Step |
Hyp |
Ref |
Expression |
1 |
|
prdstmdd.y |
⊢ 𝑌 = ( 𝑆 Xs 𝑅 ) |
2 |
|
prdstmdd.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
3 |
|
prdstmdd.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑉 ) |
4 |
|
prdstmdd.r |
⊢ ( 𝜑 → 𝑅 : 𝐼 ⟶ TopMnd ) |
5 |
|
tmdmnd |
⊢ ( 𝑥 ∈ TopMnd → 𝑥 ∈ Mnd ) |
6 |
5
|
ssriv |
⊢ TopMnd ⊆ Mnd |
7 |
|
fss |
⊢ ( ( 𝑅 : 𝐼 ⟶ TopMnd ∧ TopMnd ⊆ Mnd ) → 𝑅 : 𝐼 ⟶ Mnd ) |
8 |
4 6 7
|
sylancl |
⊢ ( 𝜑 → 𝑅 : 𝐼 ⟶ Mnd ) |
9 |
1 2 3 8
|
prdsmndd |
⊢ ( 𝜑 → 𝑌 ∈ Mnd ) |
10 |
|
tmdtps |
⊢ ( 𝑥 ∈ TopMnd → 𝑥 ∈ TopSp ) |
11 |
10
|
ssriv |
⊢ TopMnd ⊆ TopSp |
12 |
|
fss |
⊢ ( ( 𝑅 : 𝐼 ⟶ TopMnd ∧ TopMnd ⊆ TopSp ) → 𝑅 : 𝐼 ⟶ TopSp ) |
13 |
4 11 12
|
sylancl |
⊢ ( 𝜑 → 𝑅 : 𝐼 ⟶ TopSp ) |
14 |
1 3 2 13
|
prdstps |
⊢ ( 𝜑 → 𝑌 ∈ TopSp ) |
15 |
|
eqid |
⊢ ( Base ‘ 𝑌 ) = ( Base ‘ 𝑌 ) |
16 |
3
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( Base ‘ 𝑌 ) ∧ 𝑔 ∈ ( Base ‘ 𝑌 ) ) → 𝑆 ∈ 𝑉 ) |
17 |
2
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( Base ‘ 𝑌 ) ∧ 𝑔 ∈ ( Base ‘ 𝑌 ) ) → 𝐼 ∈ 𝑊 ) |
18 |
4
|
ffnd |
⊢ ( 𝜑 → 𝑅 Fn 𝐼 ) |
19 |
18
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( Base ‘ 𝑌 ) ∧ 𝑔 ∈ ( Base ‘ 𝑌 ) ) → 𝑅 Fn 𝐼 ) |
20 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( Base ‘ 𝑌 ) ∧ 𝑔 ∈ ( Base ‘ 𝑌 ) ) → 𝑓 ∈ ( Base ‘ 𝑌 ) ) |
21 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( Base ‘ 𝑌 ) ∧ 𝑔 ∈ ( Base ‘ 𝑌 ) ) → 𝑔 ∈ ( Base ‘ 𝑌 ) ) |
22 |
|
eqid |
⊢ ( +g ‘ 𝑌 ) = ( +g ‘ 𝑌 ) |
23 |
1 15 16 17 19 20 21 22
|
prdsplusgval |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ ( Base ‘ 𝑌 ) ∧ 𝑔 ∈ ( Base ‘ 𝑌 ) ) → ( 𝑓 ( +g ‘ 𝑌 ) 𝑔 ) = ( 𝑘 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑘 ) ( +g ‘ ( 𝑅 ‘ 𝑘 ) ) ( 𝑔 ‘ 𝑘 ) ) ) ) |
24 |
23
|
mpoeq3dva |
