Step |
Hyp |
Ref |
Expression |
1 |
|
lmisfree.j |
⊢ 𝐽 = ( LBasis ‘ 𝑊 ) |
2 |
|
lmisfree.f |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
3 |
|
n0 |
⊢ ( 𝐽 ≠ ∅ ↔ ∃ 𝑗 𝑗 ∈ 𝐽 ) |
4 |
|
vex |
⊢ 𝑗 ∈ V |
5 |
4
|
enref |
⊢ 𝑗 ≈ 𝑗 |
6 |
2 1
|
lbslcic |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑗 ∈ 𝐽 ∧ 𝑗 ≈ 𝑗 ) → 𝑊 ≃𝑚 ( 𝐹 freeLMod 𝑗 ) ) |
7 |
5 6
|
mp3an3 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑗 ∈ 𝐽 ) → 𝑊 ≃𝑚 ( 𝐹 freeLMod 𝑗 ) ) |
8 |
|
oveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝐹 freeLMod 𝑘 ) = ( 𝐹 freeLMod 𝑗 ) ) |
9 |
8
|
breq2d |
⊢ ( 𝑘 = 𝑗 → ( 𝑊 ≃𝑚 ( 𝐹 freeLMod 𝑘 ) ↔ 𝑊 ≃𝑚 ( 𝐹 freeLMod 𝑗 ) ) ) |
10 |
4 9
|
spcev |
⊢ ( 𝑊 ≃𝑚 ( 𝐹 freeLMod 𝑗 ) → ∃ 𝑘 𝑊 ≃𝑚 ( 𝐹 freeLMod 𝑘 ) ) |
11 |
7 10
|
syl |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑗 ∈ 𝐽 ) → ∃ 𝑘 𝑊 ≃𝑚 ( 𝐹 freeLMod 𝑘 ) ) |
12 |
11
|
ex |
⊢ ( 𝑊 ∈ LMod → ( 𝑗 ∈ 𝐽 → ∃ 𝑘 𝑊 ≃𝑚 ( 𝐹 freeLMod 𝑘 ) ) ) |
13 |
12
|
exlimdv |
⊢ ( 𝑊 ∈ LMod → ( ∃ 𝑗 𝑗 ∈ 𝐽 → ∃ 𝑘 𝑊 ≃𝑚 ( 𝐹 freeLMod 𝑘 ) ) ) |
14 |
3 13
|
syl5bi |
⊢ ( 𝑊 ∈ LMod → ( 𝐽 ≠ ∅ → ∃ 𝑘 𝑊 ≃𝑚 ( 𝐹 freeLMod 𝑘 ) ) ) |
15 |
|
lmicsym |
⊢ ( 𝑊 ≃𝑚 ( 𝐹 freeLMod 𝑘 ) → ( 𝐹 freeLMod 𝑘 ) ≃𝑚 𝑊 ) |
16 |
|
lmiclcl |
⊢ ( 𝑊 ≃𝑚 ( 𝐹 freeLMod 𝑘 ) → 𝑊 ∈ LMod ) |
17 |
2
|
lmodring |
⊢ ( 𝑊 ∈ LMod → 𝐹 ∈ Ring ) |
18 |
|
vex |
⊢ 𝑘 ∈ V |
19 |
|
eqid |
⊢ ( 𝐹 freeLMod 𝑘 ) = ( 𝐹 freeLMod 𝑘 ) |
20 |
|
eqid |
⊢ ( 𝐹 unitVec 𝑘 ) = ( 𝐹 unitVec 𝑘 ) |
21 |
|
eqid |
⊢ ( LBasis ‘ ( 𝐹 freeLMod 𝑘 ) ) = ( LBasis ‘ ( 𝐹 freeLMod 𝑘 ) ) |
22 |
19 20 21
|
frlmlbs |
⊢ ( ( 𝐹 ∈ Ring ∧ 𝑘 ∈ V ) → ran ( 𝐹 unitVec 𝑘 ) ∈ ( LBasis ‘ ( 𝐹 freeLMod 𝑘 ) ) ) |
23 |
17 18 22
|
sylancl |
⊢ ( 𝑊 ∈ LMod → ran ( 𝐹 unitVec 𝑘 ) ∈ ( LBasis ‘ ( 𝐹 freeLMod 𝑘 ) ) ) |
24 |
23
|
ne0d |
⊢ ( 𝑊 ∈ LMod → ( LBasis ‘ ( 𝐹 freeLMod 𝑘 ) ) ≠ ∅ ) |
25 |
16 24
|
syl |
⊢ ( 𝑊 ≃𝑚 ( 𝐹 freeLMod 𝑘 ) → ( LBasis ‘ ( 𝐹 freeLMod 𝑘 ) ) ≠ ∅ ) |
26 |
21 1
|
lmiclbs |
⊢ ( ( 𝐹 freeLMod 𝑘 ) ≃𝑚 𝑊 → ( ( LBasis ‘ ( 𝐹 freeLMod 𝑘 ) ) ≠ ∅ → 𝐽 ≠ ∅ ) ) |
27 |
15 25 26
|
sylc |
⊢ ( 𝑊 ≃𝑚 ( 𝐹 freeLMod 𝑘 ) → 𝐽 ≠ ∅ ) |
28 |
27
|
exlimiv |
⊢ ( ∃ 𝑘 𝑊 ≃𝑚 ( 𝐹 freeLMod 𝑘 ) → 𝐽 ≠ ∅ ) |
29 |
14 28
|
impbid1 |
⊢ ( 𝑊 ∈ LMod → ( 𝐽 ≠ ∅ ↔ ∃ 𝑘 𝑊 ≃𝑚 ( 𝐹 freeLMod 𝑘 ) ) ) |