Step |
Hyp |
Ref |
Expression |
1 |
|
dsmmcl.p |
⊢ 𝑃 = ( 𝑆 Xs 𝑅 ) |
2 |
|
dsmmcl.h |
⊢ 𝐻 = ( Base ‘ ( 𝑆 ⊕m 𝑅 ) ) |
3 |
|
dsmmcl.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
4 |
|
dsmmcl.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑉 ) |
5 |
|
dsmmcl.r |
⊢ ( 𝜑 → 𝑅 : 𝐼 ⟶ Mnd ) |
6 |
|
dsmm0cl.z |
⊢ 0 = ( 0g ‘ 𝑃 ) |
7 |
1 3 4 5
|
prdsmndd |
⊢ ( 𝜑 → 𝑃 ∈ Mnd ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
9 |
8 6
|
mndidcl |
⊢ ( 𝑃 ∈ Mnd → 0 ∈ ( Base ‘ 𝑃 ) ) |
10 |
7 9
|
syl |
⊢ ( 𝜑 → 0 ∈ ( Base ‘ 𝑃 ) ) |
11 |
1 3 4 5
|
prds0g |
⊢ ( 𝜑 → ( 0g ∘ 𝑅 ) = ( 0g ‘ 𝑃 ) ) |
12 |
11 6
|
eqtr4di |
⊢ ( 𝜑 → ( 0g ∘ 𝑅 ) = 0 ) |
13 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐼 ) → ( 0g ∘ 𝑅 ) = 0 ) |
14 |
13
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐼 ) → ( ( 0g ∘ 𝑅 ) ‘ 𝑎 ) = ( 0 ‘ 𝑎 ) ) |
15 |
5
|
ffnd |
⊢ ( 𝜑 → 𝑅 Fn 𝐼 ) |
16 |
|
fvco2 |
⊢ ( ( 𝑅 Fn 𝐼 ∧ 𝑎 ∈ 𝐼 ) → ( ( 0g ∘ 𝑅 ) ‘ 𝑎 ) = ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) ) |
17 |
15 16
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐼 ) → ( ( 0g ∘ 𝑅 ) ‘ 𝑎 ) = ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) ) |
18 |
14 17
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐼 ) → ( 0 ‘ 𝑎 ) = ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) ) |
19 |
|
nne |
⊢ ( ¬ ( 0 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) ↔ ( 0 ‘ 𝑎 ) = ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) ) |
20 |
18 19
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝐼 ) → ¬ ( 0 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) ) |
21 |
20
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑎 ∈ 𝐼 ¬ ( 0 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) ) |
22 |
|
rabeq0 |
⊢ ( { 𝑎 ∈ 𝐼 ∣ ( 0 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } = ∅ ↔ ∀ 𝑎 ∈ 𝐼 ¬ ( 0 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) ) |
23 |
21 22
|
sylibr |
⊢ ( 𝜑 → { 𝑎 ∈ 𝐼 ∣ ( 0 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } = ∅ ) |
24 |
|
0fin |
⊢ ∅ ∈ Fin |
25 |
23 24
|
eqeltrdi |
⊢ ( 𝜑 → { 𝑎 ∈ 𝐼 ∣ ( 0 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin ) |
26 |
|
eqid |
⊢ ( 𝑆 ⊕m 𝑅 ) = ( 𝑆 ⊕m 𝑅 ) |
27 |
1 26 8 2 3 15
|
dsmmelbas |
⊢ ( 𝜑 → ( 0 ∈ 𝐻 ↔ ( 0 ∈ ( Base ‘ 𝑃 ) ∧ { 𝑎 ∈ 𝐼 ∣ ( 0 ‘ 𝑎 ) ≠ ( 0g ‘ ( 𝑅 ‘ 𝑎 ) ) } ∈ Fin ) ) ) |
28 |
10 25 27
|
mpbir2and |
⊢ ( 𝜑 → 0 ∈ 𝐻 ) |