Step |
Hyp |
Ref |
Expression |
1 |
|
dprd2d2.1 |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ) ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
2 |
|
dprd2d2.2 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → 𝐺 dom DProd ( 𝑗 ∈ 𝐽 ↦ 𝑆 ) ) |
3 |
|
dprd2d2.3 |
⊢ ( 𝜑 → 𝐺 dom DProd ( 𝑖 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑗 ∈ 𝐽 ↦ 𝑆 ) ) ) ) |
4 |
|
relxp |
⊢ Rel ( { 𝑖 } × 𝐽 ) |
5 |
4
|
rgenw |
⊢ ∀ 𝑖 ∈ 𝐼 Rel ( { 𝑖 } × 𝐽 ) |
6 |
|
reliun |
⊢ ( Rel ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) ↔ ∀ 𝑖 ∈ 𝐼 Rel ( { 𝑖 } × 𝐽 ) ) |
7 |
5 6
|
mpbir |
⊢ Rel ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) |
8 |
7
|
a1i |
⊢ ( 𝜑 → Rel ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) ) |
9 |
1
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑖 ∈ 𝐼 ∀ 𝑗 ∈ 𝐽 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
10 |
|
eqid |
⊢ ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) = ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) |
11 |
10
|
fmpox |
⊢ ( ∀ 𝑖 ∈ 𝐼 ∀ 𝑗 ∈ 𝐽 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ↔ ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) : ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) ⟶ ( SubGrp ‘ 𝐺 ) ) |
12 |
9 11
|
sylib |
⊢ ( 𝜑 → ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) : ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) ⟶ ( SubGrp ‘ 𝐺 ) ) |
13 |
|
dmiun |
⊢ dom ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) = ∪ 𝑖 ∈ 𝐼 dom ( { 𝑖 } × 𝐽 ) |
14 |
|
dmxpss |
⊢ dom ( { 𝑖 } × 𝐽 ) ⊆ { 𝑖 } |
15 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → 𝑖 ∈ 𝐼 ) |
16 |
15
|
snssd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → { 𝑖 } ⊆ 𝐼 ) |
17 |
14 16
|
sstrid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → dom ( { 𝑖 } × 𝐽 ) ⊆ 𝐼 ) |
18 |
17
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑖 ∈ 𝐼 dom ( { 𝑖 } × 𝐽 ) ⊆ 𝐼 ) |
19 |
|
iunss |
⊢ ( ∪ 𝑖 ∈ 𝐼 dom ( { 𝑖 } × 𝐽 ) ⊆ 𝐼 ↔ ∀ 𝑖 ∈ 𝐼 dom ( { 𝑖 } × 𝐽 ) ⊆ 𝐼 ) |
20 |
18 19
|
sylibr |
⊢ ( 𝜑 → ∪ 𝑖 ∈ 𝐼 dom ( { 𝑖 } × 𝐽 ) ⊆ 𝐼 ) |
21 |
13 20
|
eqsstrid |
⊢ ( 𝜑 → dom ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) ⊆ 𝐼 ) |
22 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ) ) → 𝑖 ∈ 𝐼 ) |
23 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ) ) → 𝑗 ∈ 𝐽 ) |
24 |
10
|
ovmpt4g |
⊢ ( ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ∧ 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) = 𝑆 ) |
25 |
22 23 1 24
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ) ) → ( 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) = 𝑆 ) |
26 |
25
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) ∧ 𝑗 ∈ 𝐽 ) → ( 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) = 𝑆 ) |
27 |
26
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( 𝑗 ∈ 𝐽 ↦ ( 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) = ( 𝑗 ∈ 𝐽 ↦ 𝑆 ) ) |
28 |
2 27
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → 𝐺 dom DProd ( 𝑗 ∈ 𝐽 ↦ ( 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) |
29 |
28
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑖 ∈ 𝐼 𝐺 dom DProd ( 𝑗 ∈ 𝐽 ↦ ( 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) |
30 |
|
nfcv |
⊢ Ⅎ 𝑖 𝐺 |
31 |
|
nfcv |
⊢ Ⅎ 𝑖 dom DProd |
32 |
|
nfcsb1v |
⊢ Ⅎ 𝑖 ⦋ 𝑥 / 𝑖 ⦌ 𝐽 |
33 |
|
nfcv |
⊢ Ⅎ 𝑖 𝑥 |
34 |
|
nfmpo1 |
⊢ Ⅎ 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) |
35 |
|
nfcv |
⊢ Ⅎ 𝑖 𝑗 |
36 |
33 34 35
|
nfov |
⊢ Ⅎ 𝑖 ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) |
37 |
32 36
|
nfmpt |
⊢ Ⅎ 𝑖 ( 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) |
38 |
30 31 37
|
nfbr |
