Step |
Hyp |
Ref |
Expression |
0 |
|
cnlm |
⊢ NrmMod |
1 |
|
vw |
⊢ 𝑤 |
2 |
|
cngp |
⊢ NrmGrp |
3 |
|
clmod |
⊢ LMod |
4 |
2 3
|
cin |
⊢ ( NrmGrp ∩ LMod ) |
5 |
|
csca |
⊢ Scalar |
6 |
1
|
cv |
⊢ 𝑤 |
7 |
6 5
|
cfv |
⊢ ( Scalar ‘ 𝑤 ) |
8 |
|
vf |
⊢ 𝑓 |
9 |
8
|
cv |
⊢ 𝑓 |
10 |
|
cnrg |
⊢ NrmRing |
11 |
9 10
|
wcel |
⊢ 𝑓 ∈ NrmRing |
12 |
|
vx |
⊢ 𝑥 |
13 |
|
cbs |
⊢ Base |
14 |
9 13
|
cfv |
⊢ ( Base ‘ 𝑓 ) |
15 |
|
vy |
⊢ 𝑦 |
16 |
6 13
|
cfv |
⊢ ( Base ‘ 𝑤 ) |
17 |
|
cnm |
⊢ norm |
18 |
6 17
|
cfv |
⊢ ( norm ‘ 𝑤 ) |
19 |
12
|
cv |
⊢ 𝑥 |
20 |
|
cvsca |
⊢ ·𝑠 |
21 |
6 20
|
cfv |
⊢ ( ·𝑠 ‘ 𝑤 ) |
22 |
15
|
cv |
⊢ 𝑦 |
23 |
19 22 21
|
co |
⊢ ( 𝑥 ( ·𝑠 ‘ 𝑤 ) 𝑦 ) |
24 |
23 18
|
cfv |
⊢ ( ( norm ‘ 𝑤 ) ‘ ( 𝑥 ( ·𝑠 ‘ 𝑤 ) 𝑦 ) ) |
25 |
9 17
|
cfv |
⊢ ( norm ‘ 𝑓 ) |
26 |
19 25
|
cfv |
⊢ ( ( norm ‘ 𝑓 ) ‘ 𝑥 ) |
27 |
|
cmul |
⊢ · |
28 |
22 18
|
cfv |
⊢ ( ( norm ‘ 𝑤 ) ‘ 𝑦 ) |
29 |
26 28 27
|
co |
⊢ ( ( ( norm ‘ 𝑓 ) ‘ 𝑥 ) · ( ( norm ‘ 𝑤 ) ‘ 𝑦 ) ) |
30 |
24 29
|
wceq |
⊢ ( ( norm ‘ 𝑤 ) ‘ ( 𝑥 ( ·𝑠 ‘ 𝑤 ) 𝑦 ) ) = ( ( ( norm ‘ 𝑓 ) ‘ 𝑥 ) · ( ( norm ‘ 𝑤 ) ‘ 𝑦 ) ) |
31 |
30 15 16
|
wral |
⊢ ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) ( ( norm ‘ 𝑤 ) ‘ ( 𝑥 ( ·𝑠 ‘ 𝑤 ) 𝑦 ) ) = ( ( ( norm ‘ 𝑓 ) ‘ 𝑥 ) · ( ( norm ‘ 𝑤 ) ‘ 𝑦 ) ) |
32 |
31 12 14
|
wral |
⊢ ∀ 𝑥 ∈ ( Base ‘ 𝑓 ) ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) ( ( norm ‘ 𝑤 ) ‘ ( 𝑥 ( ·𝑠 ‘ 𝑤 ) 𝑦 ) ) = ( ( ( norm ‘ 𝑓 ) ‘ 𝑥 ) · ( ( norm ‘ 𝑤 ) ‘ 𝑦 ) ) |
33 |
11 32
|
wa |
⊢ ( 𝑓 ∈ NrmRing ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑓 ) ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) ( ( norm ‘ 𝑤 ) ‘ ( 𝑥 ( ·𝑠 ‘ 𝑤 ) 𝑦 ) ) = ( ( ( norm ‘ 𝑓 ) ‘ 𝑥 ) · ( ( norm ‘ 𝑤 ) ‘ 𝑦 ) ) ) |
34 |
33 8 7
|
wsbc |
⊢ [ ( Scalar ‘ 𝑤 ) / 𝑓 ] ( 𝑓 ∈ NrmRing ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑓 ) ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) ( ( norm ‘ 𝑤 ) ‘ ( 𝑥 ( ·𝑠 ‘ 𝑤 ) 𝑦 ) ) = ( ( ( norm ‘ 𝑓 ) ‘ 𝑥 ) · ( ( norm ‘ 𝑤 ) ‘ 𝑦 ) ) ) |
35 |
34 1 4
|
crab |
⊢ { 𝑤 ∈ ( NrmGrp ∩ LMod ) ∣ [ ( Scalar ‘ 𝑤 ) / 𝑓 ] ( 𝑓 ∈ NrmRing ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑓 ) ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) ( ( norm ‘ 𝑤 ) ‘ ( 𝑥 ( ·𝑠 ‘ 𝑤 ) 𝑦 ) ) = ( ( ( norm ‘ 𝑓 ) ‘ 𝑥 ) · ( ( norm ‘ 𝑤 ) ‘ 𝑦 ) ) ) } |
36 |
0 35
|
wceq |
⊢ NrmMod = { 𝑤 ∈ ( NrmGrp ∩ LMod ) ∣ [ ( Scalar ‘ 𝑤 ) / 𝑓 ] ( 𝑓 ∈ NrmRing ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑓 ) ∀ 𝑦 ∈ ( Base ‘ 𝑤 ) ( ( norm ‘ 𝑤 ) ‘ ( 𝑥 ( ·𝑠 ‘ 𝑤 ) 𝑦 ) ) = ( ( ( norm ‘ 𝑓 ) ‘ 𝑥 ) · ( ( norm ‘ 𝑤 ) ‘ 𝑦 ) ) ) } |