Step |
Hyp |
Ref |
Expression |
1 |
|
slmdvs0.f |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
2 |
|
slmdvs0.s |
⊢ · = ( ·𝑠 ‘ 𝑊 ) |
3 |
|
slmdvs0.k |
⊢ 𝐾 = ( Base ‘ 𝐹 ) |
4 |
|
slmdvs0.z |
⊢ 0 = ( 0g ‘ 𝑊 ) |
5 |
1
|
slmdsrg |
⊢ ( 𝑊 ∈ SLMod → 𝐹 ∈ SRing ) |
6 |
|
eqid |
⊢ ( .r ‘ 𝐹 ) = ( .r ‘ 𝐹 ) |
7 |
|
eqid |
⊢ ( 0g ‘ 𝐹 ) = ( 0g ‘ 𝐹 ) |
8 |
3 6 7
|
srgrz |
⊢ ( ( 𝐹 ∈ SRing ∧ 𝑋 ∈ 𝐾 ) → ( 𝑋 ( .r ‘ 𝐹 ) ( 0g ‘ 𝐹 ) ) = ( 0g ‘ 𝐹 ) ) |
9 |
5 8
|
sylan |
⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ) → ( 𝑋 ( .r ‘ 𝐹 ) ( 0g ‘ 𝐹 ) ) = ( 0g ‘ 𝐹 ) ) |
10 |
9
|
oveq1d |
⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ) → ( ( 𝑋 ( .r ‘ 𝐹 ) ( 0g ‘ 𝐹 ) ) · 0 ) = ( ( 0g ‘ 𝐹 ) · 0 ) ) |
11 |
|
simpl |
⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ) → 𝑊 ∈ SLMod ) |
12 |
|
simpr |
⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ) → 𝑋 ∈ 𝐾 ) |
13 |
5
|
adantr |
⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ) → 𝐹 ∈ SRing ) |
14 |
3 7
|
srg0cl |
⊢ ( 𝐹 ∈ SRing → ( 0g ‘ 𝐹 ) ∈ 𝐾 ) |
15 |
13 14
|
syl |
⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ) → ( 0g ‘ 𝐹 ) ∈ 𝐾 ) |
16 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
17 |
16 4
|
slmd0vcl |
⊢ ( 𝑊 ∈ SLMod → 0 ∈ ( Base ‘ 𝑊 ) ) |
18 |
17
|
adantr |
⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ) → 0 ∈ ( Base ‘ 𝑊 ) ) |
19 |
16 1 2 3 6
|
slmdvsass |
⊢ ( ( 𝑊 ∈ SLMod ∧ ( 𝑋 ∈ 𝐾 ∧ ( 0g ‘ 𝐹 ) ∈ 𝐾 ∧ 0 ∈ ( Base ‘ 𝑊 ) ) ) → ( ( 𝑋 ( .r ‘ 𝐹 ) ( 0g ‘ 𝐹 ) ) · 0 ) = ( 𝑋 · ( ( 0g ‘ 𝐹 ) · 0 ) ) ) |
20 |
11 12 15 18 19
|
syl13anc |
⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ) → ( ( 𝑋 ( .r ‘ 𝐹 ) ( 0g ‘ 𝐹 ) ) · 0 ) = ( 𝑋 · ( ( 0g ‘ 𝐹 ) · 0 ) ) ) |
21 |
16 1 2 7 4
|
slmd0vs |
⊢ ( ( 𝑊 ∈ SLMod ∧ 0 ∈ ( Base ‘ 𝑊 ) ) → ( ( 0g ‘ 𝐹 ) · 0 ) = 0 ) |
22 |
18 21
|
syldan |
⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ) → ( ( 0g ‘ 𝐹 ) · 0 ) = 0 ) |
23 |
22
|
oveq2d |
⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ) → ( 𝑋 · ( ( 0g ‘ 𝐹 ) · 0 ) ) = ( 𝑋 · 0 ) ) |
24 |
20 23
|
eqtrd |
⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ) → ( ( 𝑋 ( .r ‘ 𝐹 ) ( 0g ‘ 𝐹 ) ) · 0 ) = ( 𝑋 · 0 ) ) |
25 |
10 24 22
|
3eqtr3d |
⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ) → ( 𝑋 · 0 ) = 0 ) |