⊢ ( 𝜑 → ( 𝑓 ∈ ( Base ‘ 𝑌 ) , 𝑔 ∈ ( Base ‘ 𝑌 ) ↦ ( 𝑓 ( +g ‘ 𝑌 ) 𝑔 ) ) = ( 𝑓 ∈ ( Base ‘ 𝑌 ) , 𝑔 ∈ ( Base ‘ 𝑌 ) ↦ ( 𝑘 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑘 ) ( +g ‘ ( 𝑅 ‘ 𝑘 ) ) ( 𝑔 ‘ 𝑘 ) ) ) ) ) |
25 |
|
eqid |
⊢ ( +𝑓 ‘ 𝑌 ) = ( +𝑓 ‘ 𝑌 ) |
26 |
15 22 25
|
plusffval |
⊢ ( +𝑓 ‘ 𝑌 ) = ( 𝑓 ∈ ( Base ‘ 𝑌 ) , 𝑔 ∈ ( Base ‘ 𝑌 ) ↦ ( 𝑓 ( +g ‘ 𝑌 ) 𝑔 ) ) |
27 |
|
vex |
⊢ 𝑓 ∈ V |
28 |
|
vex |
⊢ 𝑔 ∈ V |
29 |
27 28
|
op1std |
⊢ ( 𝑧 = 〈 𝑓 , 𝑔 〉 → ( 1st ‘ 𝑧 ) = 𝑓 ) |
30 |
29
|
fveq1d |
⊢ ( 𝑧 = 〈 𝑓 , 𝑔 〉 → ( ( 1st ‘ 𝑧 ) ‘ 𝑘 ) = ( 𝑓 ‘ 𝑘 ) ) |
31 |
27 28
|
op2ndd |
⊢ ( 𝑧 = 〈 𝑓 , 𝑔 〉 → ( 2nd ‘ 𝑧 ) = 𝑔 ) |
32 |
31
|
fveq1d |
⊢ ( 𝑧 = 〈 𝑓 , 𝑔 〉 → ( ( 2nd ‘ 𝑧 ) ‘ 𝑘 ) = ( 𝑔 ‘ 𝑘 ) ) |
33 |
30 32
|
oveq12d |
⊢ ( 𝑧 = 〈 𝑓 , 𝑔 〉 → ( ( ( 1st ‘ 𝑧 ) ‘ 𝑘 ) ( +g ‘ ( 𝑅 ‘ 𝑘 ) ) ( ( 2nd ‘ 𝑧 ) ‘ 𝑘 ) ) = ( ( 𝑓 ‘ 𝑘 ) ( +g ‘ ( 𝑅 ‘ 𝑘 ) ) ( 𝑔 ‘ 𝑘 ) ) ) |
34 |
33
|
mpteq2dv |
⊢ ( 𝑧 = 〈 𝑓 , 𝑔 〉 → ( 𝑘 ∈ 𝐼 ↦ ( ( ( 1st ‘ 𝑧 ) ‘ 𝑘 ) ( +g ‘ ( 𝑅 ‘ 𝑘 ) ) ( ( 2nd ‘ 𝑧 ) ‘ 𝑘 ) ) ) = ( 𝑘 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑘 ) ( +g ‘ ( 𝑅 ‘ 𝑘 ) ) ( 𝑔 ‘ 𝑘 ) ) ) ) |
35 |
34
|
mpompt |
⊢ ( 𝑧 ∈ ( ( Base ‘ 𝑌 ) × ( Base ‘ 𝑌 ) ) ↦ ( 𝑘 ∈ 𝐼 ↦ ( ( ( 1st ‘ 𝑧 ) ‘ 𝑘 ) ( +g ‘ ( 𝑅 ‘ 𝑘 ) ) ( ( 2nd ‘ 𝑧 ) ‘ 𝑘 ) ) ) ) = ( 𝑓 ∈ ( Base ‘ 𝑌 ) , 𝑔 ∈ ( Base ‘ 𝑌 ) ↦ ( 𝑘 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑘 ) ( +g ‘ ( 𝑅 ‘ 𝑘 ) ) ( 𝑔 ‘ 𝑘 ) ) ) ) |
36 |
24 26 35
|
3eqtr4g |
⊢ ( 𝜑 → ( +𝑓 ‘ 𝑌 ) = ( 𝑧 ∈ ( ( Base ‘ 𝑌 ) × ( Base ‘ 𝑌 ) ) ↦ ( 𝑘 ∈ 𝐼 ↦ ( ( ( 1st ‘ 𝑧 ) ‘ 𝑘 ) ( +g ‘ ( 𝑅 ‘ 𝑘 ) ) ( ( 2nd ‘ 𝑧 ) ‘ 𝑘 ) ) ) ) ) |
37 |
|
eqid |
⊢ ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) = ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) |
38 |
|
eqid |
⊢ ( TopOpen ‘ 𝑌 ) = ( TopOpen ‘ 𝑌 ) |
39 |
15 38
|
istps |
⊢ ( 𝑌 ∈ TopSp ↔ ( TopOpen ‘ 𝑌 ) ∈ ( TopOn ‘ ( Base ‘ 𝑌 ) ) ) |
40 |
14 39
|
sylib |
⊢ ( 𝜑 → ( TopOpen ‘ 𝑌 ) ∈ ( TopOn ‘ ( Base ‘ 𝑌 ) ) ) |
