Step |
Hyp |
Ref |
Expression |
1 |
|
gsum2d2.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
gsum2d2.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
gsum2d2.g |
⊢ ( 𝜑 → 𝐺 ∈ CMnd ) |
4 |
|
gsum2d2.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
5 |
|
gsum2d2.r |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → 𝐶 ∈ 𝑊 ) |
6 |
|
gsum2d2.f |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ) → 𝑋 ∈ 𝐵 ) |
7 |
|
gsum2d2.u |
⊢ ( 𝜑 → 𝑈 ∈ Fin ) |
8 |
|
gsum2d2.n |
⊢ ( ( 𝜑 ∧ ( ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ∧ ¬ 𝑗 𝑈 𝑘 ) ) → 𝑋 = 0 ) |
9 |
|
snex |
⊢ { 𝑗 } ∈ V |
10 |
|
xpexg |
⊢ ( ( { 𝑗 } ∈ V ∧ 𝐶 ∈ 𝑊 ) → ( { 𝑗 } × 𝐶 ) ∈ V ) |
11 |
9 5 10
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ( { 𝑗 } × 𝐶 ) ∈ V ) |
12 |
11
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) ∈ V ) |
13 |
|
iunexg |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) ∈ V ) → ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) ∈ V ) |
14 |
4 12 13
|
syl2anc |
⊢ ( 𝜑 → ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) ∈ V ) |
15 |
|
relxp |
⊢ Rel ( { 𝑗 } × 𝐶 ) |
16 |
15
|
rgenw |
⊢ ∀ 𝑗 ∈ 𝐴 Rel ( { 𝑗 } × 𝐶 ) |
17 |
|
reliun |
⊢ ( Rel ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) ↔ ∀ 𝑗 ∈ 𝐴 Rel ( { 𝑗 } × 𝐶 ) ) |
18 |
16 17
|
mpbir |
⊢ Rel ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) |
19 |
18
|
a1i |
⊢ ( 𝜑 → Rel ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) ) |
20 |
|
vex |
⊢ 𝑥 ∈ V |
21 |
20
|
eldm2 |
⊢ ( 𝑥 ∈ dom ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) ↔ ∃ 𝑦 〈 𝑥 , 𝑦 〉 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) ) |
22 |
|
eliunxp |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) ↔ ∃ 𝑗 ∃ 𝑘 ( 〈 𝑥 , 𝑦 〉 = 〈 𝑗 , 𝑘 〉 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ) ) |
23 |
|
vex |
⊢ 𝑦 ∈ V |
24 |
20 23
|
opth1 |
⊢ ( 〈 𝑥 , 𝑦 〉 = 〈 𝑗 , 𝑘 〉 → 𝑥 = 𝑗 ) |
25 |
24
|
ad2antrl |
⊢ ( ( 𝜑 ∧ ( 〈 𝑥 , 𝑦 〉 = 〈 𝑗 , 𝑘 〉 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ) ) → 𝑥 = 𝑗 ) |
26 |
|
simprrl |
⊢ ( ( 𝜑 ∧ ( 〈 𝑥 , 𝑦 〉 = 〈 𝑗 , 𝑘 〉 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ) ) → 𝑗 ∈ 𝐴 ) |
27 |
25 26
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 〈 𝑥 , 𝑦 〉 = 〈 𝑗 , 𝑘 〉 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ) ) → 𝑥 ∈ 𝐴 ) |
28 |
27
|
ex |
⊢ ( 𝜑 → ( ( 〈 𝑥 , 𝑦 〉 = 〈 𝑗 , 𝑘 〉 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ) → 𝑥 ∈ 𝐴 ) ) |
29 |
28
|
exlimdvv |
⊢ ( 𝜑 → ( ∃ 𝑗 ∃ 𝑘 ( 〈 𝑥 , 𝑦 〉 = 〈 𝑗 , 𝑘 〉 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ) → 𝑥 ∈ 𝐴 ) ) |
30 |
22 29
|
syl5bi |
⊢ ( 𝜑 → ( 〈 𝑥 , 𝑦 〉 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) → 𝑥 ∈ 𝐴 ) ) |
31 |
30
|
exlimdv |
⊢ ( 𝜑 → ( ∃ 𝑦 〈 𝑥 , 𝑦 〉 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) → 𝑥 ∈ 𝐴 ) ) |
32 |
21 31
|
syl5bi |
⊢ ( 𝜑 → ( 𝑥 ∈ dom ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) → 𝑥 ∈ 𝐴 ) ) |
33 |
32
|
ssrdv |
⊢ ( 𝜑 → dom ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) ⊆ 𝐴 ) |
34 |
6
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑗 ∈ 𝐴 ∀ 𝑘 ∈ 𝐶 𝑋 ∈ 𝐵 ) |
35 |
|
eqid |
⊢ ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) = ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) |
36 |
35
|
fmpox |
⊢ ( ∀ 𝑗 ∈ 𝐴 ∀ 𝑘 ∈ 𝐶 𝑋 ∈ 𝐵 ↔ ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) : ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) ⟶ 𝐵 ) |
