Step |
Hyp |
Ref |
Expression |
1 |
|
dprdcntz.1 |
⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 ) |
2 |
|
dprdcntz.2 |
⊢ ( 𝜑 → dom 𝑆 = 𝐼 ) |
3 |
|
dprdcntz.3 |
⊢ ( 𝜑 → 𝑋 ∈ 𝐼 ) |
4 |
|
dprddisj.0 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
5 |
|
dprddisj.k |
⊢ 𝐾 = ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) |
6 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝑆 ‘ 𝑥 ) = ( 𝑆 ‘ 𝑋 ) ) |
7 |
|
sneq |
⊢ ( 𝑥 = 𝑋 → { 𝑥 } = { 𝑋 } ) |
8 |
7
|
difeq2d |
⊢ ( 𝑥 = 𝑋 → ( 𝐼 ∖ { 𝑥 } ) = ( 𝐼 ∖ { 𝑋 } ) ) |
9 |
8
|
imaeq2d |
⊢ ( 𝑥 = 𝑋 → ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) = ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) ) |
10 |
9
|
unieqd |
⊢ ( 𝑥 = 𝑋 → ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) = ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) ) |
11 |
10
|
fveq2d |
⊢ ( 𝑥 = 𝑋 → ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) = ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) ) ) |
12 |
6 11
|
ineq12d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = ( ( 𝑆 ‘ 𝑋 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) ) ) ) |
13 |
12
|
eqeq1d |
⊢ ( 𝑥 = 𝑋 → ( ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ↔ ( ( 𝑆 ‘ 𝑋 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) ) ) = { 0 } ) ) |
14 |
1 2
|
dprddomcld |
⊢ ( 𝜑 → 𝐼 ∈ V ) |
15 |
|
eqid |
⊢ ( Cntz ‘ 𝐺 ) = ( Cntz ‘ 𝐺 ) |
16 |
15 4 5
|
dmdprd |
⊢ ( ( 𝐼 ∈ V ∧ dom 𝑆 = 𝐼 ) → ( 𝐺 dom DProd 𝑆 ↔ ( 𝐺 ∈ Grp ∧ 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ) ) |
17 |
14 2 16
|
syl2anc |
⊢ ( 𝜑 → ( 𝐺 dom DProd 𝑆 ↔ ( 𝐺 ∈ Grp ∧ 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ) ) |
18 |
1 17
|
mpbid |
⊢ ( 𝜑 → ( 𝐺 ∈ Grp ∧ 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ) |
19 |
18
|
simp3d |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) |
20 |
|
simpr |
⊢ ( ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) → ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) |
21 |
20
|
ralimi |
⊢ ( ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) → ∀ 𝑥 ∈ 𝐼 ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) |
22 |
19 21
|
syl |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) |
23 |
13 22 3
|
rspcdva |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑋 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑋 } ) ) ) ) = { 0 } ) |