Step |
Hyp |
Ref |
Expression |
1 |
|
dmdprd.z |
⊢ 𝑍 = ( Cntz ‘ 𝐺 ) |
2 |
|
dmdprd.0 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
dmdprd.k |
⊢ 𝐾 = ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) |
4 |
|
elex |
⊢ ( 𝑆 ∈ { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) } → 𝑆 ∈ V ) |
5 |
4
|
a1i |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) → ( 𝑆 ∈ { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) } → 𝑆 ∈ V ) ) |
6 |
|
fex |
⊢ ( ( 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ∧ 𝐼 ∈ 𝑉 ) → 𝑆 ∈ V ) |
7 |
6
|
expcom |
⊢ ( 𝐼 ∈ 𝑉 → ( 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) → 𝑆 ∈ V ) ) |
8 |
7
|
adantr |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) → ( 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) → 𝑆 ∈ V ) ) |
9 |
8
|
adantrd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) → ( ( 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) → 𝑆 ∈ V ) ) |
10 |
|
df-sbc |
⊢ ( [ 𝑆 / ℎ ] ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ↔ 𝑆 ∈ { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) } ) |
11 |
|
simpr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ 𝑆 ∈ V ) → 𝑆 ∈ V ) |
12 |
|
simpr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → ℎ = 𝑆 ) |
13 |
12
|
dmeqd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → dom ℎ = dom 𝑆 ) |
14 |
|
simplr |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → dom 𝑆 = 𝐼 ) |
15 |
13 14
|
eqtrd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → dom ℎ = 𝐼 ) |
16 |
12 15
|
feq12d |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝐺 ) ↔ 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ) ) |
17 |
15
|
difeq1d |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → ( dom ℎ ∖ { 𝑥 } ) = ( 𝐼 ∖ { 𝑥 } ) ) |
18 |
12
|
fveq1d |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → ( ℎ ‘ 𝑥 ) = ( 𝑆 ‘ 𝑥 ) ) |
19 |
12
|
fveq1d |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → ( ℎ ‘ 𝑦 ) = ( 𝑆 ‘ 𝑦 ) ) |
20 |
19
|
fveq2d |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) = ( 𝑍 ‘ ( 𝑆 ‘ 𝑦 ) ) ) |
21 |
18 20
|
sseq12d |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → ( ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ↔ ( 𝑆 ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑦 ) ) ) ) |
22 |
17 21
|
raleqbidv |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑦 ) ) ) ) |
23 |
12 17
|
imaeq12d |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) = ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) |
24 |
23
|
unieqd |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) = ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) |
25 |
24
|
fveq2d |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) = ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) |
26 |
18 25
|
ineq12d |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) ) |
27 |
26
|
eqeq1d |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → ( ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ↔ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) |
28 |
22 27
|
anbi12d |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → ( ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ↔ ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ) |
29 |
15 28
|
raleqbidv |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → ( ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ↔ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ) |
30 |
16 29
|
anbi12d |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ ℎ = 𝑆 ) → ( ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ↔ ( 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ) ) |
31 |
30
|
adantlr |
⊢ ( ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ 𝑆 ∈ V ) ∧ ℎ = 𝑆 ) → ( ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ↔ ( 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ) ) |
32 |
11 31
|
sbcied |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ 𝑆 ∈ V ) → ( [ 𝑆 / ℎ ] ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ↔ ( 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ) ) |