41 |
|
txtopon |
⊢ ( ( ( TopOpen ‘ 𝑌 ) ∈ ( TopOn ‘ ( Base ‘ 𝑌 ) ) ∧ ( TopOpen ‘ 𝑌 ) ∈ ( TopOn ‘ ( Base ‘ 𝑌 ) ) ) → ( ( TopOpen ‘ 𝑌 ) ×t ( TopOpen ‘ 𝑌 ) ) ∈ ( TopOn ‘ ( ( Base ‘ 𝑌 ) × ( Base ‘ 𝑌 ) ) ) ) |
42 |
40 40 41
|
syl2anc |
⊢ ( 𝜑 → ( ( TopOpen ‘ 𝑌 ) ×t ( TopOpen ‘ 𝑌 ) ) ∈ ( TopOn ‘ ( ( Base ‘ 𝑌 ) × ( Base ‘ 𝑌 ) ) ) ) |
43 |
|
topnfn |
⊢ TopOpen Fn V |
44 |
|
ssv |
⊢ TopSp ⊆ V |
45 |
|
fnssres |
⊢ ( ( TopOpen Fn V ∧ TopSp ⊆ V ) → ( TopOpen ↾ TopSp ) Fn TopSp ) |
46 |
43 44 45
|
mp2an |
⊢ ( TopOpen ↾ TopSp ) Fn TopSp |
47 |
|
fvres |
⊢ ( 𝑥 ∈ TopSp → ( ( TopOpen ↾ TopSp ) ‘ 𝑥 ) = ( TopOpen ‘ 𝑥 ) ) |
48 |
|
eqid |
⊢ ( TopOpen ‘ 𝑥 ) = ( TopOpen ‘ 𝑥 ) |
49 |
48
|
tpstop |
⊢ ( 𝑥 ∈ TopSp → ( TopOpen ‘ 𝑥 ) ∈ Top ) |
50 |
47 49
|
eqeltrd |
⊢ ( 𝑥 ∈ TopSp → ( ( TopOpen ↾ TopSp ) ‘ 𝑥 ) ∈ Top ) |
51 |
50
|
rgen |
⊢ ∀ 𝑥 ∈ TopSp ( ( TopOpen ↾ TopSp ) ‘ 𝑥 ) ∈ Top |
52 |
|
ffnfv |
⊢ ( ( TopOpen ↾ TopSp ) : TopSp ⟶ Top ↔ ( ( TopOpen ↾ TopSp ) Fn TopSp ∧ ∀ 𝑥 ∈ TopSp ( ( TopOpen ↾ TopSp ) ‘ 𝑥 ) ∈ Top ) ) |
53 |
46 51 52
|
mpbir2an |
⊢ ( TopOpen ↾ TopSp ) : TopSp ⟶ Top |
54 |
|
fco2 |
⊢ ( ( ( TopOpen ↾ TopSp ) : TopSp ⟶ Top ∧ 𝑅 : 𝐼 ⟶ TopSp ) → ( TopOpen ∘ 𝑅 ) : 𝐼 ⟶ Top ) |
55 |
53 13 54
|
sylancr |
⊢ ( 𝜑 → ( TopOpen ∘ 𝑅 ) : 𝐼 ⟶ Top ) |
56 |
33
|
mpompt |
⊢ ( 𝑧 ∈ ( ( Base ‘ 𝑌 ) × ( Base ‘ 𝑌 ) ) ↦ ( ( ( 1st ‘ 𝑧 ) ‘ 𝑘 ) ( +g ‘ ( 𝑅 ‘ 𝑘 ) ) ( ( 2nd ‘ 𝑧 ) ‘ 𝑘 ) ) ) = ( 𝑓 ∈ ( Base ‘ 𝑌 ) , 𝑔 ∈ ( Base ‘ 𝑌 ) ↦ ( ( 𝑓 ‘ 𝑘 ) ( +g ‘ ( 𝑅 ‘ 𝑘 ) ) ( 𝑔 ‘ 𝑘 ) ) ) |
57 |
|
eqid |
⊢ ( TopOpen ‘ ( 𝑅 ‘ 𝑘 ) ) = ( TopOpen ‘ ( 𝑅 ‘ 𝑘 ) ) |
58 |
|
eqid |
⊢ ( +g ‘ ( 𝑅 ‘ 𝑘 ) ) = ( +g ‘ ( 𝑅 ‘ 𝑘 ) ) |
59 |
4
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( 𝑅 ‘ 𝑘 ) ∈ TopMnd ) |
60 |
40
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( TopOpen ‘ 𝑌 ) ∈ ( TopOn ‘ ( Base ‘ 𝑌 ) ) ) |
