Step |
Hyp |
Ref |
Expression |
1 |
|
tsmsadd.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
tsmsadd.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
tsmsadd.1 |
⊢ ( 𝜑 → 𝐺 ∈ CMnd ) |
4 |
|
tsmsadd.2 |
⊢ ( 𝜑 → 𝐺 ∈ TopMnd ) |
5 |
|
tsmsadd.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
6 |
|
tsmsadd.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
7 |
|
tsmsadd.h |
⊢ ( 𝜑 → 𝐻 : 𝐴 ⟶ 𝐵 ) |
8 |
|
tsmsadd.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝐺 tsums 𝐹 ) ) |
9 |
|
tsmsadd.y |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝐺 tsums 𝐻 ) ) |
10 |
|
tmdtps |
⊢ ( 𝐺 ∈ TopMnd → 𝐺 ∈ TopSp ) |
11 |
4 10
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ TopSp ) |
12 |
1 3 11 5 6
|
tsmscl |
⊢ ( 𝜑 → ( 𝐺 tsums 𝐹 ) ⊆ 𝐵 ) |
13 |
12 8
|
sseldd |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
14 |
1 3 11 5 7
|
tsmscl |
⊢ ( 𝜑 → ( 𝐺 tsums 𝐻 ) ⊆ 𝐵 ) |
15 |
14 9
|
sseldd |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
16 |
|
eqid |
⊢ ( +𝑓 ‘ 𝐺 ) = ( +𝑓 ‘ 𝐺 ) |
17 |
1 2 16
|
plusfval |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ( +𝑓 ‘ 𝐺 ) 𝑌 ) = ( 𝑋 + 𝑌 ) ) |
18 |
13 15 17
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 ( +𝑓 ‘ 𝐺 ) 𝑌 ) = ( 𝑋 + 𝑌 ) ) |
19 |
|
eqid |
⊢ ( TopOpen ‘ 𝐺 ) = ( TopOpen ‘ 𝐺 ) |
20 |
1 19
|
istps |
⊢ ( 𝐺 ∈ TopSp ↔ ( TopOpen ‘ 𝐺 ) ∈ ( TopOn ‘ 𝐵 ) ) |
21 |
11 20
|
sylib |
⊢ ( 𝜑 → ( TopOpen ‘ 𝐺 ) ∈ ( TopOn ‘ 𝐵 ) ) |
22 |
|
eqid |
⊢ ( 𝒫 𝐴 ∩ Fin ) = ( 𝒫 𝐴 ∩ Fin ) |
23 |
|
eqid |
⊢ ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) = ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) |
24 |
|
eqid |
⊢ ran ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) = ran ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) |
25 |
22 23 24 5
|
tsmsfbas |
⊢ ( 𝜑 → ran ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) ∈ ( fBas ‘ ( 𝒫 𝐴 ∩ Fin ) ) ) |
26 |
|
fgcl |
⊢ ( ran ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) ∈ ( fBas ‘ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( 𝒫 𝐴 ∩ Fin ) filGen ran ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) ) ∈ ( Fil ‘ ( 𝒫 𝐴 ∩ Fin ) ) ) |
27 |
25 26
|
syl |
⊢ ( 𝜑 → ( ( 𝒫 𝐴 ∩ Fin ) filGen ran ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) ) ∈ ( Fil ‘ ( 𝒫 𝐴 ∩ Fin ) ) ) |
28 |
1 22 3 5 6
|
tsmslem1 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝐺 Σg ( 𝐹 ↾ 𝑧 ) ) ∈ 𝐵 ) |
29 |
1 22 3 5 7
|
tsmslem1 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝐺 Σg ( 𝐻 ↾ 𝑧 ) ) ∈ 𝐵 ) |
30 |
