Step |
Hyp |
Ref |
Expression |
1 |
|
prdstgpd.y |
⊢ 𝑌 = ( 𝑆 Xs 𝑅 ) |
2 |
|
prdstgpd.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
3 |
|
prdstgpd.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑉 ) |
4 |
|
prdstgpd.r |
⊢ ( 𝜑 → 𝑅 : 𝐼 ⟶ TopGrp ) |
5 |
|
tgpgrp |
⊢ ( 𝑥 ∈ TopGrp → 𝑥 ∈ Grp ) |
6 |
5
|
ssriv |
⊢ TopGrp ⊆ Grp |
7 |
|
fss |
⊢ ( ( 𝑅 : 𝐼 ⟶ TopGrp ∧ TopGrp ⊆ Grp ) → 𝑅 : 𝐼 ⟶ Grp ) |
8 |
4 6 7
|
sylancl |
⊢ ( 𝜑 → 𝑅 : 𝐼 ⟶ Grp ) |
9 |
1 2 3 8
|
prdsgrpd |
⊢ ( 𝜑 → 𝑌 ∈ Grp ) |
10 |
|
tgptmd |
⊢ ( 𝑥 ∈ TopGrp → 𝑥 ∈ TopMnd ) |
11 |
10
|
ssriv |
⊢ TopGrp ⊆ TopMnd |
12 |
|
fss |
⊢ ( ( 𝑅 : 𝐼 ⟶ TopGrp ∧ TopGrp ⊆ TopMnd ) → 𝑅 : 𝐼 ⟶ TopMnd ) |
13 |
4 11 12
|
sylancl |
⊢ ( 𝜑 → 𝑅 : 𝐼 ⟶ TopMnd ) |
14 |
1 2 3 13
|
prdstmdd |
⊢ ( 𝜑 → 𝑌 ∈ TopMnd ) |
15 |
|
eqid |
⊢ ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) = ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) |
16 |
|
eqid |
⊢ ( TopOpen ‘ 𝑌 ) = ( TopOpen ‘ 𝑌 ) |
17 |
|
eqid |
⊢ ( Base ‘ 𝑌 ) = ( Base ‘ 𝑌 ) |
18 |
16 17
|
tmdtopon |
⊢ ( 𝑌 ∈ TopMnd → ( TopOpen ‘ 𝑌 ) ∈ ( TopOn ‘ ( Base ‘ 𝑌 ) ) ) |
19 |
14 18
|
syl |
⊢ ( 𝜑 → ( TopOpen ‘ 𝑌 ) ∈ ( TopOn ‘ ( Base ‘ 𝑌 ) ) ) |
20 |
|
topnfn |
⊢ TopOpen Fn V |
21 |
4
|
ffnd |
⊢ ( 𝜑 → 𝑅 Fn 𝐼 ) |
22 |
|
dffn2 |
⊢ ( 𝑅 Fn 𝐼 ↔ 𝑅 : 𝐼 ⟶ V ) |
23 |
21 22
|
sylib |
⊢ ( 𝜑 → 𝑅 : 𝐼 ⟶ V ) |
24 |
|
fnfco |
⊢ ( ( TopOpen Fn V ∧ 𝑅 : 𝐼 ⟶ V ) → ( TopOpen ∘ 𝑅 ) Fn 𝐼 ) |
25 |
20 23 24
|
sylancr |
⊢ ( 𝜑 → ( TopOpen ∘ 𝑅 ) Fn 𝐼 ) |
26 |
|
fvco3 |
⊢ ( ( 𝑅 : 𝐼 ⟶ TopGrp ∧ 𝑦 ∈ 𝐼 ) → ( ( TopOpen ∘ 𝑅 ) ‘ 𝑦 ) = ( TopOpen ‘ ( 𝑅 ‘ 𝑦 ) ) ) |
27 |
4 26
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( ( TopOpen ∘ 𝑅 ) ‘ 𝑦 ) = ( TopOpen ‘ ( 𝑅 ‘ 𝑦 ) ) ) |
28 |
4
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( 𝑅 ‘ 𝑦 ) ∈ TopGrp ) |
29 |
|
eqid |
⊢ ( TopOpen ‘ ( 𝑅 ‘ 𝑦 ) ) = ( TopOpen ‘ ( 𝑅 ‘ 𝑦 ) ) |
30 |
|
eqid |
⊢ ( Base ‘ ( 𝑅 ‘ 𝑦 ) ) = ( Base ‘ ( 𝑅 ‘ 𝑦 ) ) |
31 |
29 30
|
tgptopon |
