Step |
Hyp |
Ref |
Expression |
1 |
|
lssssr.o |
⊢ 0 = ( 0g ‘ 𝑊 ) |
2 |
|
lssssr.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
3 |
|
lssssr.w |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
4 |
|
lssssr.t |
⊢ ( 𝜑 → 𝑇 ⊆ 𝑉 ) |
5 |
|
lssssr.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑆 ) |
6 |
|
lssssr.1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ) → ( 𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈 ) ) |
7 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 = 0 ) → 𝑥 = 0 ) |
8 |
1 2
|
lss0cl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ) → 0 ∈ 𝑈 ) |
9 |
3 5 8
|
syl2anc |
⊢ ( 𝜑 → 0 ∈ 𝑈 ) |
10 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 = 0 ) → 0 ∈ 𝑈 ) |
11 |
7 10
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑥 = 0 ) → 𝑥 ∈ 𝑈 ) |
12 |
11
|
a1d |
⊢ ( ( 𝜑 ∧ 𝑥 = 0 ) → ( 𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈 ) ) |
13 |
4
|
sseld |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑉 ) ) |
14 |
13
|
ancrd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑇 → ( 𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇 ) ) ) |
15 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ≠ 0 ) → ( 𝑥 ∈ 𝑇 → ( 𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇 ) ) ) |
16 |
|
eldifsn |
⊢ ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ↔ ( 𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 ) ) |
17 |
16 6
|
sylan2br |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 ) ) → ( 𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈 ) ) |
18 |
17
|
exp32 |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑉 → ( 𝑥 ≠ 0 → ( 𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈 ) ) ) ) |
19 |
18
|
com23 |
⊢ ( 𝜑 → ( 𝑥 ≠ 0 → ( 𝑥 ∈ 𝑉 → ( 𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈 ) ) ) ) |
20 |
19
|
imp4b |
⊢ ( ( 𝜑 ∧ 𝑥 ≠ 0 ) → ( ( 𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇 ) → 𝑥 ∈ 𝑈 ) ) |
21 |
15 20
|
syld |
⊢ ( ( 𝜑 ∧ 𝑥 ≠ 0 ) → ( 𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈 ) ) |
22 |
12 21
|
pm2.61dane |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈 ) ) |
23 |
22
|
ssrdv |
⊢ ( 𝜑 → 𝑇 ⊆ 𝑈 ) |