Step |
Hyp |
Ref |
Expression |
1 |
|
nlmmul0or.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
nlmmul0or.s |
⊢ · = ( ·𝑠 ‘ 𝑊 ) |
3 |
|
nlmmul0or.z |
⊢ 0 = ( 0g ‘ 𝑊 ) |
4 |
|
nlmmul0or.f |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
5 |
|
nlmmul0or.k |
⊢ 𝐾 = ( Base ‘ 𝐹 ) |
6 |
|
nlmmul0or.o |
⊢ 𝑂 = ( 0g ‘ 𝐹 ) |
7 |
4
|
nlmngp2 |
⊢ ( 𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp ) |
8 |
7
|
3ad2ant1 |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → 𝐹 ∈ NrmGrp ) |
9 |
|
simp2 |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → 𝐴 ∈ 𝐾 ) |
10 |
|
eqid |
⊢ ( norm ‘ 𝐹 ) = ( norm ‘ 𝐹 ) |
11 |
5 10
|
nmcl |
⊢ ( ( 𝐹 ∈ NrmGrp ∧ 𝐴 ∈ 𝐾 ) → ( ( norm ‘ 𝐹 ) ‘ 𝐴 ) ∈ ℝ ) |
12 |
8 9 11
|
syl2anc |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → ( ( norm ‘ 𝐹 ) ‘ 𝐴 ) ∈ ℝ ) |
13 |
12
|
recnd |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → ( ( norm ‘ 𝐹 ) ‘ 𝐴 ) ∈ ℂ ) |
14 |
|
nlmngp |
⊢ ( 𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp ) |
15 |
14
|
3ad2ant1 |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → 𝑊 ∈ NrmGrp ) |
16 |
|
simp3 |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → 𝐵 ∈ 𝑉 ) |
17 |
|
eqid |
⊢ ( norm ‘ 𝑊 ) = ( norm ‘ 𝑊 ) |
18 |
1 17
|
nmcl |
⊢ ( ( 𝑊 ∈ NrmGrp ∧ 𝐵 ∈ 𝑉 ) → ( ( norm ‘ 𝑊 ) ‘ 𝐵 ) ∈ ℝ ) |
19 |
15 16 18
|
syl2anc |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → ( ( norm ‘ 𝑊 ) ‘ 𝐵 ) ∈ ℝ ) |
20 |
19
|
recnd |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → ( ( norm ‘ 𝑊 ) ‘ 𝐵 ) ∈ ℂ ) |
21 |
13 20
|
mul0ord |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → ( ( ( ( norm ‘ 𝐹 ) ‘ 𝐴 ) · ( ( norm ‘ 𝑊 ) ‘ 𝐵 ) ) = 0 ↔ ( ( ( norm ‘ 𝐹 ) ‘ 𝐴 ) = 0 ∨ ( ( norm ‘ 𝑊 ) ‘ 𝐵 ) = 0 ) ) ) |
22 |
1 17 2 4 5 10
|
nmvs |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → ( ( norm ‘ 𝑊 ) ‘ ( 𝐴 · 𝐵 ) ) = ( ( ( norm ‘ 𝐹 ) ‘ 𝐴 ) · ( ( norm ‘ 𝑊 ) ‘ 𝐵 ) ) ) |
23 |
22
|
eqeq1d |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → ( ( ( norm ‘ 𝑊 ) ‘ ( 𝐴 · 𝐵 ) ) = 0 ↔ ( ( ( norm ‘ 𝐹 ) ‘ 𝐴 ) · ( ( norm ‘ 𝑊 ) ‘ 𝐵 ) ) = 0 ) ) |
24 |
|
nlmlmod |
⊢ ( 𝑊 ∈ NrmMod → 𝑊 ∈ LMod ) |
25 |
1 4 2 5
|
lmodvscl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → ( 𝐴 · 𝐵 ) ∈ 𝑉 ) |
26 |
24 25
|
syl3an1 |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → ( 𝐴 · 𝐵 ) ∈ 𝑉 ) |
27 |
1 17 3
|
nmeq0 |
⊢ ( ( 𝑊 ∈ NrmGrp ∧ ( 𝐴 · 𝐵 ) ∈ 𝑉 ) → ( ( ( norm ‘ 𝑊 ) ‘ ( 𝐴 · 𝐵 ) ) = 0 ↔ ( 𝐴 · 𝐵 ) = 0 ) ) |
28 |
15 26 27
|
syl2anc |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → ( ( ( norm ‘ 𝑊 ) ‘ ( 𝐴 · 𝐵 ) ) = 0 ↔ ( 𝐴 · 𝐵 ) = 0 ) ) |
29 |
23 28
|
bitr3d |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → ( ( ( ( norm ‘ 𝐹 ) ‘ 𝐴 ) · ( ( norm ‘ 𝑊 ) ‘ 𝐵 ) ) = 0 ↔ ( 𝐴 · 𝐵 ) = 0 ) ) |
30 |
5 10 6
|
nmeq0 |
⊢ ( ( 𝐹 ∈ NrmGrp ∧ 𝐴 ∈ 𝐾 ) → ( ( ( norm ‘ 𝐹 ) ‘ 𝐴 ) = 0 ↔ 𝐴 = 𝑂 ) ) |
31 |
8 9 30
|
syl2anc |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → ( ( ( norm ‘ 𝐹 ) ‘ 𝐴 ) = 0 ↔ 𝐴 = 𝑂 ) ) |
32 |
1 17 3
|
nmeq0 |
⊢ ( ( 𝑊 ∈ NrmGrp ∧ 𝐵 ∈ 𝑉 ) → ( ( ( norm ‘ 𝑊 ) ‘ 𝐵 ) = 0 ↔ 𝐵 = 0 ) ) |
33 |
15 16 32
|
syl2anc |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → ( ( ( norm ‘ 𝑊 ) ‘ 𝐵 ) = 0 ↔ 𝐵 = 0 ) ) |
34 |
31 33
|
orbi12d |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → ( ( ( ( norm ‘ 𝐹 ) ‘ 𝐴 ) = 0 ∨ ( ( norm ‘ 𝑊 ) ‘ 𝐵 ) = 0 ) ↔ ( 𝐴 = 𝑂 ∨ 𝐵 = 0 ) ) ) |
35 |
21 29 34
|
3bitr3d |
⊢ ( ( 𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝐴 · 𝐵 ) = 0 ↔ ( 𝐴 = 𝑂 ∨ 𝐵 = 0 ) ) ) |