Step | Hyp | Ref
| Expression |
1 | | simpr 476 |
. . 3
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑁:𝑋⟶𝐴) |
2 | 1 | feqmptd 6159 |
. 2
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑁‘𝑥))) |
3 | | tngnm.x |
. . . . . . . 8
⊢ 𝑋 = (Base‘𝐺) |
4 | | eqid 2610 |
. . . . . . . 8
⊢
(-g‘𝐺) = (-g‘𝐺) |
5 | 3, 4 | grpsubf 17317 |
. . . . . . 7
⊢ (𝐺 ∈ Grp →
(-g‘𝐺):(𝑋 × 𝑋)⟶𝑋) |
6 | 5 | ad2antrr 758 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → (-g‘𝐺):(𝑋 × 𝑋)⟶𝑋) |
7 | | simpr 476 |
. . . . . . 7
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
8 | | eqid 2610 |
. . . . . . . . 9
⊢
(0g‘𝐺) = (0g‘𝐺) |
9 | 3, 8 | grpidcl 17273 |
. . . . . . . 8
⊢ (𝐺 ∈ Grp →
(0g‘𝐺)
∈ 𝑋) |
10 | 9 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → (0g‘𝐺) ∈ 𝑋) |
11 | | opelxpi 5072 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝑋 ∧ (0g‘𝐺) ∈ 𝑋) → 〈𝑥, (0g‘𝐺)〉 ∈ (𝑋 × 𝑋)) |
12 | 7, 10, 11 | syl2anc 691 |
. . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → 〈𝑥, (0g‘𝐺)〉 ∈ (𝑋 × 𝑋)) |
13 | | fvco3 6185 |
. . . . . 6
⊢
(((-g‘𝐺):(𝑋 × 𝑋)⟶𝑋 ∧ 〈𝑥, (0g‘𝐺)〉 ∈ (𝑋 × 𝑋)) → ((𝑁 ∘ (-g‘𝐺))‘〈𝑥, (0g‘𝐺)〉) = (𝑁‘((-g‘𝐺)‘〈𝑥, (0g‘𝐺)〉))) |
14 | 6, 12, 13 | syl2anc 691 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → ((𝑁 ∘ (-g‘𝐺))‘〈𝑥, (0g‘𝐺)〉) = (𝑁‘((-g‘𝐺)‘〈𝑥, (0g‘𝐺)〉))) |
15 | | df-ov 6552 |
. . . . 5
⊢ (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺)) = ((𝑁 ∘ (-g‘𝐺))‘〈𝑥, (0g‘𝐺)〉) |
16 | | df-ov 6552 |
. . . . . 6
⊢ (𝑥(-g‘𝐺)(0g‘𝐺)) = ((-g‘𝐺)‘〈𝑥, (0g‘𝐺)〉) |
17 | 16 | fveq2i 6106 |
. . . . 5
⊢ (𝑁‘(𝑥(-g‘𝐺)(0g‘𝐺))) = (𝑁‘((-g‘𝐺)‘〈𝑥, (0g‘𝐺)〉)) |
18 | 14, 15, 17 | 3eqtr4g 2669 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺)) = (𝑁‘(𝑥(-g‘𝐺)(0g‘𝐺)))) |
19 | 3, 8, 4 | grpsubid1 17323 |
. . . . . 6
⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑥(-g‘𝐺)(0g‘𝐺)) = 𝑥) |
20 | 19 | adantlr 747 |
. . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → (𝑥(-g‘𝐺)(0g‘𝐺)) = 𝑥) |
21 | 20 | fveq2d 6107 |
. . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑥(-g‘𝐺)(0g‘𝐺))) = (𝑁‘𝑥)) |
22 | 18, 21 | eqtr2d 2645 |
. . 3
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → (𝑁‘𝑥) = (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺))) |
23 | 22 | mpteq2dva 4672 |
. 2
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑁‘𝑥)) = (𝑥 ∈ 𝑋 ↦ (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺)))) |
24 | | fvex 6113 |
. . . . . . . 8
⊢
(Base‘𝐺)
∈ V |
25 | 3, 24 | eqeltri 2684 |
. . . . . . 7
⊢ 𝑋 ∈ V |
26 | | tngnm.a |
. . . . . . 7
⊢ 𝐴 ∈ V |
27 | | fex2 7014 |
. . . . . . 7
⊢ ((𝑁:𝑋⟶𝐴 ∧ 𝑋 ∈ V ∧ 𝐴 ∈ V) → 𝑁 ∈ V) |
28 | 25, 26, 27 | mp3an23 1408 |
. . . . . 6
⊢ (𝑁:𝑋⟶𝐴 → 𝑁 ∈ V) |
29 | 28 | adantl 481 |
. . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑁 ∈ V) |
30 | | tngnm.t |
. . . . . 6
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
31 | 30, 3 | tngbas 22255 |
. . . . 5
⊢ (𝑁 ∈ V → 𝑋 = (Base‘𝑇)) |
32 | 29, 31 | syl 17 |
. . . 4
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑋 = (Base‘𝑇)) |
33 | 30, 4 | tngds 22262 |
. . . . . 6
⊢ (𝑁 ∈ V → (𝑁 ∘
(-g‘𝐺)) =
(dist‘𝑇)) |
34 | 29, 33 | syl 17 |
. . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → (𝑁 ∘ (-g‘𝐺)) = (dist‘𝑇)) |
35 | | eqidd 2611 |
. . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑥 = 𝑥) |
36 | 30, 8 | tng0 22257 |
. . . . . 6
⊢ (𝑁 ∈ V →
(0g‘𝐺) =
(0g‘𝑇)) |
37 | 29, 36 | syl 17 |
. . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → (0g‘𝐺) = (0g‘𝑇)) |
38 | 34, 35, 37 | oveq123d 6570 |
. . . 4
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺)) = (𝑥(dist‘𝑇)(0g‘𝑇))) |
39 | 32, 38 | mpteq12dv 4663 |
. . 3
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺))) = (𝑥 ∈ (Base‘𝑇) ↦ (𝑥(dist‘𝑇)(0g‘𝑇)))) |
40 | | eqid 2610 |
. . . 4
⊢
(norm‘𝑇) =
(norm‘𝑇) |
41 | | eqid 2610 |
. . . 4
⊢
(Base‘𝑇) =
(Base‘𝑇) |
42 | | eqid 2610 |
. . . 4
⊢
(0g‘𝑇) = (0g‘𝑇) |
43 | | eqid 2610 |
. . . 4
⊢
(dist‘𝑇) =
(dist‘𝑇) |
44 | 40, 41, 42, 43 | nmfval 22203 |
. . 3
⊢
(norm‘𝑇) =
(𝑥 ∈ (Base‘𝑇) ↦ (𝑥(dist‘𝑇)(0g‘𝑇))) |
45 | 39, 44 | syl6eqr 2662 |
. 2
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺))) = (norm‘𝑇)) |
46 | 2, 23, 45 | 3eqtrd 2648 |
1
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑁 = (norm‘𝑇)) |