Step |
Hyp |
Ref |
Expression |
1 |
|
tngnrg.t |
⊢ 𝑇 = ( 𝑅 toNrmGrp 𝐹 ) |
2 |
|
tngnrg.a |
⊢ 𝐴 = ( AbsVal ‘ 𝑅 ) |
3 |
2
|
abvrcl |
⊢ ( 𝐹 ∈ 𝐴 → 𝑅 ∈ Ring ) |
4 |
|
ringgrp |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp ) |
5 |
3 4
|
syl |
⊢ ( 𝐹 ∈ 𝐴 → 𝑅 ∈ Grp ) |
6 |
|
eqid |
⊢ ( -g ‘ 𝑅 ) = ( -g ‘ 𝑅 ) |
7 |
1 6
|
tngds |
⊢ ( 𝐹 ∈ 𝐴 → ( 𝐹 ∘ ( -g ‘ 𝑅 ) ) = ( dist ‘ 𝑇 ) ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
9 |
8 2 6
|
abvmet |
⊢ ( 𝐹 ∈ 𝐴 → ( 𝐹 ∘ ( -g ‘ 𝑅 ) ) ∈ ( Met ‘ ( Base ‘ 𝑅 ) ) ) |
10 |
7 9
|
eqeltrrd |
⊢ ( 𝐹 ∈ 𝐴 → ( dist ‘ 𝑇 ) ∈ ( Met ‘ ( Base ‘ 𝑅 ) ) ) |
11 |
2 8
|
abvf |
⊢ ( 𝐹 ∈ 𝐴 → 𝐹 : ( Base ‘ 𝑅 ) ⟶ ℝ ) |
12 |
|
eqid |
⊢ ( dist ‘ 𝑇 ) = ( dist ‘ 𝑇 ) |
13 |
1 8 12
|
tngngp2 |
⊢ ( 𝐹 : ( Base ‘ 𝑅 ) ⟶ ℝ → ( 𝑇 ∈ NrmGrp ↔ ( 𝑅 ∈ Grp ∧ ( dist ‘ 𝑇 ) ∈ ( Met ‘ ( Base ‘ 𝑅 ) ) ) ) ) |
14 |
11 13
|
syl |
⊢ ( 𝐹 ∈ 𝐴 → ( 𝑇 ∈ NrmGrp ↔ ( 𝑅 ∈ Grp ∧ ( dist ‘ 𝑇 ) ∈ ( Met ‘ ( Base ‘ 𝑅 ) ) ) ) ) |
15 |
5 10 14
|
mpbir2and |
⊢ ( 𝐹 ∈ 𝐴 → 𝑇 ∈ NrmGrp ) |
16 |
|
reex |
⊢ ℝ ∈ V |
17 |
1 8 16
|
tngnm |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝐹 : ( Base ‘ 𝑅 ) ⟶ ℝ ) → 𝐹 = ( norm ‘ 𝑇 ) ) |
18 |
5 11 17
|
syl2anc |
⊢ ( 𝐹 ∈ 𝐴 → 𝐹 = ( norm ‘ 𝑇 ) ) |
19 |
|
eqidd |
⊢ ( 𝐹 ∈ 𝐴 → ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) ) |
20 |
1 8
|
tngbas |
⊢ ( 𝐹 ∈ 𝐴 → ( Base ‘ 𝑅 ) = ( Base ‘ 𝑇 ) ) |
21 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
22 |
1 21
|
tngplusg |
⊢ ( 𝐹 ∈ 𝐴 → ( +g ‘ 𝑅 ) = ( +g ‘ 𝑇 ) ) |
23 |
22
|
oveqdr |
⊢ ( ( 𝐹 ∈ 𝐴 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) ) → ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑇 ) 𝑦 ) ) |
24 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
25 |
1 24
|
tngmulr |
⊢ ( 𝐹 ∈ 𝐴 → ( .r ‘ 𝑅 ) = ( .r ‘ 𝑇 ) ) |
26 |
25
|
oveqdr |
⊢ ( ( 𝐹 ∈ 𝐴 ∧ ( 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = ( 𝑥 ( .r ‘ 𝑇 ) 𝑦 ) ) |
27 |
19 20 23 26
|
abvpropd |
⊢ ( 𝐹 ∈ 𝐴 → ( AbsVal ‘ 𝑅 ) = ( AbsVal ‘ 𝑇 ) ) |
28 |
2 27
|
syl5eq |
⊢ ( 𝐹 ∈ 𝐴 → 𝐴 = ( AbsVal ‘ 𝑇 ) ) |
29 |
18 28
|
eleq12d |
⊢ ( 𝐹 ∈ 𝐴 → ( 𝐹 ∈ 𝐴 ↔ ( norm ‘ 𝑇 ) ∈ ( AbsVal ‘ 𝑇 ) ) ) |
30 |
29
|
ibi |
⊢ ( 𝐹 ∈ 𝐴 → ( norm ‘ 𝑇 ) ∈ ( AbsVal ‘ 𝑇 ) ) |
31 |
|
eqid |
⊢ ( norm ‘ 𝑇 ) = ( norm ‘ 𝑇 ) |
32 |
|
eqid |
⊢ ( AbsVal ‘ 𝑇 ) = ( AbsVal ‘ 𝑇 ) |
33 |
31 32
|
isnrg |
⊢ ( 𝑇 ∈ NrmRing ↔ ( 𝑇 ∈ NrmGrp ∧ ( norm ‘ 𝑇 ) ∈ ( AbsVal ‘ 𝑇 ) ) ) |
34 |
15 30 33
|
sylanbrc |
⊢ ( 𝐹 ∈ 𝐴 → 𝑇 ∈ NrmRing ) |