1 19 22 24 3 5 6
|
tsmsval |
⊢ ( 𝜑 → ( 𝐺 tsums 𝐹 ) = ( ( ( TopOpen ‘ 𝐺 ) fLimf ( ( 𝒫 𝐴 ∩ Fin ) filGen ran ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) ) ) ‘ ( 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( 𝐺 Σg ( 𝐹 ↾ 𝑧 ) ) ) ) ) |
31 |
8 30
|
eleqtrd |
⊢ ( 𝜑 → 𝑋 ∈ ( ( ( TopOpen ‘ 𝐺 ) fLimf ( ( 𝒫 𝐴 ∩ Fin ) filGen ran ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) ) ) ‘ ( 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( 𝐺 Σg ( 𝐹 ↾ 𝑧 ) ) ) ) ) |
32 |
1 19 22 24 3 5 7
|
tsmsval |
⊢ ( 𝜑 → ( 𝐺 tsums 𝐻 ) = ( ( ( TopOpen ‘ 𝐺 ) fLimf ( ( 𝒫 𝐴 ∩ Fin ) filGen ran ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) ) ) ‘ ( 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( 𝐺 Σg ( 𝐻 ↾ 𝑧 ) ) ) ) ) |
33 |
9 32
|
eleqtrd |
⊢ ( 𝜑 → 𝑌 ∈ ( ( ( TopOpen ‘ 𝐺 ) fLimf ( ( 𝒫 𝐴 ∩ Fin ) filGen ran ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) ) ) ‘ ( 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( 𝐺 Σg ( 𝐻 ↾ 𝑧 ) ) ) ) ) |
34 |
19 16
|
tmdcn |
⊢ ( 𝐺 ∈ TopMnd → ( +𝑓 ‘ 𝐺 ) ∈ ( ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) Cn ( TopOpen ‘ 𝐺 ) ) ) |
35 |
4 34
|
syl |
⊢ ( 𝜑 → ( +𝑓 ‘ 𝐺 ) ∈ ( ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) Cn ( TopOpen ‘ 𝐺 ) ) ) |
36 |
13 15
|
opelxpd |
⊢ ( 𝜑 → 〈 𝑋 , 𝑌 〉 ∈ ( 𝐵 × 𝐵 ) ) |
37 |
|
txtopon |
⊢ ( ( ( TopOpen ‘ 𝐺 ) ∈ ( TopOn ‘ 𝐵 ) ∧ ( TopOpen ‘ 𝐺 ) ∈ ( TopOn ‘ 𝐵 ) ) → ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) ∈ ( TopOn ‘ ( 𝐵 × 𝐵 ) ) ) |
38 |
21 21 37
|
syl2anc |
⊢ ( 𝜑 → ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) ∈ ( TopOn ‘ ( 𝐵 × 𝐵 ) ) ) |
39 |
|
toponuni |
⊢ ( ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) ∈ ( TopOn ‘ ( 𝐵 × 𝐵 ) ) → ( 𝐵 × 𝐵 ) = ∪ ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) ) |
40 |
38 39
|
syl |
⊢ ( 𝜑 → ( 𝐵 × 𝐵 ) = ∪ ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) ) |
41 |
36 40
|
eleqtrd |
⊢ ( 𝜑 → 〈 𝑋 , 𝑌 〉 ∈ ∪ ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) ) |
42 |
|
eqid |
⊢ ∪ ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) = ∪ ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) |
43 |
42
|
cncnpi |
⊢ ( ( ( +𝑓 ‘ 𝐺 ) ∈ ( ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) Cn ( TopOpen ‘ 𝐺 ) ) ∧ 〈 𝑋 , 𝑌 〉 ∈ ∪ ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) ) → ( +𝑓 ‘ 𝐺 ) ∈ ( ( ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) CnP ( TopOpen ‘ 𝐺 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) ) |
44 |
35 41 43
|
syl2anc |
