥 + 𝑦 ) ) = ( ( 𝐹𝑥 ) ( 𝐹𝑦 ) ) ∧ ( 𝐹0 ) = 𝑍 ) ) )
15 7 8 13 14 syl21anbrc ( 𝜑𝐹 ∈ ( 𝑆 MndHom 𝑇 ) )