33 |
10 32
|
bitr3id |
⊢ ( ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) ∧ 𝑆 ∈ V ) → ( 𝑆 ∈ { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) } ↔ ( 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ) ) |
34 |
33
|
ex |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) → ( 𝑆 ∈ V → ( 𝑆 ∈ { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) } ↔ ( 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ) ) ) |
35 |
5 9 34
|
pm5.21ndd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) → ( 𝑆 ∈ { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) } ↔ ( 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ) ) |
36 |
35
|
anbi2d |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) → ( ( 𝐺 ∈ Grp ∧ 𝑆 ∈ { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) } ) ↔ ( 𝐺 ∈ Grp ∧ ( 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ) ) ) |
37 |
|
df-br |
⊢ ( 𝐺 dom DProd 𝑆 ↔ 〈 𝐺 , 𝑆 〉 ∈ dom DProd ) |
38 |
|
fvex |
⊢ ( 𝑠 ‘ 𝑥 ) ∈ V |
39 |
38
|
rgenw |
⊢ ∀ 𝑥 ∈ dom 𝑠 ( 𝑠 ‘ 𝑥 ) ∈ V |
40 |
|
ixpexg |
⊢ ( ∀ 𝑥 ∈ dom 𝑠 ( 𝑠 ‘ 𝑥 ) ∈ V → X 𝑥 ∈ dom 𝑠 ( 𝑠 ‘ 𝑥 ) ∈ V ) |
41 |
39 40
|
ax-mp |
⊢ X 𝑥 ∈ dom 𝑠 ( 𝑠 ‘ 𝑥 ) ∈ V |
42 |
41
|
mptrabex |
⊢ ( 𝑓 ∈ { ℎ ∈ X 𝑥 ∈ dom 𝑠 ( 𝑠 ‘ 𝑥 ) ∣ ℎ finSupp ( 0g ‘ 𝑔 ) } ↦ ( 𝑔 Σg 𝑓 ) ) ∈ V |
43 |
42
|
rnex |
⊢ ran ( 𝑓 ∈ { ℎ ∈ X 𝑥 ∈ dom 𝑠 ( 𝑠 ‘ 𝑥 ) ∣ ℎ finSupp ( 0g ‘ 𝑔 ) } ↦ ( 𝑔 Σg 𝑓 ) ) ∈ V |
44 |
43
|
rgen2w |
⊢ ∀ 𝑔 ∈ Grp ∀ 𝑠 ∈ { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝑔 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝑔 ) ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝑔 ) ) ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝑔 ) } ) ) } ran ( 𝑓 ∈ { ℎ ∈ X 𝑥 ∈ dom 𝑠 ( 𝑠 ‘ 𝑥 ) ∣ ℎ finSupp ( 0g ‘ 𝑔 ) } ↦ ( 𝑔 Σg 𝑓 ) ) ∈ V |
45 |
|
df-dprd |
⊢ DProd = ( 𝑔 ∈ Grp , 𝑠 ∈ { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝑔 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝑔 ) ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝑔 ) ) ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝑔 ) } ) ) } ↦ ran ( 𝑓 ∈ { ℎ ∈ X 𝑥 ∈ dom 𝑠 ( 𝑠 ‘ 𝑥 ) ∣ ℎ finSupp ( 0g ‘ 𝑔 ) } ↦ ( 𝑔 Σg 𝑓 ) ) ) |
46 |
45
|
fmpox |
⊢ ( ∀ 𝑔 ∈ Grp ∀ 𝑠 ∈ { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝑔 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝑔 ) ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝑔 ) ) ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝑔 ) } ) ) } ran ( 𝑓 ∈ { ℎ ∈ X 𝑥 ∈ dom 𝑠 ( 𝑠 ‘ 𝑥 ) ∣ ℎ finSupp ( 0g ‘ 𝑔 ) } ↦ ( 𝑔 Σg 𝑓 ) ) ∈ V ↔ DProd : ∪ 𝑔 ∈ Grp ( { 𝑔 } × { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝑔 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝑔 ) ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝑔 ) ) ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝑔 ) } ) ) } ) ⟶ V ) |
47 |
44 46
|
mpbi |
⊢ DProd : ∪ 𝑔 ∈ Grp ( { 𝑔 } × { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝑔 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝑔 ) ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝑔 ) ) ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝑔 ) } ) ) } ) ⟶ V |
48 |
47
|
fdmi |
⊢ dom DProd = ∪ 𝑔 ∈ Grp ( { 𝑔 } × { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝑔 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝑔 ) ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝑔 ) ) ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝑔 ) } ) ) } ) |
49 |
48
|
eleq2i |
⊢ ( 〈 𝐺 , 𝑆 〉 ∈ dom DProd ↔ 〈 𝐺 , 𝑆 〉 ∈ ∪ 𝑔 ∈ Grp ( { 𝑔 } × { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝑔 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝑔 ) ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝑔 ) ) ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝑔 ) } ) ) } ) ) |
50 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( SubGrp ‘ 𝑔 ) = ( SubGrp ‘ 𝐺 ) ) |
51 |
50
|
feq3d |
⊢ ( 𝑔 = 𝐺 → ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝑔 ) ↔ ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝐺 ) ) ) |
52 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( Cntz ‘ 𝑔 ) = ( Cntz ‘ 𝐺 ) ) |