⊢ Ⅎ 𝑖 𝐺 dom DProd ( 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) |
39 |
|
csbeq1a |
⊢ ( 𝑖 = 𝑥 → 𝐽 = ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ) |
40 |
|
oveq1 |
⊢ ( 𝑖 = 𝑥 → ( 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) = ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) |
41 |
39 40
|
mpteq12dv |
⊢ ( 𝑖 = 𝑥 → ( 𝑗 ∈ 𝐽 ↦ ( 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) = ( 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) |
42 |
41
|
breq2d |
⊢ ( 𝑖 = 𝑥 → ( 𝐺 dom DProd ( 𝑗 ∈ 𝐽 ↦ ( 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ↔ 𝐺 dom DProd ( 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) ) |
43 |
38 42
|
rspc |
⊢ ( 𝑥 ∈ 𝐼 → ( ∀ 𝑖 ∈ 𝐼 𝐺 dom DProd ( 𝑗 ∈ 𝐽 ↦ ( 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) → 𝐺 dom DProd ( 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) ) |
44 |
29 43
|
mpan9 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝐺 dom DProd ( 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) |
45 |
|
nfcv |
⊢ Ⅎ 𝑦 ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) |
46 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑥 |
47 |
|
nfmpo2 |
⊢ Ⅎ 𝑗 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) |
48 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑦 |
49 |
46 47 48
|
nfov |
⊢ Ⅎ 𝑗 ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑦 ) |
50 |
|
oveq2 |
⊢ ( 𝑗 = 𝑦 → ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) = ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑦 ) ) |
51 |
45 49 50
|
cbvmpt |
⊢ ( 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) = ( 𝑦 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑦 ) ) |
52 |
|
nfv |
⊢ Ⅎ 𝑖 𝑗 = 𝑧 |
53 |
32
|
nfcri |
⊢ Ⅎ 𝑖 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 |
54 |
52 53
|
nfan |
⊢ Ⅎ 𝑖 ( 𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ) |
55 |
39
|
eleq2d |
⊢ ( 𝑖 = 𝑥 → ( 𝑗 ∈ 𝐽 ↔ 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ) ) |
56 |
55
|
anbi2d |
⊢ ( 𝑖 = 𝑥 → ( ( 𝑗 = 𝑧 ∧ 𝑗 ∈ 𝐽 ) ↔ ( 𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ) ) ) |
57 |
54 56
|
equsexv |
⊢ ( ∃ 𝑖 ( 𝑖 = 𝑥 ∧ ( 𝑗 = 𝑧 ∧ 𝑗 ∈ 𝐽 ) ) ↔ ( 𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ) ) |
58 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) ∧ ( 𝑖 = 𝑥 ∧ 𝑗 = 𝑧 ) ) → 𝑖 = 𝑥 ) |
59 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) ∧ ( 𝑖 = 𝑥 ∧ 𝑗 = 𝑧 ) ) → 𝑥 ∈ 𝐼 ) |
60 |
58 59
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) ∧ ( 𝑖 = 𝑥 ∧ 𝑗 = 𝑧 ) ) → 𝑖 ∈ 𝐼 ) |
61 |
60
|
biantrurd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) ∧ ( 𝑖 = 𝑥 ∧ 𝑗 = 𝑧 ) ) → ( 𝑗 ∈ 𝐽 ↔ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ) ) ) |
62 |
61
|
pm5.32da |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ( ( 𝑖 = 𝑥 ∧ 𝑗 = 𝑧 ) ∧ 𝑗 ∈ 𝐽 ) ↔ ( ( 𝑖 = 𝑥 ∧ 𝑗 = 𝑧 ) ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ) ) ) ) |
63 |
|
anass |
⊢ ( ( ( 𝑖 = 𝑥 ∧ 𝑗 = 𝑧 ) ∧ 𝑗 ∈ 𝐽 ) ↔ ( 𝑖 = 𝑥 ∧ ( 𝑗 = 𝑧 ∧ 𝑗 ∈ 𝐽 ) ) ) |
64 |
|
eqcom |
⊢ ( 〈 𝑥 , 𝑧 〉 = 〈 𝑖 , 𝑗 〉 ↔ 〈 𝑖 , 𝑗 〉 = 〈 𝑥 , 𝑧 〉 ) |
65 |
|
vex |
⊢ 𝑖 ∈ V |
66 |
|
vex |
⊢ 𝑗 ∈ V |
67 |
65 66
|
opth |
⊢ ( 〈 𝑖 , 𝑗 〉 = 〈 𝑥 , 𝑧 〉 ↔ ( 𝑖 = 𝑥 ∧ 𝑗 = 𝑧 ) ) |
68 |
64 67
|
bitr2i |
⊢ ( ( 𝑖 = 𝑥 ∧ 𝑗 = 𝑧 ) ↔ 〈 𝑥 , 𝑧 〉 = 〈 𝑖 , 𝑗 〉 ) |
69 |
68
|
anbi1i |
⊢ ( ( ( 𝑖 = 𝑥 ∧ 𝑗 = 𝑧 ) ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ) ) ↔ ( 〈 𝑥 , 𝑧 〉 = 〈 𝑖 , 𝑗 〉 ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ) ) ) |
70 |
62 63 69
|
3bitr3g |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝑖 = 𝑥 ∧ ( 𝑗 = 𝑧 ∧ 𝑗 ∈ 𝐽 ) ) ↔ ( 〈 𝑥 , 𝑧 〉 = 〈 𝑖 , 𝑗 〉 ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ) ) ) ) |
71 |
70