37 |
34 36
|
sylib |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) : ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) ⟶ 𝐵 ) |
38 |
1 2 3 4 5 6 7 8
|
gsum2d2lem |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) finSupp 0 ) |
39 |
1 2 3 14 19 4 33 37 38
|
gsum2d |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) ) = ( 𝐺 Σg ( 𝑚 ∈ 𝐴 ↦ ( 𝐺 Σg ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑚 } ) ↦ ( 𝑚 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) ) ) ) ) |
40 |
|
nfcv |
⊢ Ⅎ 𝑗 𝐺 |
41 |
|
nfcv |
⊢ Ⅎ 𝑗 Σg |
42 |
|
nfiu1 |
⊢ Ⅎ 𝑗 ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) |
43 |
|
nfcv |
⊢ Ⅎ 𝑗 { 𝑚 } |
44 |
42 43
|
nfima |
⊢ Ⅎ 𝑗 ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑚 } ) |
45 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑚 |
46 |
|
nfmpo1 |
⊢ Ⅎ 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) |
47 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑛 |
48 |
45 46 47
|
nfov |
⊢ Ⅎ 𝑗 ( 𝑚 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) |
49 |
44 48
|
nfmpt |
⊢ Ⅎ 𝑗 ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑚 } ) ↦ ( 𝑚 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) |
50 |
40 41 49
|
nfov |
⊢ Ⅎ 𝑗 ( 𝐺 Σg ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑚 } ) ↦ ( 𝑚 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) ) |
51 |
|
nfcv |
⊢ Ⅎ 𝑚 ( 𝐺 Σg ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑗 } ) ↦ ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) ) |
52 |
|
sneq |
⊢ ( 𝑚 = 𝑗 → { 𝑚 } = { 𝑗 } ) |
53 |
52
|
imaeq2d |
⊢ ( 𝑚 = 𝑗 → ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑚 } ) = ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑗 } ) ) |
54 |
|
oveq1 |
⊢ ( 𝑚 = 𝑗 → ( 𝑚 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) = ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) |
55 |
53 54
|
mpteq12dv |
⊢ ( 𝑚 = 𝑗 → ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑚 } ) ↦ ( 𝑚 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) = ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑗 } ) ↦ ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) ) |
56 |
55
|
oveq2d |
⊢ ( 𝑚 = 𝑗 → ( 𝐺 Σg ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑚 } ) ↦ ( 𝑚 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) ) = ( 𝐺 Σg ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑗 } ) ↦ ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) ) ) |
57 |
50 51 56
|
cbvmpt |
⊢ ( 𝑚 ∈ 𝐴 ↦ ( 𝐺 Σg ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑚 } ) ↦ ( 𝑚 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) ) ) = ( 𝑗 ∈ 𝐴 ↦ ( 𝐺 Σg ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑗 } ) ↦ ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) ) ) |
58 |
|
vex |
⊢ 𝑗 ∈ V |
59 |
|
vex |
⊢ 𝑘 ∈ V |
60 |
58 59
|
elimasn |
⊢ ( 𝑘 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑗 } ) ↔ 〈 𝑗 , 𝑘 〉 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) ) |
61 |
|
opeliunxp |
⊢ ( 〈 𝑗 , 𝑘 〉 ∈ ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) ↔ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ) |
62 |
60 61
|
bitri |
⊢ ( 𝑘 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑗 } ) ↔ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ) |
63 |
62
|
baib |
⊢ ( 𝑗 ∈ 𝐴 → ( 𝑘 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑗 } ) ↔ 𝑘 ∈ 𝐶 ) ) |