53 |
52 1
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( Cntz ‘ 𝑔 ) = 𝑍 ) |
54 |
53
|
fveq1d |
⊢ ( 𝑔 = 𝐺 → ( ( Cntz ‘ 𝑔 ) ‘ ( ℎ ‘ 𝑦 ) ) = ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ) |
55 |
54
|
sseq2d |
⊢ ( 𝑔 = 𝐺 → ( ( ℎ ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝑔 ) ‘ ( ℎ ‘ 𝑦 ) ) ↔ ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ) ) |
56 |
55
|
ralbidv |
⊢ ( 𝑔 = 𝐺 → ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝑔 ) ‘ ( ℎ ‘ 𝑦 ) ) ↔ ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ) ) |
57 |
50
|
fveq2d |
⊢ ( 𝑔 = 𝐺 → ( mrCls ‘ ( SubGrp ‘ 𝑔 ) ) = ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) ) |
58 |
57 3
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( mrCls ‘ ( SubGrp ‘ 𝑔 ) ) = 𝐾 ) |
59 |
58
|
fveq1d |
⊢ ( 𝑔 = 𝐺 → ( ( mrCls ‘ ( SubGrp ‘ 𝑔 ) ) ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) = ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) |
60 |
59
|
ineq2d |
⊢ ( 𝑔 = 𝐺 → ( ( ℎ ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝑔 ) ) ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) ) |
61 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( 0g ‘ 𝑔 ) = ( 0g ‘ 𝐺 ) ) |
62 |
61 2
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( 0g ‘ 𝑔 ) = 0 ) |
63 |
62
|
sneqd |
⊢ ( 𝑔 = 𝐺 → { ( 0g ‘ 𝑔 ) } = { 0 } ) |
64 |
60 63
|
eqeq12d |
⊢ ( 𝑔 = 𝐺 → ( ( ( ℎ ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝑔 ) ) ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝑔 ) } ↔ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) |
65 |
56 64
|
anbi12d |
⊢ ( 𝑔 = 𝐺 → ( ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝑔 ) ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝑔 ) ) ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝑔 ) } ) ↔ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ) |
66 |
65
|
ralbidv |
⊢ ( 𝑔 = 𝐺 → ( ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝑔 ) ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝑔 ) ) ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝑔 ) } ) ↔ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ) |
67 |
51 66
|
anbi12d |
⊢ ( 𝑔 = 𝐺 → ( ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝑔 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝑔 ) ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝑔 ) ) ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝑔 ) } ) ) ↔ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ) ) |
68 |
67
|
abbidv |
⊢ ( 𝑔 = 𝐺 → { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝑔 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝑔 ) ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝑔 ) ) ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝑔 ) } ) ) } = { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) } ) |
69 |
68
|
opeliunxp2 |
⊢ ( 〈 𝐺 , 𝑆 〉 ∈ ∪ 𝑔 ∈ Grp ( { 𝑔 } × { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝑔 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( ( Cntz ‘ 𝑔 ) ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( ( mrCls ‘ ( SubGrp ‘ 𝑔 ) ) ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { ( 0g ‘ 𝑔 ) } ) ) } ) ↔ ( 𝐺 ∈ Grp ∧ 𝑆 ∈ { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) } ) ) |
70 |
37 49 69
|
3bitri |
⊢ ( 𝐺 dom DProd 𝑆 ↔ ( 𝐺 ∈ Grp ∧ 𝑆 ∈ { ℎ ∣ ( ℎ : dom ℎ ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ dom ℎ ( ∀ 𝑦 ∈ ( dom ℎ ∖ { 𝑥 } ) ( ℎ ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( ℎ ‘ 𝑦 ) ) ∧ ( ( ℎ ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( ℎ “ ( dom ℎ ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) } ) ) |
71 |
|
3anass |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ↔ ( 𝐺 ∈ Grp ∧ ( 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ) ) |
72 |
36 70 71
|
3bitr4g |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ dom 𝑆 = 𝐼 ) → ( 𝐺 dom DProd 𝑆 ↔ ( 𝐺 ∈ Grp ∧ 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ∧ ∀ 𝑥 ∈ 𝐼 ( ∀ 𝑦 ∈ ( 𝐼 ∖ { 𝑥 } ) ( 𝑆 ‘ 𝑥 ) ⊆ ( 𝑍 ‘ ( 𝑆 ‘ 𝑦 ) ) ∧ ( ( 𝑆 ‘ 𝑥 ) ∩ ( 𝐾 ‘ ∪ ( 𝑆 “ ( 𝐼 ∖ { 𝑥 } ) ) ) ) = { 0 } ) ) ) ) |