|
exbidv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ∃ 𝑖 ( 𝑖 = 𝑥 ∧ ( 𝑗 = 𝑧 ∧ 𝑗 ∈ 𝐽 ) ) ↔ ∃ 𝑖 ( 〈 𝑥 , 𝑧 〉 = 〈 𝑖 , 𝑗 〉 ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ) ) ) ) |
72 |
57 71
|
bitr3id |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ) ↔ ∃ 𝑖 ( 〈 𝑥 , 𝑧 〉 = 〈 𝑖 , 𝑗 〉 ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ) ) ) ) |
73 |
72
|
exbidv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ∃ 𝑗 ( 𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ) ↔ ∃ 𝑗 ∃ 𝑖 ( 〈 𝑥 , 𝑧 〉 = 〈 𝑖 , 𝑗 〉 ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ) ) ) ) |
74 |
|
vex |
⊢ 𝑧 ∈ V |
75 |
|
eleq1w |
⊢ ( 𝑗 = 𝑧 → ( 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↔ 𝑧 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ) ) |
76 |
74 75
|
ceqsexv |
⊢ ( ∃ 𝑗 ( 𝑗 = 𝑧 ∧ 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ) ↔ 𝑧 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ) |
77 |
|
excom |
⊢ ( ∃ 𝑗 ∃ 𝑖 ( 〈 𝑥 , 𝑧 〉 = 〈 𝑖 , 𝑗 〉 ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ) ) ↔ ∃ 𝑖 ∃ 𝑗 ( 〈 𝑥 , 𝑧 〉 = 〈 𝑖 , 𝑗 〉 ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ) ) ) |
78 |
73 76 77
|
3bitr3g |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑧 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↔ ∃ 𝑖 ∃ 𝑗 ( 〈 𝑥 , 𝑧 〉 = 〈 𝑖 , 𝑗 〉 ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ) ) ) ) |
79 |
|
elrelimasn |
⊢ ( Rel ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) → ( 𝑧 ∈ ( ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) “ { 𝑥 } ) ↔ 𝑥 ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) 𝑧 ) ) |
80 |
7 79
|
ax-mp |
⊢ ( 𝑧 ∈ ( ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) “ { 𝑥 } ) ↔ 𝑥 ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) 𝑧 ) |
81 |
|
df-br |
⊢ ( 𝑥 ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) 𝑧 ↔ 〈 𝑥 , 𝑧 〉 ∈ ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) ) |
82 |
|
eliunxp |
⊢ ( 〈 𝑥 , 𝑧 〉 ∈ ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) ↔ ∃ 𝑖 ∃ 𝑗 ( 〈 𝑥 , 𝑧 〉 = 〈 𝑖 , 𝑗 〉 ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ) ) ) |
83 |
80 81 82
|
3bitri |
⊢ ( 𝑧 ∈ ( ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) “ { 𝑥 } ) ↔ ∃ 𝑖 ∃ 𝑗 ( 〈 𝑥 , 𝑧 〉 = 〈 𝑖 , 𝑗 〉 ∧ ( 𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 𝐽 ) ) ) |
84 |
78 83
|
bitr4di |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑧 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↔ 𝑧 ∈ ( ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) “ { 𝑥 } ) ) ) |
85 |
84
|
eqrdv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ⦋ 𝑥 / 𝑖 ⦌ 𝐽 = ( ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) “ { 𝑥 } ) ) |
86 |
85
|
mpteq1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑦 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑦 ) ) = ( 𝑦 ∈ ( ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) “ { 𝑥 } ) ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑦 ) ) ) |
87 |
51 86
|
syl5eq |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) = ( 𝑦 ∈ ( ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) “ { 𝑥 } ) ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑦 ) ) ) |
88 |
44 87
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝐺 dom DProd ( 𝑦 ∈ ( ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) “ { 𝑥 } ) ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑦 ) ) ) |
89 |
27
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( 𝐺 DProd ( 𝑗 ∈ 𝐽 ↦ ( 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) = ( 𝐺 DProd ( 𝑗 ∈ 𝐽 ↦ 𝑆 ) ) ) |
90 |
89
