Step |
Hyp |
Ref |
Expression |
1 |
|
lcvat.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
2 |
|
lcvat.p |
⊢ ⊕ = ( LSSum ‘ 𝑊 ) |
3 |
|
lcvat.a |
⊢ 𝐴 = ( LSAtoms ‘ 𝑊 ) |
4 |
|
icvat.c |
⊢ 𝐶 = ( ⋖L ‘ 𝑊 ) |
5 |
|
lcvat.w |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
6 |
|
lcvat.t |
⊢ ( 𝜑 → 𝑇 ∈ 𝑆 ) |
7 |
|
lcvat.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑆 ) |
8 |
|
lcvat.l |
⊢ ( 𝜑 → 𝑇 𝐶 𝑈 ) |
9 |
1 4 5 6 7 8
|
lcvpss |
⊢ ( 𝜑 → 𝑇 ⊊ 𝑈 ) |
10 |
1 2 3 5 6 7 9
|
lrelat |
⊢ ( 𝜑 → ∃ 𝑞 ∈ 𝐴 ( 𝑇 ⊊ ( 𝑇 ⊕ 𝑞 ) ∧ ( 𝑇 ⊕ 𝑞 ) ⊆ 𝑈 ) ) |
11 |
5
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ∧ ( 𝑇 ⊊ ( 𝑇 ⊕ 𝑞 ) ∧ ( 𝑇 ⊕ 𝑞 ) ⊆ 𝑈 ) ) → 𝑊 ∈ LMod ) |
12 |
6
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ∧ ( 𝑇 ⊊ ( 𝑇 ⊕ 𝑞 ) ∧ ( 𝑇 ⊕ 𝑞 ) ⊆ 𝑈 ) ) → 𝑇 ∈ 𝑆 ) |
13 |
7
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ∧ ( 𝑇 ⊊ ( 𝑇 ⊕ 𝑞 ) ∧ ( 𝑇 ⊕ 𝑞 ) ⊆ 𝑈 ) ) → 𝑈 ∈ 𝑆 ) |
14 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ∧ ( 𝑇 ⊊ ( 𝑇 ⊕ 𝑞 ) ∧ ( 𝑇 ⊕ 𝑞 ) ⊆ 𝑈 ) ) → 𝑞 ∈ 𝐴 ) |
15 |
1 3 11 14
|
lsatlssel |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ∧ ( 𝑇 ⊊ ( 𝑇 ⊕ 𝑞 ) ∧ ( 𝑇 ⊕ 𝑞 ) ⊆ 𝑈 ) ) → 𝑞 ∈ 𝑆 ) |
16 |
1 2
|
lsmcl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑞 ∈ 𝑆 ) → ( 𝑇 ⊕ 𝑞 ) ∈ 𝑆 ) |
17 |
11 12 15 16
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ∧ ( 𝑇 ⊊ ( 𝑇 ⊕ 𝑞 ) ∧ ( 𝑇 ⊕ 𝑞 ) ⊆ 𝑈 ) ) → ( 𝑇 ⊕ 𝑞 ) ∈ 𝑆 ) |
18 |
8
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ∧ ( 𝑇 ⊊ ( 𝑇 ⊕ 𝑞 ) ∧ ( 𝑇 ⊕ 𝑞 ) ⊆ 𝑈 ) ) → 𝑇 𝐶 𝑈 ) |
19 |
|
simp3l |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ∧ ( 𝑇 ⊊ ( 𝑇 ⊕ 𝑞 ) ∧ ( 𝑇 ⊕ 𝑞 ) ⊆ 𝑈 ) ) → 𝑇 ⊊ ( 𝑇 ⊕ 𝑞 ) ) |
20 |
|
simp3r |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ∧ ( 𝑇 ⊊ ( 𝑇 ⊕ 𝑞 ) ∧ ( 𝑇 ⊕ 𝑞 ) ⊆ 𝑈 ) ) → ( 𝑇 ⊕ 𝑞 ) ⊆ 𝑈 ) |
21 |
1 4 11 12 13 17 18 19 20
|
lcvnbtwn2 |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ∧ ( 𝑇 ⊊ ( 𝑇 ⊕ 𝑞 ) ∧ ( 𝑇 ⊕ 𝑞 ) ⊆ 𝑈 ) ) → ( 𝑇 ⊕ 𝑞 ) = 𝑈 ) |
22 |
21
|
3exp |
⊢ ( 𝜑 → ( 𝑞 ∈ 𝐴 → ( ( 𝑇 ⊊ ( 𝑇 ⊕ 𝑞 ) ∧ ( 𝑇 ⊕ 𝑞 ) ⊆ 𝑈 ) → ( 𝑇 ⊕ 𝑞 ) = 𝑈 ) ) ) |
23 |
22
|
reximdvai |
⊢ ( 𝜑 → ( ∃ 𝑞 ∈ 𝐴 ( 𝑇 ⊊ ( 𝑇 ⊕ 𝑞 ) ∧ ( 𝑇 ⊕ 𝑞 ) ⊆ 𝑈 ) → ∃ 𝑞 ∈ 𝐴 ( 𝑇 ⊕ 𝑞 ) = 𝑈 ) ) |
24 |
10 23
|
mpd |
⊢ ( 𝜑 → ∃ 𝑞 ∈ 𝐴 ( 𝑇 ⊕ 𝑞 ) = 𝑈 ) |