| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpr 484 | . . 3
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑁:𝑋⟶𝐴) | 
| 2 | 1 | feqmptd 6977 | . 2
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑁‘𝑥))) | 
| 3 |  | tngnm.x | . . . . . . . 8
⊢ 𝑋 = (Base‘𝐺) | 
| 4 |  | eqid 2737 | . . . . . . . 8
⊢
(-g‘𝐺) = (-g‘𝐺) | 
| 5 | 3, 4 | grpsubf 19037 | . . . . . . 7
⊢ (𝐺 ∈ Grp →
(-g‘𝐺):(𝑋 × 𝑋)⟶𝑋) | 
| 6 | 5 | ad2antrr 726 | . . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → (-g‘𝐺):(𝑋 × 𝑋)⟶𝑋) | 
| 7 |  | simpr 484 | . . . . . . 7
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) | 
| 8 |  | eqid 2737 | . . . . . . . . 9
⊢
(0g‘𝐺) = (0g‘𝐺) | 
| 9 | 3, 8 | grpidcl 18983 | . . . . . . . 8
⊢ (𝐺 ∈ Grp →
(0g‘𝐺)
∈ 𝑋) | 
| 10 | 9 | ad2antrr 726 | . . . . . . 7
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → (0g‘𝐺) ∈ 𝑋) | 
| 11 | 7, 10 | opelxpd 5724 | . . . . . 6
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → 〈𝑥, (0g‘𝐺)〉 ∈ (𝑋 × 𝑋)) | 
| 12 |  | fvco3 7008 | . . . . . 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 584 | . . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → ((𝑁 ∘ (-g‘𝐺))‘〈𝑥, (0g‘𝐺)〉) = (𝑁‘((-g‘𝐺)‘〈𝑥, (0g‘𝐺)〉))) | 
| 14 |  | df-ov 7434 | . . . . 5
⊢ (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺)) = ((𝑁 ∘ (-g‘𝐺))‘〈𝑥, (0g‘𝐺)〉) | 
| 15 |  | df-ov 7434 | . . . . . 6
⊢ (𝑥(-g‘𝐺)(0g‘𝐺)) = ((-g‘𝐺)‘〈𝑥, (0g‘𝐺)〉) | 
| 16 | 15 | fveq2i 6909 | . . . . 5
⊢ (𝑁‘(𝑥(-g‘𝐺)(0g‘𝐺))) = (𝑁‘((-g‘𝐺)‘〈𝑥, (0g‘𝐺)〉)) | 
| 17 | 13, 14, 16 | 3eqtr4g 2802 | . . . 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 19043 | . . . . . 6
⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋) → (𝑥(-g‘𝐺)(0g‘𝐺)) = 𝑥) | 
| 19 | 18 | adantlr 715 | . . . . 5
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → (𝑥(-g‘𝐺)(0g‘𝐺)) = 𝑥) | 
| 20 | 19 | fveq2d 6910 | . . . 4
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑥(-g‘𝐺)(0g‘𝐺))) = (𝑁‘𝑥)) | 
| 21 | 17, 20 | eqtr2d 2778 | . . 3
⊢ (((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) ∧ 𝑥 ∈ 𝑋) → (𝑁‘𝑥) = (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺))) | 
| 22 | 21 | mpteq2dva 5242 | . 2
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑁‘𝑥)) = (𝑥 ∈ 𝑋 ↦ (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺)))) | 
| 23 | 3 | fvexi 6920 | . . . . . . 7
⊢ 𝑋 ∈ V | 
| 24 |  | tngnm.a | . . . . . . 7
⊢ 𝐴 ∈ V | 
| 25 |  | fex2 7958 | . . . . . . 7
⊢ ((𝑁:𝑋⟶𝐴 ∧ 𝑋 ∈ V ∧ 𝐴 ∈ V) → 𝑁 ∈ V) | 
| 26 | 23, 24, 25 | mp3an23 1455 | . . . . . 6
⊢ (𝑁:𝑋⟶𝐴 → 𝑁 ∈ V) | 
| 27 | 26 | adantl 481 | . . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑁 ∈ V) | 
| 28 |  | tngnm.t | . . . . . 6
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | 
| 29 | 28, 3 | tngbas 24655 | . . . . 5
⊢ (𝑁 ∈ V → 𝑋 = (Base‘𝑇)) | 
| 30 | 27, 29 | syl 17 | . . . 4
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑋 = (Base‘𝑇)) | 
| 31 | 28, 4 | tngds 24668 | . . . . . 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 2738 | . . . . 5
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑥 = 𝑥) | 
| 34 | 28, 8 | tng0 24659 | . . . . . 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 7452 | . . . 4
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺)) = (𝑥(dist‘𝑇)(0g‘𝑇))) | 
| 37 | 30, 36 | mpteq12dv 5233 | . . 3
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺))) = (𝑥 ∈ (Base‘𝑇) ↦ (𝑥(dist‘𝑇)(0g‘𝑇)))) | 
| 38 |  | eqid 2737 | . . . 4
⊢
(norm‘𝑇) =
(norm‘𝑇) | 
| 39 |  | eqid 2737 | . . . 4
⊢
(Base‘𝑇) =
(Base‘𝑇) | 
| 40 |  | eqid 2737 | . . . 4
⊢
(0g‘𝑇) = (0g‘𝑇) | 
| 41 |  | eqid 2737 | . . . 4
⊢
(dist‘𝑇) =
(dist‘𝑇) | 
| 42 | 38, 39, 40, 41 | nmfval 24601 | . . 3
⊢
(norm‘𝑇) =
(𝑥 ∈ (Base‘𝑇) ↦ (𝑥(dist‘𝑇)(0g‘𝑇))) | 
| 43 | 37, 42 | eqtr4di 2795 | . 2
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → (𝑥 ∈ 𝑋 ↦ (𝑥(𝑁 ∘ (-g‘𝐺))(0g‘𝐺))) = (norm‘𝑇)) | 
| 44 | 2, 22, 43 | 3eqtrd 2781 | 1
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶𝐴) → 𝑁 = (norm‘𝑇)) |