Metamath Proof Explorer


Theorem resmgmhm2

Description: One direction of resmgmhm2b . (Contributed by AV, 26-Feb-2020)

Ref Expression
Hypothesis resmgmhm2.u 𝑈 = ( 𝑇s 𝑋 )
Assertion resmgmhm2 ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) → 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) )

Proof

Step Hyp Ref Expression
1 resmgmhm2.u 𝑈 = ( 𝑇s 𝑋 )
2 mgmhmrcl ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) → ( 𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm ) )
3 2 simpld ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) → 𝑆 ∈ Mgm )
4 submgmrcl ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) → 𝑇 ∈ Mgm )
5 3 4 anim12i ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) → ( 𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm ) )
6 eqid ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
7 eqid ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
8 6 7 mgmhmf ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑈 ) )
9 1 submgmbas ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) → 𝑋 = ( Base ‘ 𝑈 ) )
10 eqid ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 )
11 10 submgmss ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) → 𝑋 ⊆ ( Base ‘ 𝑇 ) )
12 9 11 eqsstrrd ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) → ( Base ‘ 𝑈 ) ⊆ ( Base ‘ 𝑇 ) )
13 fss ( ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑈 ) ∧ ( Base ‘ 𝑈 ) ⊆ ( Base ‘ 𝑇 ) ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) )
14 8 12 13 syl2an ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) )
15 eqid ( +g𝑆 ) = ( +g𝑆 )
16 eqid ( +g𝑈 ) = ( +g𝑈 )
17 6 15 16 mgmhmlin ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝐹 ‘ ( 𝑥 ( +g𝑆 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( +g𝑈 ) ( 𝐹𝑦 ) ) )
18 17 3expb ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( +g𝑆 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( +g𝑈 ) ( 𝐹𝑦 ) ) )
19 18 adantlr ( ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( +g𝑆 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( +g𝑈 ) ( 𝐹𝑦 ) ) )
20 eqid ( +g𝑇 ) = ( +g𝑇 )
21 1 20 ressplusg ( 𝑋 ∈ ( SubMgm ‘ 𝑇 ) → ( +g𝑇 ) = ( +g𝑈 ) )
22 21 ad2antlr ( ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( +g𝑇 ) = ( +g𝑈 ) )
23 22 oveqd ( ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( ( 𝐹𝑥 ) ( +g𝑇 ) ( 𝐹𝑦 ) ) = ( ( 𝐹𝑥 ) ( +g𝑈 ) ( 𝐹𝑦 ) ) )
24 19 23 eqtr4d ( ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( +g𝑆 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( +g𝑇 ) ( 𝐹𝑦 ) ) )
25 24 ralrimivva ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑥 ( +g𝑆 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( +g𝑇 ) ( 𝐹𝑦 ) ) )
26 14 25 jca ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) → ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑥 ( +g𝑆 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( +g𝑇 ) ( 𝐹𝑦 ) ) ) )
27 6 10 15 20 ismgmhm ( 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) ↔ ( ( 𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm ) ∧ ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( 𝐹 ‘ ( 𝑥 ( +g𝑆 ) 𝑦 ) ) = ( ( 𝐹𝑥 ) ( +g𝑇 ) ( 𝐹𝑦 ) ) ) ) )
28 5 26 27 sylanbrc ( ( 𝐹 ∈ ( 𝑆 MgmHom 𝑈 ) ∧ 𝑋 ∈ ( SubMgm ‘ 𝑇 ) ) → 𝐹 ∈ ( 𝑆 MgmHom 𝑇 ) )