Step | Hyp | Ref
| Expression |
1 | | simpr 487 |
. . 3
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑁:𝑋⟶𝐴) |
2 | 1 | feqmptd 6733 |
. 2
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑁‘𝑥))) |
3 | | tngnm.x |
. . . . . . . 8
⊢ 𝑋 = (Base‘𝐺) |
4 | | eqid 2821 |
. . . . . . . 8
⊢
(-g‘𝐺) = (-g‘𝐺) |
5 | 3, 4 | grpsubf 18178 |
. . . . . . 7
⊢ (𝐺 ∈ Grp →
(-g‘𝐺):(𝑋 × 𝑋)⟶𝑋) |
6 | 5 | ad2antrr 724 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → (-g‘𝐺):(𝑋 × 𝑋)⟶𝑋) |
7 | | simpr 487 |
. . . . . . 7
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
8 | | eqid 2821 |
. . . . . . . . 9
⊢
(0g‘𝐺) = (0g‘𝐺) |
9 | 3, 8 | grpidcl 18131 |
. . . . . . . 8
⊢ (𝐺 ∈ Grp →
(0g‘𝐺)
∈ 𝑋) |
10 | 9 | ad2antrr 724 |
. . . . . . 7
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → (0g‘𝐺) ∈ 𝑋) |
11 | 7, 10 | opelxpd 5593 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → 〈𝑥, (0g‘𝐺)〉 ∈ (𝑋 × 𝑋)) |
12 | | fvco3 6760 |
. . . . . 6
⊢
(((-g‘𝐺):(𝑋 × 𝑋)⟶𝑋 ∧ 〈𝑥, (0g‘𝐺)〉 ∈ (𝑋 × 𝑋)) → ((𝑁 ∘ (-g‘𝐺))‘〈𝑥, (0g‘𝐺)〉) = (𝑁‘((-g‘𝐺)‘〈𝑥, (0g‘𝐺)〉))) |
13 | 6, 11, 12 | syl2anc 586 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → ((𝑁 ∘ (-g‘𝐺))‘〈𝑥, (0g‘𝐺)〉) = (𝑁‘((-g‘𝐺)‘〈𝑥, (0g‘𝐺)〉))) |
14 | | df-ov 7159 |
. . . . 5
⊢ (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺)) = ((𝑁 ∘ (-g‘𝐺))‘〈𝑥, (0g‘𝐺)〉) |
15 | | df-ov 7159 |
. . . . . 6
⊢ (𝑥(-g‘𝐺)(0g‘𝐺)) = ((-g‘𝐺)‘〈𝑥, (0g‘𝐺)〉) |
16 | 15 | fveq2i 6673 |
. . . . 5
⊢ (𝑁‘(𝑥(-g‘𝐺)(0g‘𝐺))) = (𝑁‘((-g‘𝐺)‘〈𝑥, (0g‘𝐺)〉)) |
17 | 13, 14, 16 | 3eqtr4g 2881 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺)) = (𝑁‘(𝑥(-g‘𝐺)(0g‘𝐺)))) |
18 | 3, 8, 4 | grpsubid1 18184 |
. . . . . 6
⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑥(-g‘𝐺)(0g‘𝐺)) = 𝑥) |
19 | 18 | adantlr 713 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → (𝑥(-g‘𝐺)(0g‘𝐺)) = 𝑥) |
20 | 19 | fveq2d 6674 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑥(-g‘𝐺)(0g‘𝐺))) = (𝑁‘𝑥)) |
21 | 17, 20 | eqtr2d 2857 |
. . 3
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → (𝑁‘𝑥) = (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺))) |
22 | 21 | mpteq2dva 5161 |
. 2
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑁‘𝑥)) = (𝑥 ∈ 𝑋 ↦ (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺)))) |
23 | 3 | fvexi 6684 |
. . . . . . 7
⊢ 𝑋 ∈ V |
24 | | tngnm.a |
. . . . . . 7
⊢ 𝐴 ∈ V |
25 | | fex2 7638 |
. . . . . . 7
⊢ ((𝑁:𝑋⟶𝐴 ∧ 𝑋 ∈ V ∧ 𝐴 ∈ V) → 𝑁 ∈ V) |
26 | 23, 24, 25 | mp3an23 1449 |
. . . . . 6
⊢ (𝑁:𝑋⟶𝐴 → 𝑁 ∈ V) |
27 | 26 | adantl 484 |
. . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑁 ∈ V) |
28 | | tngnm.t |
. . . . . 6
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
29 | 28, 3 | tngbas 23250 |
. . . . 5
⊢ (𝑁 ∈ V → 𝑋 = (Base‘𝑇)) |
30 | 27, 29 | syl 17 |
. . . 4
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑋 = (Base‘𝑇)) |
31 | 28, 4 | tngds 23257 |
. . . . . 6
⊢ (𝑁 ∈ V → (𝑁 ∘
(-g‘𝐺)) =
(dist‘𝑇)) |
32 | 27, 31 | syl 17 |
. . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → (𝑁 ∘ (-g‘𝐺)) = (dist‘𝑇)) |
33 | | eqidd 2822 |
. . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑥 = 𝑥) |
34 | 28, 8 | tng0 23252 |
. . . . . 6
⊢ (𝑁 ∈ V →
(0g‘𝐺) =
(0g‘𝑇)) |
35 | 27, 34 | syl 17 |
. . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → (0g‘𝐺) = (0g‘𝑇)) |
36 | 32, 33, 35 | oveq123d 7177 |
. . . 4
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺)) = (𝑥(dist‘𝑇)(0g‘𝑇))) |
37 | 30, 36 | mpteq12dv 5151 |
. . 3
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺))) = (𝑥 ∈ (Base‘𝑇) ↦ (𝑥(dist‘𝑇)(0g‘𝑇)))) |
38 | | eqid 2821 |
. . . 4
⊢
(norm‘𝑇) =
(norm‘𝑇) |
39 | | eqid 2821 |
. . . 4
⊢
(Base‘𝑇) =
(Base‘𝑇) |
40 | | eqid 2821 |
. . . 4
⊢
(0g‘𝑇) = (0g‘𝑇) |
41 | | eqid 2821 |
. . . 4
⊢
(dist‘𝑇) =
(dist‘𝑇) |
42 | 38, 39, 40, 41 | nmfval 23198 |
. . 3
⊢
(norm‘𝑇) =
(𝑥 ∈ (Base‘𝑇) ↦ (𝑥(dist‘𝑇)(0g‘𝑇))) |
43 | 37, 42 | syl6eqr 2874 |
. 2
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺))) = (norm‘𝑇)) |
44 | 2, 22, 43 | 3eqtrd 2860 |
1
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑁 = (norm‘𝑇)) |