| Step | Hyp | Ref
| Expression |
| 1 | | tngngp.t |
. . . . 5
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
| 2 | | tngngp.x |
. . . . 5
⊢ 𝑋 = (Base‘𝐺) |
| 3 | | eqid 2737 |
. . . . 5
⊢
(dist‘𝑇) =
(dist‘𝑇) |
| 4 | 1, 2, 3 | tngngp2 24673 |
. . . 4
⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘𝑋)))) |
| 5 | 4 | simprbda 498 |
. . 3
⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp) |
| 6 | | simplr 769 |
. . . . . . 7
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑇 ∈ NrmGrp) |
| 7 | | simpr 484 |
. . . . . . . 8
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) |
| 8 | 2 | fvexi 6920 |
. . . . . . . . . . 11
⊢ 𝑋 ∈ V |
| 9 | | reex 11246 |
. . . . . . . . . . 11
⊢ ℝ
∈ V |
| 10 | | fex2 7958 |
. . . . . . . . . . 11
⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑋 ∈ V ∧ ℝ ∈ V) →
𝑁 ∈
V) |
| 11 | 8, 9, 10 | mp3an23 1455 |
. . . . . . . . . 10
⊢ (𝑁:𝑋⟶ℝ → 𝑁 ∈ V) |
| 12 | 11 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑁 ∈ V) |
| 13 | 1, 2 | tngbas 24655 |
. . . . . . . . 9
⊢ (𝑁 ∈ V → 𝑋 = (Base‘𝑇)) |
| 14 | 12, 13 | syl 17 |
. . . . . . . 8
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑋 = (Base‘𝑇)) |
| 15 | 7, 14 | eleqtrd 2843 |
. . . . . . 7
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ (Base‘𝑇)) |
| 16 | | eqid 2737 |
. . . . . . . 8
⊢
(Base‘𝑇) =
(Base‘𝑇) |
| 17 | | eqid 2737 |
. . . . . . . 8
⊢
(norm‘𝑇) =
(norm‘𝑇) |
| 18 | | eqid 2737 |
. . . . . . . 8
⊢
(0g‘𝑇) = (0g‘𝑇) |
| 19 | 16, 17, 18 | nmeq0 24631 |
. . . . . . 7
⊢ ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇)) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑇))) |
| 20 | 6, 15, 19 | syl2anc 584 |
. . . . . 6
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (((norm‘𝑇)‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑇))) |
| 21 | 5 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝐺 ∈ Grp) |
| 22 | | simpll 767 |
. . . . . . . . 9
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑁:𝑋⟶ℝ) |
| 23 | 1, 2, 9 | tngnm 24672 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Grp ∧ 𝑁:𝑋⟶ℝ) → 𝑁 = (norm‘𝑇)) |
| 24 | 21, 22, 23 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 𝑁 = (norm‘𝑇)) |
| 25 | 24 | fveq1d 6908 |
. . . . . . 7
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑁‘𝑥) = ((norm‘𝑇)‘𝑥)) |
| 26 | 25 | eqeq1d 2739 |
. . . . . 6
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘𝑥) = 0 ↔ ((norm‘𝑇)‘𝑥) = 0)) |
| 27 | | tngngp.z |
. . . . . . . . 9
⊢ 0 =
(0g‘𝐺) |
| 28 | 1, 27 | tng0 24659 |
. . . . . . . 8
⊢ (𝑁 ∈ V → 0 =
(0g‘𝑇)) |
| 29 | 12, 28 | syl 17 |
. . . . . . 7
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → 0 =
(0g‘𝑇)) |
| 30 | 29 | eqeq2d 2748 |
. . . . . 6
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑥 = 0 ↔ 𝑥 = (0g‘𝑇))) |
| 31 | 20, 26, 30 | 3bitr4d 311 |
. . . . 5
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 )) |
| 32 | | simpllr 776 |
. . . . . . . 8
⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑇 ∈ NrmGrp) |
| 33 | 15 | adantr 480 |
