Step |
Hyp |
Ref |
Expression |
1 |
|
lincvalsc0.b |
⊢ 𝐵 = ( Base ‘ 𝑀 ) |
2 |
|
lincvalsc0.s |
⊢ 𝑆 = ( Scalar ‘ 𝑀 ) |
3 |
|
lincvalsc0.0 |
⊢ 0 = ( 0g ‘ 𝑆 ) |
4 |
|
lincvalsc0.z |
⊢ 𝑍 = ( 0g ‘ 𝑀 ) |
5 |
|
lincvalsc0.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝑉 ↦ 0 ) |
6 |
|
lcoc0.r |
⊢ 𝑅 = ( Base ‘ 𝑆 ) |
7 |
2 6 3
|
lmod0cl |
⊢ ( 𝑀 ∈ LMod → 0 ∈ 𝑅 ) |
8 |
7
|
ad2antrr |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) ∧ 𝑥 ∈ 𝑉 ) → 0 ∈ 𝑅 ) |
9 |
8 5
|
fmptd |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → 𝐹 : 𝑉 ⟶ 𝑅 ) |
10 |
6
|
fvexi |
⊢ 𝑅 ∈ V |
11 |
10
|
a1i |
⊢ ( 𝑀 ∈ LMod → 𝑅 ∈ V ) |
12 |
|
elmapg |
⊢ ( ( 𝑅 ∈ V ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝐹 ∈ ( 𝑅 ↑m 𝑉 ) ↔ 𝐹 : 𝑉 ⟶ 𝑅 ) ) |
13 |
11 12
|
sylan |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝐹 ∈ ( 𝑅 ↑m 𝑉 ) ↔ 𝐹 : 𝑉 ⟶ 𝑅 ) ) |
14 |
9 13
|
mpbird |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → 𝐹 ∈ ( 𝑅 ↑m 𝑉 ) ) |
15 |
|
eqidd |
⊢ ( 𝑥 = 𝑣 → 0 = 0 ) |
16 |
15
|
cbvmptv |
⊢ ( 𝑥 ∈ 𝑉 ↦ 0 ) = ( 𝑣 ∈ 𝑉 ↦ 0 ) |
17 |
5 16
|
eqtri |
⊢ 𝐹 = ( 𝑣 ∈ 𝑉 ↦ 0 ) |
18 |
|
simpr |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → 𝑉 ∈ 𝒫 𝐵 ) |
19 |
3
|
fvexi |
⊢ 0 ∈ V |
20 |
19
|
a1i |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → 0 ∈ V ) |
21 |
19
|
a1i |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) ∧ 𝑣 ∈ 𝑉 ) → 0 ∈ V ) |
22 |
17 18 20 21
|
mptsuppd |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝐹 supp 0 ) = { 𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } ) |
23 |
|
neirr |
⊢ ¬ 0 ≠ 0 |
24 |
23
|
a1i |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → ¬ 0 ≠ 0 ) |
25 |
24
|
ralrimivw |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → ∀ 𝑣 ∈ 𝑉 ¬ 0 ≠ 0 ) |
26 |
|
rabeq0 |
⊢ ( { 𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } = ∅ ↔ ∀ 𝑣 ∈ 𝑉 ¬ 0 ≠ 0 ) |
27 |
25 26
|
sylibr |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → { 𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } = ∅ ) |
28 |
|
0fin |
⊢ ∅ ∈ Fin |
29 |
28
|
a1i |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → ∅ ∈ Fin ) |
30 |
27 29
|
eqeltrd |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → { 𝑣 ∈ 𝑉 ∣ 0 ≠ 0 } ∈ Fin ) |
31 |
22 30
|
eqeltrd |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝐹 supp 0 ) ∈ Fin ) |
32 |
5
|
funmpt2 |
⊢ Fun 𝐹 |
33 |
32
|
a1i |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → Fun 𝐹 ) |
34 |
|
funisfsupp |
⊢ ( ( Fun 𝐹 ∧ 𝐹 ∈ ( 𝑅 ↑m 𝑉 ) ∧ 0 ∈ V ) → ( 𝐹 finSupp 0 ↔ ( 𝐹 supp 0 ) ∈ Fin ) ) |
35 |
33 14 20 34
|
syl3anc |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝐹 finSupp 0 ↔ ( 𝐹 supp 0 ) ∈ Fin ) ) |
36 |
31 35
|
mpbird |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → 𝐹 finSupp 0 ) |
37 |
1 2 3 4 5
|
lincvalsc0 |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝐹 ( linC ‘ 𝑀 ) 𝑉 ) = 𝑍 ) |
38 |
14 36 37
|
3jca |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝐹 ∈ ( 𝑅 ↑m 𝑉 ) ∧ 𝐹 finSupp 0 ∧ ( 𝐹 ( linC ‘ 𝑀 ) 𝑉 ) = 𝑍 ) ) |