61 |
60 60
|
cnmpt1st |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( 𝑓 ∈ ( Base ‘ 𝑌 ) , 𝑔 ∈ ( Base ‘ 𝑌 ) ↦ 𝑓 ) ∈ ( ( ( TopOpen ‘ 𝑌 ) ×t ( TopOpen ‘ 𝑌 ) ) Cn ( TopOpen ‘ 𝑌 ) ) ) |
62 |
1 3 2 18 38
|
prdstopn |
⊢ ( 𝜑 → ( TopOpen ‘ 𝑌 ) = ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ) |
63 |
62
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( TopOpen ‘ 𝑌 ) = ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ) |
64 |
63 60
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ∈ ( TopOn ‘ ( Base ‘ 𝑌 ) ) ) |
65 |
|
toponuni |
⊢ ( ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ∈ ( TopOn ‘ ( Base ‘ 𝑌 ) ) → ( Base ‘ 𝑌 ) = ∪ ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ) |
66 |
64 65
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( Base ‘ 𝑌 ) = ∪ ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ) |
67 |
66
|
mpteq1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( 𝑥 ∈ ( Base ‘ 𝑌 ) ↦ ( 𝑥 ‘ 𝑘 ) ) = ( 𝑥 ∈ ∪ ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ↦ ( 𝑥 ‘ 𝑘 ) ) ) |
68 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → 𝐼 ∈ 𝑊 ) |
69 |
55
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( TopOpen ∘ 𝑅 ) : 𝐼 ⟶ Top ) |
70 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → 𝑘 ∈ 𝐼 ) |
71 |
|
eqid |
⊢ ∪ ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) = ∪ ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) |
72 |
71 37
|
ptpjcn |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ ( TopOpen ∘ 𝑅 ) : 𝐼 ⟶ Top ∧ 𝑘 ∈ 𝐼 ) → ( 𝑥 ∈ ∪ ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ↦ ( 𝑥 ‘ 𝑘 ) ) ∈ ( ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) Cn ( ( TopOpen ∘ 𝑅 ) ‘ 𝑘 ) ) ) |
73 |
68 69 70 72