⊢ ( ( 𝑅 ‘ 𝑦 ) ∈ TopGrp → ( TopOpen ‘ ( 𝑅 ‘ 𝑦 ) ) ∈ ( TopOn ‘ ( Base ‘ ( 𝑅 ‘ 𝑦 ) ) ) ) |
32 |
|
topontop |
⊢ ( ( TopOpen ‘ ( 𝑅 ‘ 𝑦 ) ) ∈ ( TopOn ‘ ( Base ‘ ( 𝑅 ‘ 𝑦 ) ) ) → ( TopOpen ‘ ( 𝑅 ‘ 𝑦 ) ) ∈ Top ) |
33 |
28 31 32
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( TopOpen ‘ ( 𝑅 ‘ 𝑦 ) ) ∈ Top ) |
34 |
27 33
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( ( TopOpen ∘ 𝑅 ) ‘ 𝑦 ) ∈ Top ) |
35 |
34
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐼 ( ( TopOpen ∘ 𝑅 ) ‘ 𝑦 ) ∈ Top ) |
36 |
|
ffnfv |
⊢ ( ( TopOpen ∘ 𝑅 ) : 𝐼 ⟶ Top ↔ ( ( TopOpen ∘ 𝑅 ) Fn 𝐼 ∧ ∀ 𝑦 ∈ 𝐼 ( ( TopOpen ∘ 𝑅 ) ‘ 𝑦 ) ∈ Top ) ) |
37 |
25 35 36
|
sylanbrc |
⊢ ( 𝜑 → ( TopOpen ∘ 𝑅 ) : 𝐼 ⟶ Top ) |
38 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( TopOpen ‘ 𝑌 ) ∈ ( TopOn ‘ ( Base ‘ 𝑌 ) ) ) |
39 |
1 3 2 21 16
|
prdstopn |
⊢ ( 𝜑 → ( TopOpen ‘ 𝑌 ) = ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ) |
40 |
39
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( TopOpen ‘ 𝑌 ) = ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ) |
41 |
40
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) = ( TopOpen ‘ 𝑌 ) ) |
42 |
41 38
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ∈ ( TopOn ‘ ( Base ‘ 𝑌 ) ) ) |
43 |
|
toponuni |
⊢ ( ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ∈ ( TopOn ‘ ( Base ‘ 𝑌 ) ) → ( Base ‘ 𝑌 ) = ∪ ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ) |
44 |
|
mpteq1 |
⊢ ( ( Base ‘ 𝑌 ) = ∪ ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) → ( 𝑥 ∈ ( Base ‘ 𝑌 ) ↦ ( 𝑥 ‘ 𝑦 ) ) = ( 𝑥 ∈ ∪ ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ↦ ( 𝑥 ‘ 𝑦 ) ) ) |
45 |
42 43 44
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( 𝑥 ∈ ( Base ‘ 𝑌 ) ↦ ( 𝑥 ‘ 𝑦 ) ) = ( 𝑥 ∈ ∪ ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ↦ ( 𝑥 ‘ 𝑦 ) ) ) |
46 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → 𝐼 ∈ 𝑊 ) |