⊢ ( 𝜑 → ( +𝑓 ‘ 𝐺 ) ∈ ( ( ( ( TopOpen ‘ 𝐺 ) ×t ( TopOpen ‘ 𝐺 ) ) CnP ( TopOpen ‘ 𝐺 ) ) ‘ 〈 𝑋 , 𝑌 〉 ) ) |
45 |
21 21 27 28 29 31 33 44
|
flfcnp2 |
⊢ ( 𝜑 → ( 𝑋 ( +𝑓 ‘ 𝐺 ) 𝑌 ) ∈ ( ( ( TopOpen ‘ 𝐺 ) fLimf ( ( 𝒫 𝐴 ∩ Fin ) filGen ran ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) ) ) ‘ ( 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( 𝐺 Σg ( 𝐹 ↾ 𝑧 ) ) ( +𝑓 ‘ 𝐺 ) ( 𝐺 Σg ( 𝐻 ↾ 𝑧 ) ) ) ) ) ) |
46 |
18 45
|
eqeltrrd |
⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( ( ( TopOpen ‘ 𝐺 ) fLimf ( ( 𝒫 𝐴 ∩ Fin ) filGen ran ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) ) ) ‘ ( 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( 𝐺 Σg ( 𝐹 ↾ 𝑧 ) ) ( +𝑓 ‘ 𝐺 ) ( 𝐺 Σg ( 𝐻 ↾ 𝑧 ) ) ) ) ) ) |
47 |
|
cmnmnd |
⊢ ( 𝐺 ∈ CMnd → 𝐺 ∈ Mnd ) |
48 |
3 47
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ Mnd ) |
49 |
1 2
|
mndcl |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 ) |
50 |
49
|
3expb |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 ) |
51 |
48 50
|
sylan |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 ) |
52 |
|
inidm |
⊢ ( 𝐴 ∩ 𝐴 ) = 𝐴 |
53 |
51 6 7 5 5 52
|
off |
⊢ ( 𝜑 → ( 𝐹 ∘f + 𝐻 ) : 𝐴 ⟶ 𝐵 ) |
54 |
1 19 22 24 3 5 53
|
tsmsval |
⊢ ( 𝜑 → ( 𝐺 tsums ( 𝐹 ∘f + 𝐻 ) ) = ( ( ( TopOpen ‘ 𝐺 ) fLimf ( ( 𝒫 𝐴 ∩ Fin ) filGen ran ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) ) ) ‘ ( 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( 𝐺 Σg ( ( 𝐹 ∘f + 𝐻 ) ↾ 𝑧 ) ) ) ) ) |
55 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
56 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝐺 ∈ CMnd ) |
57 |
|
elinel2 |
⊢ ( 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) → 𝑧 ∈ Fin ) |
58 |
57
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑧 ∈ Fin ) |
59 |
|
elfpw |
⊢ ( 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ↔ ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ∈ Fin ) ) |
60 |
59
|
simplbi |
⊢ ( 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) → 𝑧 ⊆ 𝐴 ) |
61 |
|
fssres |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑧 ⊆ 𝐴 ) → ( 𝐹 ↾ 𝑧 ) : 𝑧 ⟶ 𝐵 ) |
62 |
6 60 61
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝐹 ↾ 𝑧 ) : 𝑧 ⟶ 𝐵 ) |
63 |
|
fssres |
⊢ ( ( 𝐻 : 𝐴 ⟶ 𝐵 ∧ 𝑧 ⊆ 𝐴 ) → ( 𝐻 ↾ 𝑧 ) : 𝑧 ⟶ 𝐵 ) |
64 |
7 60 63
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝐻 ↾ 𝑧 ) : 𝑧 ⟶ 𝐵 ) |
65 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 0g ‘ 𝐺 ) ∈ V ) |
66 |
62 58 65
|
fdmfifsupp |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝐹 ↾ 𝑧 ) finSupp ( 0g ‘ 𝐺 ) ) |
67 |
