Step |
Hyp |
Ref |
Expression |
1 |
|
lssats2.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
2 |
|
lssats2.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
3 |
|
lssats2.w |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
4 |
|
lssats2.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑆 ) |
5 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑈 ) → 𝑦 ∈ 𝑈 ) |
6 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑈 ) → 𝑊 ∈ LMod ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
8 |
7 1
|
lssel |
⊢ ( ( 𝑈 ∈ 𝑆 ∧ 𝑦 ∈ 𝑈 ) → 𝑦 ∈ ( Base ‘ 𝑊 ) ) |
9 |
4 8
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑈 ) → 𝑦 ∈ ( Base ‘ 𝑊 ) ) |
10 |
7 2
|
lspsnid |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑦 ∈ ( Base ‘ 𝑊 ) ) → 𝑦 ∈ ( 𝑁 ‘ { 𝑦 } ) ) |
11 |
6 9 10
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑈 ) → 𝑦 ∈ ( 𝑁 ‘ { 𝑦 } ) ) |
12 |
|
sneq |
⊢ ( 𝑥 = 𝑦 → { 𝑥 } = { 𝑦 } ) |
13 |
12
|
fveq2d |
⊢ ( 𝑥 = 𝑦 → ( 𝑁 ‘ { 𝑥 } ) = ( 𝑁 ‘ { 𝑦 } ) ) |
14 |
13
|
eleq2d |
⊢ ( 𝑥 = 𝑦 → ( 𝑦 ∈ ( 𝑁 ‘ { 𝑥 } ) ↔ 𝑦 ∈ ( 𝑁 ‘ { 𝑦 } ) ) ) |
15 |
14
|
rspcev |
⊢ ( ( 𝑦 ∈ 𝑈 ∧ 𝑦 ∈ ( 𝑁 ‘ { 𝑦 } ) ) → ∃ 𝑥 ∈ 𝑈 𝑦 ∈ ( 𝑁 ‘ { 𝑥 } ) ) |
16 |
5 11 15
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑈 ) → ∃ 𝑥 ∈ 𝑈 𝑦 ∈ ( 𝑁 ‘ { 𝑥 } ) ) |
17 |
16
|
ex |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝑈 → ∃ 𝑥 ∈ 𝑈 𝑦 ∈ ( 𝑁 ‘ { 𝑥 } ) ) ) |
18 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) → 𝑊 ∈ LMod ) |
19 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) → 𝑈 ∈ 𝑆 ) |
20 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) → 𝑥 ∈ 𝑈 ) |
21 |
1 2 18 19 20
|
lspsnel5a |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) → ( 𝑁 ‘ { 𝑥 } ) ⊆ 𝑈 ) |
22 |
21
|
sseld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑈 ) → ( 𝑦 ∈ ( 𝑁 ‘ { 𝑥 } ) → 𝑦 ∈ 𝑈 ) ) |
23 |
22
|
rexlimdva |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝑈 𝑦 ∈ ( 𝑁 ‘ { 𝑥 } ) → 𝑦 ∈ 𝑈 ) ) |
24 |
17 23
|
impbid |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝑈 ↔ ∃ 𝑥 ∈ 𝑈 𝑦 ∈ ( 𝑁 ‘ { 𝑥 } ) ) ) |
25 |
|
eliun |
⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝑈 ( 𝑁 ‘ { 𝑥 } ) ↔ ∃ 𝑥 ∈ 𝑈 𝑦 ∈ ( 𝑁 ‘ { 𝑥 } ) ) |
26 |
24 25
|
bitr4di |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝑈 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝑈 ( 𝑁 ‘ { 𝑥 } ) ) ) |
27 |
26
|
eqrdv |
⊢ ( 𝜑 → 𝑈 = ∪ 𝑥 ∈ 𝑈 ( 𝑁 ‘ { 𝑥 } ) ) |