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑖 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑗 ∈ 𝐽 ↦ ( 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑗 ∈ 𝐽 ↦ 𝑆 ) ) ) ) |
91 |
3 90
|
breqtrrd |
⊢ ( 𝜑 → 𝐺 dom DProd ( 𝑖 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑗 ∈ 𝐽 ↦ ( 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) ) ) |
92 |
|
nfcv |
⊢ Ⅎ 𝑥 ( 𝐺 DProd ( 𝑗 ∈ 𝐽 ↦ ( 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) |
93 |
|
nfcv |
⊢ Ⅎ 𝑖 DProd |
94 |
30 93 37
|
nfov |
⊢ Ⅎ 𝑖 ( 𝐺 DProd ( 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) |
95 |
41
|
oveq2d |
⊢ ( 𝑖 = 𝑥 → ( 𝐺 DProd ( 𝑗 ∈ 𝐽 ↦ ( 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) = ( 𝐺 DProd ( 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) ) |
96 |
92 94 95
|
cbvmpt |
⊢ ( 𝑖 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑗 ∈ 𝐽 ↦ ( 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) ) |
97 |
87
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 DProd ( 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) = ( 𝐺 DProd ( 𝑦 ∈ ( ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) “ { 𝑥 } ) ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑦 ) ) ) ) |
98 |
97
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑗 ∈ ⦋ 𝑥 / 𝑖 ⦌ 𝐽 ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑦 ∈ ( ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) “ { 𝑥 } ) ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑦 ) ) ) ) ) |
99 |
96 98
|
syl5eq |
⊢ ( 𝜑 → ( 𝑖 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑗 ∈ 𝐽 ↦ ( 𝑖 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑗 ) ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑦 ∈ ( ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) “ { 𝑥 } ) ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑦 ) ) ) ) ) |
100 |
91 99
|
breqtrd |
⊢ ( 𝜑 → 𝐺 dom DProd ( 𝑥 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑦 ∈ ( ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) “ { 𝑥 } ) ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑦 ) ) ) ) ) |
101 |
|
eqid |
⊢ ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) = ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) |
102 |
8 12 21 88 100 101
|
dprd2da |
⊢ ( 𝜑 → 𝐺 dom DProd ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) ) |
103 |
8 12 21 88 100 101
|
dprd2db |
⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) ) = ( 𝐺 DProd ( 𝑥 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑦 ∈ ( ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) “ { 𝑥 } ) ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑦 ) ) ) ) ) ) |
104 |
99 90
|
eqtr3d |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑦 ∈ ( ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) “ { 𝑥 } ) ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑦 ) ) ) ) = ( 𝑖 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑗 ∈ 𝐽 ↦ 𝑆 ) ) ) ) |
105 |
104
|
oveq2d |
⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑥 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑦 ∈ ( ∪ 𝑖 ∈ 𝐼 ( { 𝑖 } × 𝐽 ) “ { 𝑥 } ) ↦ ( 𝑥 ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) 𝑦 ) ) ) ) ) = ( 𝐺 DProd ( 𝑖 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑗 ∈ 𝐽 ↦ 𝑆 ) ) ) ) ) |
106 |
103 105
|
eqtrd |
⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) ) = ( 𝐺 DProd ( 𝑖 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑗 ∈ 𝐽 ↦ 𝑆 ) ) ) ) ) |
107 |
102 106
|
jca |
⊢ ( 𝜑 → ( 𝐺 dom DProd ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) ∧ ( 𝐺 DProd ( 𝑖 ∈ 𝐼 , 𝑗 ∈ 𝐽 ↦ 𝑆 ) ) = ( 𝐺 DProd ( 𝑖 ∈ 𝐼 ↦ ( 𝐺 DProd ( 𝑗 ∈ 𝐽 ↦ 𝑆 ) ) ) ) ) ) |