64 58 65
|
fdmfifsupp |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝐻 ↾ 𝑧 ) finSupp ( 0g ‘ 𝐺 ) ) |
68 |
1 55 2 56 58 62 64 66 67
|
gsumadd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝐺 Σg ( ( 𝐹 ↾ 𝑧 ) ∘f + ( 𝐻 ↾ 𝑧 ) ) ) = ( ( 𝐺 Σg ( 𝐹 ↾ 𝑧 ) ) + ( 𝐺 Σg ( 𝐻 ↾ 𝑧 ) ) ) ) |
69 |
6 5
|
fexd |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
70 |
7 5
|
fexd |
⊢ ( 𝜑 → 𝐻 ∈ V ) |
71 |
|
offres |
⊢ ( ( 𝐹 ∈ V ∧ 𝐻 ∈ V ) → ( ( 𝐹 ∘f + 𝐻 ) ↾ 𝑧 ) = ( ( 𝐹 ↾ 𝑧 ) ∘f + ( 𝐻 ↾ 𝑧 ) ) ) |
72 |
69 70 71
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐹 ∘f + 𝐻 ) ↾ 𝑧 ) = ( ( 𝐹 ↾ 𝑧 ) ∘f + ( 𝐻 ↾ 𝑧 ) ) ) |
73 |
72
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( 𝐹 ∘f + 𝐻 ) ↾ 𝑧 ) = ( ( 𝐹 ↾ 𝑧 ) ∘f + ( 𝐻 ↾ 𝑧 ) ) ) |
74 |
73
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝐺 Σg ( ( 𝐹 ∘f + 𝐻 ) ↾ 𝑧 ) ) = ( 𝐺 Σg ( ( 𝐹 ↾ 𝑧 ) ∘f + ( 𝐻 ↾ 𝑧 ) ) ) ) |
75 |
1 2 16
|
plusfval |
⊢ ( ( ( 𝐺 Σg ( 𝐹 ↾ 𝑧 ) ) ∈ 𝐵 ∧ ( 𝐺 Σg ( 𝐻 ↾ 𝑧 ) ) ∈ 𝐵 ) → ( ( 𝐺 Σg ( 𝐹 ↾ 𝑧 ) ) ( +𝑓 ‘ 𝐺 ) ( 𝐺 Σg ( 𝐻 ↾ 𝑧 ) ) ) = ( ( 𝐺 Σg ( 𝐹 ↾ 𝑧 ) ) + ( 𝐺 Σg ( 𝐻 ↾ 𝑧 ) ) ) ) |
76 |
28 29 75
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( 𝐺 Σg ( 𝐹 ↾ 𝑧 ) ) ( +𝑓 ‘ 𝐺 ) ( 𝐺 Σg ( 𝐻 ↾ 𝑧 ) ) ) = ( ( 𝐺 Σg ( 𝐹 ↾ 𝑧 ) ) + ( 𝐺 Σg ( 𝐻 ↾ 𝑧 ) ) ) ) |
77 |
68 74 76
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝐺 Σg ( ( 𝐹 ∘f + 𝐻 ) ↾ 𝑧 ) ) = ( ( 𝐺 Σg ( 𝐹 ↾ 𝑧 ) ) ( +𝑓 ‘ 𝐺 ) ( 𝐺 Σg ( 𝐻 ↾ 𝑧 ) ) ) ) |
78 |
77
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( 𝐺 Σg ( ( 𝐹 ∘f + 𝐻 ) ↾ 𝑧 ) ) ) = ( 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( 𝐺 Σg ( 𝐹 ↾ 𝑧 ) ) ( +𝑓 ‘ 𝐺 ) ( 𝐺 Σg ( 𝐻 ↾ 𝑧 ) ) ) ) ) |
79 |
78
|
fveq2d |
⊢ ( 𝜑 → ( ( ( TopOpen ‘ 𝐺 ) fLimf ( ( 𝒫 𝐴 ∩ Fin ) filGen ran ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) ) ) ‘ ( 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( 𝐺 Σg ( ( 𝐹 ∘f + 𝐻 ) ↾ 𝑧 ) ) ) ) = ( ( ( TopOpen ‘ 𝐺 ) fLimf ( ( 𝒫 𝐴 ∩ Fin ) filGen ran ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) ) ) ‘ ( 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( 𝐺 Σg ( 𝐹 ↾ 𝑧 ) ) ( +𝑓 ‘ 𝐺 ) ( 𝐺 Σg ( 𝐻 ↾ 𝑧 ) ) ) ) ) ) |
80 |
54 79
|
eqtrd |
⊢ ( 𝜑 → ( 𝐺 tsums ( 𝐹 ∘f + 𝐻 ) ) = ( ( ( TopOpen ‘ 𝐺 ) fLimf ( ( 𝒫 𝐴 ∩ Fin ) filGen ran ( 𝑦 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ { 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ∣ 𝑦 ⊆ 𝑧 } ) ) ) ‘ ( 𝑧 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( 𝐺 Σg ( 𝐹 ↾ 𝑧 ) ) ( +𝑓 ‘ 𝐺 ) ( 𝐺 Σg ( 𝐻 ↾ 𝑧 ) ) ) ) ) ) |
81 |
46 80
|
eleqtrrd |
⊢ ( 𝜑 → ( 𝑋 + 𝑌 ) ∈ ( 𝐺 tsums ( 𝐹 ∘f + 𝐻 ) ) ) |