. . . . . . . 8
⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑥 ∈ (Base‘𝑇)) |
| 34 | 14 | eleq2d 2827 |
. . . . . . . . 9
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑋 ↔ 𝑦 ∈ (Base‘𝑇))) |
| 35 | 34 | biimpa 476 |
. . . . . . . 8
⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ (Base‘𝑇)) |
| 36 | | eqid 2737 |
. . . . . . . . 9
⊢
(-g‘𝑇) = (-g‘𝑇) |
| 37 | 16, 17, 36 | nmmtri 24635 |
. . . . . . . 8
⊢ ((𝑇 ∈ NrmGrp ∧ 𝑥 ∈ (Base‘𝑇) ∧ 𝑦 ∈ (Base‘𝑇)) → ((norm‘𝑇)‘(𝑥(-g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦))) |
| 38 | 32, 33, 35, 37 | syl3anc 1373 |
. . . . . . 7
⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((norm‘𝑇)‘(𝑥(-g‘𝑇)𝑦)) ≤ (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦))) |
| 39 | | tngngp.m |
. . . . . . . . . . 11
⊢ − =
(-g‘𝐺) |
| 40 | 2, 14 | eqtr3id 2791 |
. . . . . . . . . . . 12
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (Base‘𝐺) = (Base‘𝑇)) |
| 41 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(+g‘𝐺) = (+g‘𝐺) |
| 42 | 1, 41 | tngplusg 24657 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ V →
(+g‘𝐺) =
(+g‘𝑇)) |
| 43 | 12, 42 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (+g‘𝐺) = (+g‘𝑇)) |
| 44 | 40, 43 | grpsubpropd 19063 |
. . . . . . . . . . 11
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (-g‘𝐺) = (-g‘𝑇)) |
| 45 | 39, 44 | eqtrid 2789 |
. . . . . . . . . 10
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → − =
(-g‘𝑇)) |
| 46 | 45 | oveqd 7448 |
. . . . . . . . 9
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑥 − 𝑦) = (𝑥(-g‘𝑇)𝑦)) |
| 47 | 24, 46 | fveq12d 6913 |
. . . . . . . 8
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑥 − 𝑦)) = ((norm‘𝑇)‘(𝑥(-g‘𝑇)𝑦))) |
| 48 | 47 | adantr 480 |
. . . . . . 7
⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥 − 𝑦)) = ((norm‘𝑇)‘(𝑥(-g‘𝑇)𝑦))) |
| 49 | 24 | fveq1d 6908 |
. . . . . . . . 9
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (𝑁‘𝑦) = ((norm‘𝑇)‘𝑦)) |
| 50 | 25, 49 | oveq12d 7449 |
. . . . . . . 8
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘𝑥) + (𝑁‘𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦))) |
| 51 | 50 | adantr 480 |
. . . . . . 7
⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑁‘𝑥) + (𝑁‘𝑦)) = (((norm‘𝑇)‘𝑥) + ((norm‘𝑇)‘𝑦))) |
| 52 | 38, 48, 51 | 3brtr4d 5175 |
. . . . . 6
⊢ ((((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) |
| 53 | 52 | ralrimiva 3146 |
. . . . 5
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) |
| 54 | 31, 53 | jca 511 |
. . . 4
⊢ (((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) ∧ 𝑥 ∈ 𝑋) → (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))) |
| 55 | 54 | ralrimiva 3146 |
. . 3
⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))) |
| 56 | 5, 55 | jca 511 |
. 2
⊢ ((𝑁:𝑋⟶ℝ ∧ 𝑇 ∈ NrmGrp) → (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) |
| 57 | | simprl 771 |
. . 3
⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) → 𝐺 ∈ Grp) |
| 58 | | simpl 482 |
. . 3
⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) → 𝑁:𝑋⟶ℝ) |
| 59 | | simpl 482 |
. . . . . 6
⊢ ((((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 )) |
| 60 | 59 | ralimi 3083 |
. . . . 5
⊢
(∀𝑥 ∈
𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ∀𝑥 ∈ 𝑋 ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 )) |
| 61 | 60 | ad2antll 729 |
. . . 4
⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) → ∀𝑥 ∈ 𝑋 ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 )) |
| 62 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑥 = 𝑎 → (𝑁‘𝑥) = (𝑁‘𝑎)) |
| 63 | 62 | eqeq1d 2739 |
. . . . . 6
⊢ (𝑥 = 𝑎 → ((𝑁‘𝑥) = 0 ↔ (𝑁‘𝑎) = 0)) |
| 64 | | eqeq1 2741 |
. . . . . 6
⊢ (𝑥 = 𝑎 → (𝑥 = 0 ↔ 𝑎 = 0 )) |
| 65 | 63, 64 | bibi12d 345 |
. . . . 5
⊢ (𝑥 = 𝑎 → (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 ))) |
| 66 | 65 | rspccva 3621 |
. . . 4
⊢
((∀𝑥 ∈
𝑋 ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ 𝑎 ∈ 𝑋) → ((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 )) |
| 67 | 61, 66 | sylan 580 |
. . 3
⊢ (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) ∧ 𝑎 ∈ 𝑋) → ((𝑁‘𝑎) = 0 ↔ 𝑎 = 0 )) |
| 68 | | simpr 484 |
. . . . . 6
⊢ ((((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) |
| 69 | 68 | ralimi 3083 |
. . . . 5
⊢
(∀𝑥 ∈
𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) |
| 70 | 69 | ad2antll 729 |
. . . 4
⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) |
| 71 | | fvoveq1 7454 |
. . . . . . 7
⊢ (𝑥 = 𝑎 → (𝑁‘(𝑥 − 𝑦)) = (𝑁‘(𝑎 − 𝑦))) |
| 72 | 62 | oveq1d 7446 |
. . . . . . 7
⊢ (𝑥 = 𝑎 → ((𝑁‘𝑥) + (𝑁‘𝑦)) = ((𝑁‘𝑎) + (𝑁‘𝑦))) |
| 73 | 71, 72 | breq12d 5156 |
. . . . . 6
⊢ (𝑥 = 𝑎 → ((𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ↔ (𝑁‘(𝑎 − 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦)))) |
| 74 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑦 = 𝑏 → (𝑎 − 𝑦) = (𝑎 − 𝑏)) |
| 75 | 74 | fveq2d 6910 |
. . . . . . 7
⊢ (𝑦 = 𝑏 → (𝑁‘(𝑎 − 𝑦)) = (𝑁‘(𝑎 − 𝑏))) |
| 76 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑦 = 𝑏 → (𝑁‘𝑦) = (𝑁‘𝑏)) |
| 77 | 76 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑦 = 𝑏 → ((𝑁‘𝑎) + (𝑁‘𝑦)) = ((𝑁‘𝑎) + (𝑁‘𝑏))) |
| 78 | 75, 77 | breq12d 5156 |
. . . . . 6
⊢ (𝑦 = 𝑏 → ((𝑁‘(𝑎 − 𝑦)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑦)) ↔ (𝑁‘(𝑎 − 𝑏)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏)))) |
| 79 | 73, 78 | rspc2va 3634 |
. . . . 5
⊢ (((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → (𝑁‘(𝑎 − 𝑏)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏))) |
| 80 | 79 | ancoms 458 |
. . . 4
⊢
((∀𝑥 ∈
𝑋 ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑁‘(𝑎 − 𝑏)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏))) |
| 81 | 70, 80 | sylan 580 |
. . 3
⊢ (((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑁‘(𝑎 − 𝑏)) ≤ ((𝑁‘𝑎) + (𝑁‘𝑏))) |
| 82 | 1, 2, 39, 27, 57, 58, 67, 81 | tngngpd 24674 |
. 2
⊢ ((𝑁:𝑋⟶ℝ ∧ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) → 𝑇 ∈ NrmGrp) |
| 83 | 56, 82 | impbida 801 |
1
⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))) |