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( 𝑥 ∈ ∪ ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ↦ ( 𝑥 ‘ 𝑘 ) ) ∈ ( ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) Cn ( ( TopOpen ∘ 𝑅 ) ‘ 𝑘 ) ) ) |
74 |
67 73
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( 𝑥 ∈ ( Base ‘ 𝑌 ) ↦ ( 𝑥 ‘ 𝑘 ) ) ∈ ( ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) Cn ( ( TopOpen ∘ 𝑅 ) ‘ 𝑘 ) ) ) |
75 |
63
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) = ( TopOpen ‘ 𝑌 ) ) |
76 |
|
fvco3 |
⊢ ( ( 𝑅 : 𝐼 ⟶ TopMnd ∧ 𝑘 ∈ 𝐼 ) → ( ( TopOpen ∘ 𝑅 ) ‘ 𝑘 ) = ( TopOpen ‘ ( 𝑅 ‘ 𝑘 ) ) ) |
77 |
4 76
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( ( TopOpen ∘ 𝑅 ) ‘ 𝑘 ) = ( TopOpen ‘ ( 𝑅 ‘ 𝑘 ) ) ) |
78 |
75 77
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) Cn ( ( TopOpen ∘ 𝑅 ) ‘ 𝑘 ) ) = ( ( TopOpen ‘ 𝑌 ) Cn ( TopOpen ‘ ( 𝑅 ‘ 𝑘 ) ) ) ) |
79 |
74 78
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( 𝑥 ∈ ( Base ‘ 𝑌 ) ↦ ( 𝑥 ‘ 𝑘 ) ) ∈ ( ( TopOpen ‘ 𝑌 ) Cn ( TopOpen ‘ ( 𝑅 ‘ 𝑘 ) ) ) ) |
80 |
|
fveq1 |
⊢ ( 𝑥 = 𝑓 → ( 𝑥 ‘ 𝑘 ) = ( 𝑓 ‘ 𝑘 ) ) |
81 |
60 60 61 60 79 80
|
cnmpt21 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( 𝑓 ∈ ( Base ‘ 𝑌 ) , 𝑔 ∈ ( Base ‘ 𝑌 ) ↦ ( 𝑓 ‘ 𝑘 ) ) ∈ ( ( ( TopOpen ‘ 𝑌 ) ×t ( TopOpen ‘ 𝑌 ) ) Cn ( TopOpen ‘ ( 𝑅 ‘ 𝑘 ) ) ) ) |
82 |
60 60
|
cnmpt2nd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( 𝑓 ∈ ( Base ‘ 𝑌 ) , 𝑔 ∈ ( Base ‘ 𝑌 ) ↦ 𝑔 ) ∈ ( ( ( TopOpen ‘ 𝑌 ) ×t ( TopOpen ‘ 𝑌 ) ) Cn ( TopOpen ‘ 𝑌 ) ) ) |
83 |
|
fveq1 |
⊢ ( 𝑥 = 𝑔 → ( 𝑥 ‘ 𝑘 ) = ( 𝑔 ‘ 𝑘 ) ) |
84 |
60 60 82 60 79 83
|
cnmpt21 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( 𝑓 ∈ ( Base ‘ 𝑌 ) , 𝑔 ∈ ( Base ‘ 𝑌 ) ↦ ( 𝑔 ‘ 𝑘 ) ) ∈ ( ( ( TopOpen ‘ 𝑌 ) ×t ( TopOpen ‘ 𝑌 ) ) Cn ( TopOpen ‘ ( 𝑅 ‘ 𝑘 ) ) ) ) |
85 |
57 58 59 60 60 81 84
|
