Step | Hyp | Ref
| Expression |
1 | | simp1 1061 |
. . 3
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
2 | | nsgsubg 17626 |
. . . 4
⊢ (𝑈 ∈ (NrmSGrp‘𝑆) → 𝑈 ∈ (SubGrp‘𝑆)) |
3 | 2 | 3ad2ant2 1083 |
. . 3
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝑈 ∈ (SubGrp‘𝑆)) |
4 | | ghmima 17681 |
. . 3
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑆)) → (𝐹 “ 𝑈) ∈ (SubGrp‘𝑇)) |
5 | 1, 3, 4 | syl2anc 693 |
. 2
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹 “ 𝑈) ∈ (SubGrp‘𝑇)) |
6 | 1 | adantr 481 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
7 | | ghmgrp1 17662 |
. . . . . . . . 9
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp) |
8 | 6, 7 | syl 17 |
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑆 ∈ Grp) |
9 | | simprl 794 |
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑧 ∈ (Base‘𝑆)) |
10 | | eqid 2622 |
. . . . . . . . . . . 12
⊢
(Base‘𝑆) =
(Base‘𝑆) |
11 | 10 | subgss 17595 |
. . . . . . . . . . 11
⊢ (𝑈 ∈ (SubGrp‘𝑆) → 𝑈 ⊆ (Base‘𝑆)) |
12 | 3, 11 | syl 17 |
. . . . . . . . . 10
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝑈 ⊆ (Base‘𝑆)) |
13 | 12 | adantr 481 |
. . . . . . . . 9
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑈 ⊆ (Base‘𝑆)) |
14 | | simprr 796 |
. . . . . . . . 9
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑥 ∈ 𝑈) |
15 | 13, 14 | sseldd 3604 |
. . . . . . . 8
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑥 ∈ (Base‘𝑆)) |
16 | | eqid 2622 |
. . . . . . . . 9
⊢
(+g‘𝑆) = (+g‘𝑆) |
17 | 10, 16 | grpcl 17430 |
. . . . . . . 8
⊢ ((𝑆 ∈ Grp ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝑧(+g‘𝑆)𝑥) ∈ (Base‘𝑆)) |
18 | 8, 9, 15, 17 | syl3anc 1326 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → (𝑧(+g‘𝑆)𝑥) ∈ (Base‘𝑆)) |
19 | | eqid 2622 |
. . . . . . . 8
⊢
(-g‘𝑆) = (-g‘𝑆) |
20 | | eqid 2622 |
. . . . . . . 8
⊢
(-g‘𝑇) = (-g‘𝑇) |
21 | 10, 19, 20 | ghmsub 17668 |
. . . . . . 7
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑧(+g‘𝑆)𝑥) ∈ (Base‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆)) → (𝐹‘((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧)) = ((𝐹‘(𝑧(+g‘𝑆)𝑥))(-g‘𝑇)(𝐹‘𝑧))) |
22 | 6, 18, 9, 21 | syl3anc 1326 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → (𝐹‘((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧)) = ((𝐹‘(𝑧(+g‘𝑆)𝑥))(-g‘𝑇)(𝐹‘𝑧))) |
23 | | eqid 2622 |
. . . . . . . . 9
⊢
(+g‘𝑇) = (+g‘𝑇) |
24 | 10, 16, 23 | ghmlin 17665 |
. . . . . . . 8
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ (Base‘𝑆)) → (𝐹‘(𝑧(+g‘𝑆)𝑥)) = ((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))) |
25 | 6, 9, 15, 24 | syl3anc 1326 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → (𝐹‘(𝑧(+g‘𝑆)𝑥)) = ((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))) |
26 | 25 | oveq1d 6665 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → ((𝐹‘(𝑧(+g‘𝑆)𝑥))(-g‘𝑇)(𝐹‘𝑧)) = (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧))) |
27 | 22, 26 | eqtrd 2656 |
. . . . 5
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → (𝐹‘((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧)) = (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧))) |
28 | | ghmnsgima.1 |
. . . . . . . . . 10
⊢ 𝑌 = (Base‘𝑇) |
29 | 10, 28 | ghmf 17664 |
. . . . . . . . 9
⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶𝑌) |
30 | 1, 29 | syl 17 |
. . . . . . . 8
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹:(Base‘𝑆)⟶𝑌) |
31 | 30 | adantr 481 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝐹:(Base‘𝑆)⟶𝑌) |
32 | | ffn 6045 |
. . . . . . 7
⊢ (𝐹:(Base‘𝑆)⟶𝑌 → 𝐹 Fn (Base‘𝑆)) |
33 | 31, 32 | syl 17 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝐹 Fn (Base‘𝑆)) |
34 | | simpl2 1065 |
. . . . . . 7
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → 𝑈 ∈ (NrmSGrp‘𝑆)) |
35 | 10, 16, 19 | nsgconj 17627 |
. . . . . . 7
⊢ ((𝑈 ∈ (NrmSGrp‘𝑆) ∧ 𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈) → ((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧) ∈ 𝑈) |
36 | 34, 9, 14, 35 | syl3anc 1326 |
. . . . . 6
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → ((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧) ∈ 𝑈) |
37 | | fnfvima 6496 |
. . . . . 6