47 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( TopOpen ∘ 𝑅 ) : 𝐼 ⟶ Top ) |
48 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → 𝑦 ∈ 𝐼 ) |
49 |
|
eqid |
⊢ ∪ ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) = ∪ ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) |
50 |
49 15
|
ptpjcn |
⊢ ( ( 𝐼 ∈ 𝑊 ∧ ( TopOpen ∘ 𝑅 ) : 𝐼 ⟶ Top ∧ 𝑦 ∈ 𝐼 ) → ( 𝑥 ∈ ∪ ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ↦ ( 𝑥 ‘ 𝑦 ) ) ∈ ( ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) Cn ( ( TopOpen ∘ 𝑅 ) ‘ 𝑦 ) ) ) |
51 |
46 47 48 50
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( 𝑥 ∈ ∪ ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ↦ ( 𝑥 ‘ 𝑦 ) ) ∈ ( ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) Cn ( ( TopOpen ∘ 𝑅 ) ‘ 𝑦 ) ) ) |
52 |
45 51
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( 𝑥 ∈ ( Base ‘ 𝑌 ) ↦ ( 𝑥 ‘ 𝑦 ) ) ∈ ( ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) Cn ( ( TopOpen ∘ 𝑅 ) ‘ 𝑦 ) ) ) |
53 |
41 27
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) Cn ( ( TopOpen ∘ 𝑅 ) ‘ 𝑦 ) ) = ( ( TopOpen ‘ 𝑌 ) Cn ( TopOpen ‘ ( 𝑅 ‘ 𝑦 ) ) ) ) |
54 |
52 53
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( 𝑥 ∈ ( Base ‘ 𝑌 ) ↦ ( 𝑥 ‘ 𝑦 ) ) ∈ ( ( TopOpen ‘ 𝑌 ) Cn ( TopOpen ‘ ( 𝑅 ‘ 𝑦 ) ) ) ) |
55 |
|
eqid |
⊢ ( invg ‘ ( 𝑅 ‘ 𝑦 ) ) = ( invg ‘ ( 𝑅 ‘ 𝑦 ) ) |
56 |
29 55
|
tgpinv |
⊢ ( ( 𝑅 ‘ 𝑦 ) ∈ TopGrp → ( invg ‘ ( 𝑅 ‘ 𝑦 ) ) ∈ ( ( TopOpen ‘ ( 𝑅 ‘ 𝑦 ) ) Cn ( TopOpen ‘ ( 𝑅 ‘ 𝑦 ) ) ) ) |
57 |
28 56
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( invg ‘ ( 𝑅 ‘ 𝑦 ) ) ∈ ( ( TopOpen ‘ ( 𝑅 ‘ 𝑦 ) ) Cn ( TopOpen ‘ ( 𝑅 ‘ 𝑦 ) ) ) ) |
58 |
38 54 57
|
cnmpt11f |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( 𝑥 ∈ ( Base ‘ 𝑌 ) ↦ ( ( invg ‘ ( 𝑅 ‘ 𝑦 ) ) ‘ ( 𝑥 ‘ 𝑦 ) ) ) ∈ ( ( TopOpen ‘ 𝑌 ) Cn ( TopOpen ‘ ( 𝑅 ‘ 𝑦 ) ) ) ) |
59 |
27
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( ( TopOpen ‘ 𝑌 ) Cn ( ( TopOpen ∘ 𝑅 ) ‘ 𝑦 ) ) = ( ( TopOpen ‘ 𝑌 ) Cn ( TopOpen ‘ ( 𝑅 ‘ 𝑦 ) ) ) ) |