cnmpt2plusg |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( 𝑓 ∈ ( Base ‘ 𝑌 ) , 𝑔 ∈ ( Base ‘ 𝑌 ) ↦ ( ( 𝑓 ‘ 𝑘 ) ( +g ‘ ( 𝑅 ‘ 𝑘 ) ) ( 𝑔 ‘ 𝑘 ) ) ) ∈ ( ( ( TopOpen ‘ 𝑌 ) ×t ( TopOpen ‘ 𝑌 ) ) Cn ( TopOpen ‘ ( 𝑅 ‘ 𝑘 ) ) ) ) |
86 |
77
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( ( ( TopOpen ‘ 𝑌 ) ×t ( TopOpen ‘ 𝑌 ) ) Cn ( ( TopOpen ∘ 𝑅 ) ‘ 𝑘 ) ) = ( ( ( TopOpen ‘ 𝑌 ) ×t ( TopOpen ‘ 𝑌 ) ) Cn ( TopOpen ‘ ( 𝑅 ‘ 𝑘 ) ) ) ) |
87 |
85 86
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( 𝑓 ∈ ( Base ‘ 𝑌 ) , 𝑔 ∈ ( Base ‘ 𝑌 ) ↦ ( ( 𝑓 ‘ 𝑘 ) ( +g ‘ ( 𝑅 ‘ 𝑘 ) ) ( 𝑔 ‘ 𝑘 ) ) ) ∈ ( ( ( TopOpen ‘ 𝑌 ) ×t ( TopOpen ‘ 𝑌 ) ) Cn ( ( TopOpen ∘ 𝑅 ) ‘ 𝑘 ) ) ) |
88 |
56 87
|
eqeltrid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐼 ) → ( 𝑧 ∈ ( ( Base ‘ 𝑌 ) × ( Base ‘ 𝑌 ) ) ↦ ( ( ( 1st ‘ 𝑧 ) ‘ 𝑘 ) ( +g ‘ ( 𝑅 ‘ 𝑘 ) ) ( ( 2nd ‘ 𝑧 ) ‘ 𝑘 ) ) ) ∈ ( ( ( TopOpen ‘ 𝑌 ) ×t ( TopOpen ‘ 𝑌 ) ) Cn ( ( TopOpen ∘ 𝑅 ) ‘ 𝑘 ) ) ) |
89 |
37 42 2 55 88
|
ptcn |
⊢ ( 𝜑 → ( 𝑧 ∈ ( ( Base ‘ 𝑌 ) × ( Base ‘ 𝑌 ) ) ↦ ( 𝑘 ∈ 𝐼 ↦ ( ( ( 1st ‘ 𝑧 ) ‘ 𝑘 ) ( +g ‘ ( 𝑅 ‘ 𝑘 ) ) ( ( 2nd ‘ 𝑧 ) ‘ 𝑘 ) ) ) ) ∈ ( ( ( TopOpen ‘ 𝑌 ) ×t ( TopOpen ‘ 𝑌 ) ) Cn ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ) ) |
90 |
36 89
|
eqeltrd |
⊢ ( 𝜑 → ( +𝑓 ‘ 𝑌 ) ∈ ( ( ( TopOpen ‘ 𝑌 ) ×t ( TopOpen ‘ 𝑌 ) ) Cn ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ) ) |
91 |
62
|
oveq2d |
⊢ ( 𝜑 → ( ( ( TopOpen ‘ 𝑌 ) ×t ( TopOpen ‘ 𝑌 ) ) Cn ( TopOpen ‘ 𝑌 ) ) = ( ( ( TopOpen ‘ 𝑌 ) ×t ( TopOpen ‘ 𝑌 ) ) Cn ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ) ) |
92 |
90 91
|
eleqtrrd |
⊢ ( 𝜑 → ( +𝑓 ‘ 𝑌 ) ∈ ( ( ( TopOpen ‘ 𝑌 ) ×t ( TopOpen ‘ 𝑌 ) ) Cn ( TopOpen ‘ 𝑌 ) ) ) |
93 |
25 38
|
istmd |
⊢ ( 𝑌 ∈ TopMnd ↔ ( 𝑌 ∈ Mnd ∧ 𝑌 ∈ TopSp ∧ ( +𝑓 ‘ 𝑌 ) ∈ ( ( ( TopOpen ‘ 𝑌 ) ×t ( TopOpen ‘ 𝑌 ) ) Cn ( TopOpen ‘ 𝑌 ) ) ) ) |
94 |
9 14 92 93
|
syl3anbrc |
⊢ ( 𝜑 → 𝑌 ∈ TopMnd ) |