Metamath Proof Explorer

Theorem gsum2d2

Description: Write a group sum over a two-dimensional region as a double sum. Note that C ( j ) is a function of j . (Contributed by Mario Carneiro, 28-Dec-2014)

Ref Expression
Hypotheses gsum2d2.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
gsum2d2.z ⊢ 0 = ( 0g ‘ 𝐺 )
gsum2d2.g ⊢ ( 𝜑 → 𝐺 ∈ CMnd )
gsum2d2.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
gsum2d2.r ⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → 𝐶 ∈ 𝑊 )
gsum2d2.f ⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ) → 𝑋 ∈ 𝐵 )
gsum2d2.u ⊢ ( 𝜑 → 𝑈 ∈ Fin )
gsum2d2.n ⊢ ( ( 𝜑 ∧ ( ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ∧ ¬ 𝑗 𝑈 𝑘 ) ) → 𝑋 = 0 )
Assertion gsum2d2 ( 𝜑 → ( 𝐺 Σg ( 𝑗 ∈ 𝐴 , 𝑘 ∈ 𝐶 ↦ 𝑋 ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝐴 ↦ ( 𝐺 Σg ( 𝑘 ∈ 𝐶 ↦ 𝑋 ) ) ) ) )

Proof

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 vsnex ⊢ { 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 biimtrid ⊢ ( 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 biimtrid ⊢ ( 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 ) ) ) ) )