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Metamath Proof Explorer

Theorem ghmnsgpreima

Description: The inverse image of a normal subgroup under a homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015)

Ref Expression
Assertion ghmnsgpreima ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) → ( ◡ 𝐹 “ 𝑉 ) ∈ ( NrmSGrp ‘ 𝑆 ) )

Proof

Step Hyp Ref Expression
1 nsgsubg ⊢ ( 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) → 𝑉 ∈ ( SubGrp ‘ 𝑇 ) )
2 ghmpreima ⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( SubGrp ‘ 𝑇 ) ) → ( ◡ 𝐹 “ 𝑉 ) ∈ ( SubGrp ‘ 𝑆 ) )
3 1 2 sylan2 ⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) → ( ◡ 𝐹 “ 𝑉 ) ∈ ( SubGrp ‘ 𝑆 ) )
4 ghmgrp1 ⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝑆 ∈ Grp )
5 4 ad2antrr ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → 𝑆 ∈ Grp )
6 simprl ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → 𝑥 ∈ ( Base ‘ 𝑆 ) )
7 simprr ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) )
8 simpll ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) )
9 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
10 eqid ⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 )
11 9 10 ghmf ⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) )
12 8 11 syl ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) )
13 12 ffnd ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → 𝐹 Fn ( Base ‘ 𝑆 ) )
14 elpreima ⊢ ( 𝐹 Fn ( Base ‘ 𝑆 ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ↔ ( 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑉 ) ) )
15 13 14 syl ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ↔ ( 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑉 ) ) )
16 7 15 mpbid ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → ( 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑉 ) )
17 16 simpld ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → 𝑦 ∈ ( Base ‘ 𝑆 ) )
18 eqid ⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 )
19 9 18 grpcl ⊢ ( ( 𝑆 ∈ Grp ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ∈ ( Base ‘ 𝑆 ) )
20 5 6 17 19 syl3anc ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ∈ ( Base ‘ 𝑆 ) )
21 eqid ⊢ ( -g ‘ 𝑆 ) = ( -g ‘ 𝑆 )
22 9 21 grpsubcl ⊢ ( ( 𝑆 ∈ Grp ∧ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( -g ‘ 𝑆 ) 𝑥 ) ∈ ( Base ‘ 𝑆 ) )
23 5 20 6 22 syl3anc ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( -g ‘ 𝑆 ) 𝑥 ) ∈ ( Base ‘ 𝑆 ) )
24 eqid ⊢ ( -g ‘ 𝑇 ) = ( -g ‘ 𝑇 )
25 9 21 24 ghmsub ⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) → ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( -g ‘ 𝑆 ) 𝑥 ) ) = ( ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) ( -g ‘ 𝑇 ) ( 𝐹 ‘ 𝑥 ) ) )
26 8 20 6 25 syl3anc ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( -g ‘ 𝑆 ) 𝑥 ) ) = ( ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) ( -g ‘ 𝑇 ) ( 𝐹 ‘ 𝑥 ) ) )
27 eqid ⊢ ( +g ‘ 𝑇 ) = ( +g ‘ 𝑇 )
28 9 18 27 ghmlin ⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) )
29 8 6 17 28 syl3anc ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) )
30 29 oveq1d ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → ( ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ) ( -g ‘ 𝑇 ) ( 𝐹 ‘ 𝑥 ) ) = ( ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ( -g ‘ 𝑇 ) ( 𝐹 ‘ 𝑥 ) ) )
31 26 30 eqtrd ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( -g ‘ 𝑆 ) 𝑥 ) ) = ( ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ( -g ‘ 𝑇 ) ( 𝐹 ‘ 𝑥 ) ) )
32 simplr ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) )
33 12 6 ffvelcdmd ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ( Base ‘ 𝑇 ) )
34 16 simprd ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑉 )
35 10 27 24 nsgconj ⊢ ( ( 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ ( Base ‘ 𝑇 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑉 ) → ( ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ( -g ‘ 𝑇 ) ( 𝐹 ‘ 𝑥 ) ) ∈ 𝑉 )
36 32 33 34 35 syl3anc ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → ( ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ( -g ‘ 𝑇 ) ( 𝐹 ‘ 𝑥 ) ) ∈ 𝑉 )
37 31 36 eqeltrd ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( -g ‘ 𝑆 ) 𝑥 ) ) ∈ 𝑉 )
38 elpreima ⊢ ( 𝐹 Fn ( Base ‘ 𝑆 ) → ( ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( -g ‘ 𝑆 ) 𝑥 ) ∈ ( ◡ 𝐹 “ 𝑉 ) ↔ ( ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( -g ‘ 𝑆 ) 𝑥 ) ∈ ( Base ‘ 𝑆 ) ∧ ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( -g ‘ 𝑆 ) 𝑥 ) ) ∈ 𝑉 ) ) )
39 13 38 syl ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → ( ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( -g ‘ 𝑆 ) 𝑥 ) ∈ ( ◡ 𝐹 “ 𝑉 ) ↔ ( ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( -g ‘ 𝑆 ) 𝑥 ) ∈ ( Base ‘ 𝑆 ) ∧ ( 𝐹 ‘ ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( -g ‘ 𝑆 ) 𝑥 ) ) ∈ 𝑉 ) ) )
40 23 37 39 mpbir2and ⊢ ( ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ) ) → ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( -g ‘ 𝑆 ) 𝑥 ) ∈ ( ◡ 𝐹 “ 𝑉 ) )
41 40 ralrimivva ⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( -g ‘ 𝑆 ) 𝑥 ) ∈ ( ◡ 𝐹 “ 𝑉 ) )
42 9 18 21 isnsg3 ⊢ ( ( ◡ 𝐹 “ 𝑉 ) ∈ ( NrmSGrp ‘ 𝑆 ) ↔ ( ( ◡ 𝐹 “ 𝑉 ) ∈ ( SubGrp ‘ 𝑆 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ∀ 𝑦 ∈ ( ◡ 𝐹 “ 𝑉 ) ( ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ( -g ‘ 𝑆 ) 𝑥 ) ∈ ( ◡ 𝐹 “ 𝑉 ) ) )
43 3 41 42 sylanbrc ⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ 𝑉 ∈ ( NrmSGrp ‘ 𝑇 ) ) → ( ◡ 𝐹 “ 𝑉 ) ∈ ( NrmSGrp ‘ 𝑆 ) )