Step |
Hyp |
Ref |
Expression |
1 |
|
lssatle.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
2 |
|
lssatle.a |
⊢ 𝐴 = ( LSAtoms ‘ 𝑊 ) |
3 |
|
lssatle.w |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
4 |
|
lssatle.t |
⊢ ( 𝜑 → 𝑇 ∈ 𝑆 ) |
5 |
|
lssatle.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑆 ) |
6 |
|
sstr |
⊢ ( ( 𝑝 ⊆ 𝑇 ∧ 𝑇 ⊆ 𝑈 ) → 𝑝 ⊆ 𝑈 ) |
7 |
6
|
expcom |
⊢ ( 𝑇 ⊆ 𝑈 → ( 𝑝 ⊆ 𝑇 → 𝑝 ⊆ 𝑈 ) ) |
8 |
7
|
ralrimivw |
⊢ ( 𝑇 ⊆ 𝑈 → ∀ 𝑝 ∈ 𝐴 ( 𝑝 ⊆ 𝑇 → 𝑝 ⊆ 𝑈 ) ) |
9 |
|
ss2rab |
⊢ ( { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇 } ⊆ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ↔ ∀ 𝑝 ∈ 𝐴 ( 𝑝 ⊆ 𝑇 → 𝑝 ⊆ 𝑈 ) ) |
10 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇 } ⊆ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ) → 𝑊 ∈ LMod ) |
11 |
1 2
|
lsatlss |
⊢ ( 𝑊 ∈ LMod → 𝐴 ⊆ 𝑆 ) |
12 |
|
rabss2 |
⊢ ( 𝐴 ⊆ 𝑆 → { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ⊆ { 𝑝 ∈ 𝑆 ∣ 𝑝 ⊆ 𝑈 } ) |
13 |
|
uniss |
⊢ ( { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ⊆ { 𝑝 ∈ 𝑆 ∣ 𝑝 ⊆ 𝑈 } → ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ⊆ ∪ { 𝑝 ∈ 𝑆 ∣ 𝑝 ⊆ 𝑈 } ) |
14 |
3 11 12 13
|
4syl |
⊢ ( 𝜑 → ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ⊆ ∪ { 𝑝 ∈ 𝑆 ∣ 𝑝 ⊆ 𝑈 } ) |
15 |
|
unimax |
⊢ ( 𝑈 ∈ 𝑆 → ∪ { 𝑝 ∈ 𝑆 ∣ 𝑝 ⊆ 𝑈 } = 𝑈 ) |
16 |
5 15
|
syl |
⊢ ( 𝜑 → ∪ { 𝑝 ∈ 𝑆 ∣ 𝑝 ⊆ 𝑈 } = 𝑈 ) |
17 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
18 |
17 1
|
lssss |
⊢ ( 𝑈 ∈ 𝑆 → 𝑈 ⊆ ( Base ‘ 𝑊 ) ) |
19 |
5 18
|
syl |
⊢ ( 𝜑 → 𝑈 ⊆ ( Base ‘ 𝑊 ) ) |
20 |
16 19
|
eqsstrd |
⊢ ( 𝜑 → ∪ { 𝑝 ∈ 𝑆 ∣ 𝑝 ⊆ 𝑈 } ⊆ ( Base ‘ 𝑊 ) ) |
21 |
14 20
|
sstrd |
⊢ ( 𝜑 → ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ⊆ ( Base ‘ 𝑊 ) ) |
22 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇 } ⊆ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ) → ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ⊆ ( Base ‘ 𝑊 ) ) |
23 |
|
uniss |
⊢ ( { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇 } ⊆ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } → ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇 } ⊆ ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ) |
24 |
23
|
adantl |
⊢ ( ( 𝜑 ∧ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇 } ⊆ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ) → ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇 } ⊆ ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ) |
25 |
|
eqid |
⊢ ( LSpan ‘ 𝑊 ) = ( LSpan ‘ 𝑊 ) |
26 |
17 25
|
lspss |
⊢ ( ( 𝑊 ∈ LMod ∧ ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ⊆ ( Base ‘ 𝑊 ) ∧ ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇 } ⊆ ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ) → ( ( LSpan ‘ 𝑊 ) ‘ ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ) ) |
27 |
10 22 24 26
|
syl3anc |
⊢ ( ( 𝜑 ∧ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇 } ⊆ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ) → ( ( LSpan ‘ 𝑊 ) ‘ ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ) ) |
28 |
27
|
ex |
⊢ ( 𝜑 → ( { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇 } ⊆ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } → ( ( LSpan ‘ 𝑊 ) ‘ ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ) ) ) |
29 |
1 25 2
|
lssats |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ) → 𝑇 = ( ( LSpan ‘ 𝑊 ) ‘ ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇 } ) ) |
30 |
3 4 29
|
syl2anc |
⊢ ( 𝜑 → 𝑇 = ( ( LSpan ‘ 𝑊 ) ‘ ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇 } ) ) |
31 |
1 25 2
|
lssats |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ) → 𝑈 = ( ( LSpan ‘ 𝑊 ) ‘ ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ) ) |
32 |
3 5 31
|
syl2anc |
⊢ ( 𝜑 → 𝑈 = ( ( LSpan ‘ 𝑊 ) ‘ ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ) ) |
33 |
30 32
|
sseq12d |
⊢ ( 𝜑 → ( 𝑇 ⊆ 𝑈 ↔ ( ( LSpan ‘ 𝑊 ) ‘ ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇 } ) ⊆ ( ( LSpan ‘ 𝑊 ) ‘ ∪ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } ) ) ) |
34 |
28 33
|
sylibrd |
⊢ ( 𝜑 → ( { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇 } ⊆ { 𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈 } → 𝑇 ⊆ 𝑈 ) ) |
35 |
9 34
|
syl5bir |
⊢ ( 𝜑 → ( ∀ 𝑝 ∈ 𝐴 ( 𝑝 ⊆ 𝑇 → 𝑝 ⊆ 𝑈 ) → 𝑇 ⊆ 𝑈 ) ) |
36 |
8 35
|
impbid2 |
⊢ ( 𝜑 → ( 𝑇 ⊆ 𝑈 ↔ ∀ 𝑝 ∈ 𝐴 ( 𝑝 ⊆ 𝑇 → 𝑝 ⊆ 𝑈 ) ) ) |