64 |
63
|
eqrdv |
⊢ ( 𝑗 ∈ 𝐴 → ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑗 } ) = 𝐶 ) |
65 |
64
|
mpteq1d |
⊢ ( 𝑗 ∈ 𝐴 → ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑗 } ) ↦ ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) = ( 𝑛 ∈ 𝐶 ↦ ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) ) |
66 |
|
nfcv |
⊢ Ⅎ 𝑘 𝑗 |
67 |
|
nfmpo2 |
⊢ Ⅎ 𝑘 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) |
68 |
|
nfcv |
⊢ Ⅎ 𝑘 𝑛 |
69 |
66 67 68
|
nfov |
⊢ Ⅎ 𝑘 ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) |
70 |
|
nfcv |
⊢ Ⅎ 𝑛 ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑘 ) |
71 |
|
oveq2 |
⊢ ( 𝑛 = 𝑘 → ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) = ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑘 ) ) |
72 |
69 70 71
|
cbvmpt |
⊢ ( 𝑛 ∈ 𝐶 ↦ ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) = ( 𝑘 ∈ 𝐶 ↦ ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑘 ) ) |
73 |
65 72
|
eqtrdi |
⊢ ( 𝑗 ∈ 𝐴 → ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑗 } ) ↦ ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) = ( 𝑘 ∈ 𝐶 ↦ ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑘 ) ) ) |
74 |
73
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑗 } ) ↦ ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) = ( 𝑘 ∈ 𝐶 ↦ ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑘 ) ) ) |
75 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ) → 𝑗 ∈ 𝐴 ) |
76 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ) → 𝑘 ∈ 𝐶 ) |
77 |
35
|
ovmpt4g |
⊢ ( ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑘 ) = 𝑋 ) |
78 |
75 76 6 77
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ) → ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑘 ) = 𝑋 ) |
79 |
78
|
anassrs |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑘 ∈ 𝐶 ) → ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑘 ) = 𝑋 ) |
80 |
79
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ( 𝑘 ∈ 𝐶 ↦ ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑘 ) ) = ( 𝑘 ∈ 𝐶 ↦ 𝑋 ) ) |
81 |
74 80
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑗 } ) ↦ ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) = ( 𝑘 ∈ 𝐶 ↦ 𝑋 ) ) |
82 |
81
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ( 𝐺 Σg ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑗 } ) ↦ ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) ) = ( 𝐺 Σg ( 𝑘 ∈ 𝐶 ↦ 𝑋 ) ) ) |
83 |
82
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝐴 ↦ ( 𝐺 Σg ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑗 } ) ↦ ( 𝑗 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) ) ) = ( 𝑗 ∈ 𝐴 ↦ ( 𝐺 Σg ( 𝑘 ∈ 𝐶 ↦ 𝑋 ) ) ) ) |
84 |
57 83
|
eqtrid |
⊢ ( 𝜑 → ( 𝑚 ∈ 𝐴 ↦ ( 𝐺 Σg ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑚 } ) ↦ ( 𝑚 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) ) ) = ( 𝑗 ∈ 𝐴 ↦ ( 𝐺 Σg ( 𝑘 ∈ 𝐶 ↦ 𝑋 ) ) ) ) |
85 |
84
|
oveq2d |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑚 ∈ 𝐴 ↦ ( 𝐺 Σg ( 𝑛 ∈ ( ∪ 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐶 ) “ { 𝑚 } ) ↦ ( 𝑚 ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) 𝑛 ) ) ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝐴 ↦ ( 𝐺 Σg ( 𝑘 ∈ 𝐶 ↦ 𝑋 ) ) ) ) ) |
86 |
39 85
|
eqtrd |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝐴 ↦ ( 𝐺 Σg ( 𝑘 ∈ 𝐶 ↦ 𝑋 ) ) ) ) ) |