⊢ ((𝐹 Fn (Base‘𝑆) ∧ 𝑈 ⊆ (Base‘𝑆) ∧ ((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧) ∈ 𝑈) → (𝐹‘((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧)) ∈ (𝐹 “ 𝑈)) |
38 | 33, 13, 36, 37 | syl3anc 1326 |
. . . . 5
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → (𝐹‘((𝑧(+g‘𝑆)𝑥)(-g‘𝑆)𝑧)) ∈ (𝐹 “ 𝑈)) |
39 | 27, 38 | eqeltrrd 2702 |
. . . 4
⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) ∧ (𝑧 ∈ (Base‘𝑆) ∧ 𝑥 ∈ 𝑈)) → (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈)) |
40 | 39 | ralrimivva 2971 |
. . 3
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ∀𝑧 ∈ (Base‘𝑆)∀𝑥 ∈ 𝑈 (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈)) |
41 | 30, 32 | syl 17 |
. . . . 5
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → 𝐹 Fn (Base‘𝑆)) |
42 | | oveq1 6657 |
. . . . . . . . 9
⊢ (𝑥 = (𝐹‘𝑧) → (𝑥(+g‘𝑇)𝑦) = ((𝐹‘𝑧)(+g‘𝑇)𝑦)) |
43 | | id 22 |
. . . . . . . . 9
⊢ (𝑥 = (𝐹‘𝑧) → 𝑥 = (𝐹‘𝑧)) |
44 | 42, 43 | oveq12d 6668 |
. . . . . . . 8
⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) = (((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧))) |
45 | 44 | eleq1d 2686 |
. . . . . . 7
⊢ (𝑥 = (𝐹‘𝑧) → (((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈) ↔ (((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈))) |
46 | 45 | ralbidv 2986 |
. . . . . 6
⊢ (𝑥 = (𝐹‘𝑧) → (∀𝑦 ∈ (𝐹 “ 𝑈)((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈) ↔ ∀𝑦 ∈ (𝐹 “ 𝑈)(((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈))) |
47 | 46 | ralrn 6362 |
. . . . 5
⊢ (𝐹 Fn (Base‘𝑆) → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ (𝐹 “ 𝑈)((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈) ↔ ∀𝑧 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹 “ 𝑈)(((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈))) |
48 | 41, 47 | syl 17 |
. . . 4
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ (𝐹 “ 𝑈)((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈) ↔ ∀𝑧 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹 “ 𝑈)(((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈))) |
49 | | simp3 1063 |
. . . . 5
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ran 𝐹 = 𝑌) |
50 | 49 | raleqdv 3144 |
. . . 4
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ (𝐹 “ 𝑈)((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈) ↔ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ (𝐹 “ 𝑈)((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈))) |
51 | | oveq2 6658 |
. . . . . . . . 9
⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑧)(+g‘𝑇)𝑦) = ((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))) |
52 | 51 | oveq1d 6665 |
. . . . . . . 8
⊢ (𝑦 = (𝐹‘𝑥) → (((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) = (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧))) |
53 | 52 | eleq1d 2686 |
. . . . . . 7
⊢ (𝑦 = (𝐹‘𝑥) → ((((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈) ↔ (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈))) |
54 | 53 | ralima 6498 |
. . . . . 6
⊢ ((𝐹 Fn (Base‘𝑆) ∧ 𝑈 ⊆ (Base‘𝑆)) → (∀𝑦 ∈ (𝐹 “ 𝑈)(((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈) ↔ ∀𝑥 ∈ 𝑈 (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈))) |
55 | 41, 12, 54 | syl2anc 693 |
. . . . 5
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (∀𝑦 ∈ (𝐹 “ 𝑈)(((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈) ↔ ∀𝑥 ∈ 𝑈 (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈))) |
56 | 55 | ralbidv 2986 |
. . . 4
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (∀𝑧 ∈ (Base‘𝑆)∀𝑦 ∈ (𝐹 “ 𝑈)(((𝐹‘𝑧)(+g‘𝑇)𝑦)(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈) ↔ ∀𝑧 ∈ (Base‘𝑆)∀𝑥 ∈ 𝑈 (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈))) |
57 | 48, 50, 56 | 3bitr3d 298 |
. . 3
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (∀𝑥 ∈ 𝑌 ∀𝑦 ∈ (𝐹 “ 𝑈)((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈) ↔ ∀𝑧 ∈ (Base‘𝑆)∀𝑥 ∈ 𝑈 (((𝐹‘𝑧)(+g‘𝑇)(𝐹‘𝑥))(-g‘𝑇)(𝐹‘𝑧)) ∈ (𝐹 “ 𝑈))) |
58 | 40, 57 | mpbird 247 |
. 2
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ (𝐹 “ 𝑈)((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈)) |
59 | 28, 23, 20 | isnsg3 17628 |
. 2
⊢ ((𝐹 “ 𝑈) ∈ (NrmSGrp‘𝑇) ↔ ((𝐹 “ 𝑈) ∈ (SubGrp‘𝑇) ∧ ∀𝑥 ∈ 𝑌 ∀𝑦 ∈ (𝐹 “ 𝑈)((𝑥(+g‘𝑇)𝑦)(-g‘𝑇)𝑥) ∈ (𝐹 “ 𝑈))) |
60 | 5, 58, 59 | sylanbrc 698 |
1
⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (NrmSGrp‘𝑆) ∧ ran 𝐹 = 𝑌) → (𝐹 “ 𝑈) ∈ (NrmSGrp‘𝑇)) |