60 |
58 59
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐼 ) → ( 𝑥 ∈ ( Base ‘ 𝑌 ) ↦ ( ( invg ‘ ( 𝑅 ‘ 𝑦 ) ) ‘ ( 𝑥 ‘ 𝑦 ) ) ) ∈ ( ( TopOpen ‘ 𝑌 ) Cn ( ( TopOpen ∘ 𝑅 ) ‘ 𝑦 ) ) ) |
61 |
15 19 2 37 60
|
ptcn |
⊢ ( 𝜑 → ( 𝑥 ∈ ( Base ‘ 𝑌 ) ↦ ( 𝑦 ∈ 𝐼 ↦ ( ( invg ‘ ( 𝑅 ‘ 𝑦 ) ) ‘ ( 𝑥 ‘ 𝑦 ) ) ) ) ∈ ( ( TopOpen ‘ 𝑌 ) Cn ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ) ) |
62 |
|
eqid |
⊢ ( invg ‘ 𝑌 ) = ( invg ‘ 𝑌 ) |
63 |
17 62
|
grpinvf |
⊢ ( 𝑌 ∈ Grp → ( invg ‘ 𝑌 ) : ( Base ‘ 𝑌 ) ⟶ ( Base ‘ 𝑌 ) ) |
64 |
9 63
|
syl |
⊢ ( 𝜑 → ( invg ‘ 𝑌 ) : ( Base ‘ 𝑌 ) ⟶ ( Base ‘ 𝑌 ) ) |
65 |
64
|
feqmptd |
⊢ ( 𝜑 → ( invg ‘ 𝑌 ) = ( 𝑥 ∈ ( Base ‘ 𝑌 ) ↦ ( ( invg ‘ 𝑌 ) ‘ 𝑥 ) ) ) |
66 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑌 ) ) → 𝐼 ∈ 𝑊 ) |
67 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑌 ) ) → 𝑆 ∈ 𝑉 ) |
68 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑌 ) ) → 𝑅 : 𝐼 ⟶ Grp ) |
69 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑌 ) ) → 𝑥 ∈ ( Base ‘ 𝑌 ) ) |
70 |
1 66 67 68 17 62 69
|
prdsinvgd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑌 ) ) → ( ( invg ‘ 𝑌 ) ‘ 𝑥 ) = ( 𝑦 ∈ 𝐼 ↦ ( ( invg ‘ ( 𝑅 ‘ 𝑦 ) ) ‘ ( 𝑥 ‘ 𝑦 ) ) ) ) |
71 |
70
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ ( Base ‘ 𝑌 ) ↦ ( ( invg ‘ 𝑌 ) ‘ 𝑥 ) ) = ( 𝑥 ∈ ( Base ‘ 𝑌 ) ↦ ( 𝑦 ∈ 𝐼 ↦ ( ( invg ‘ ( 𝑅 ‘ 𝑦 ) ) ‘ ( 𝑥 ‘ 𝑦 ) ) ) ) ) |
72 |
65 71
|
eqtrd |
⊢ ( 𝜑 → ( invg ‘ 𝑌 ) = ( 𝑥 ∈ ( Base ‘ 𝑌 ) ↦ ( 𝑦 ∈ 𝐼 ↦ ( ( invg ‘ ( 𝑅 ‘ 𝑦 ) ) ‘ ( 𝑥 ‘ 𝑦 ) ) ) ) ) |
73 |
39
|
oveq2d |
⊢ ( 𝜑 → ( ( TopOpen ‘ 𝑌 ) Cn ( TopOpen ‘ 𝑌 ) ) = ( ( TopOpen ‘ 𝑌 ) Cn ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ) ) |
74 |
61 72 73
|
3eltr4d |
⊢ ( 𝜑 → ( invg ‘ 𝑌 ) ∈ ( ( TopOpen ‘ 𝑌 ) Cn ( TopOpen ‘ 𝑌 ) ) ) |
75 |
16 62
|
istgp |
⊢ ( 𝑌 ∈ TopGrp ↔ ( 𝑌 ∈ Grp ∧ 𝑌 ∈ TopMnd ∧ ( invg ‘ 𝑌 ) ∈ ( ( TopOpen ‘ 𝑌 ) Cn ( TopOpen ‘ 𝑌 ) ) ) ) |
76 |
9 14 74 75
|
syl3anbrc |
⊢ ( 𝜑 → 𝑌 ∈ TopGrp ) |