Step |
Hyp |
Ref |
Expression |
1 |
|
prdsbas.p |
⊢ 𝑃 = ( 𝑆 Xs 𝑅 ) |
2 |
|
prdsbas.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑉 ) |
3 |
|
prdsbas.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝑊 ) |
4 |
|
prdsbas.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
5 |
|
prdsbas.i |
⊢ ( 𝜑 → dom 𝑅 = 𝐼 ) |
6 |
|
prdstset.l |
⊢ 𝑂 = ( TopSet ‘ 𝑃 ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
8 |
1 2 3 4 5
|
prdsbas |
⊢ ( 𝜑 → 𝐵 = X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
9 |
|
eqid |
⊢ ( +g ‘ 𝑃 ) = ( +g ‘ 𝑃 ) |
10 |
1 2 3 4 5 9
|
prdsplusg |
⊢ ( 𝜑 → ( +g ‘ 𝑃 ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) |
11 |
|
eqid |
⊢ ( .r ‘ 𝑃 ) = ( .r ‘ 𝑃 ) |
12 |
1 2 3 4 5 11
|
prdsmulr |
⊢ ( 𝜑 → ( .r ‘ 𝑃 ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( .r ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) |
13 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑃 ) = ( ·𝑠 ‘ 𝑃 ) |
14 |
1 2 3 4 5 7 13
|
prdsvsca |
⊢ ( 𝜑 → ( ·𝑠 ‘ 𝑃 ) = ( 𝑓 ∈ ( Base ‘ 𝑆 ) , 𝑔 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( 𝑓 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) |
15 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) ) |
16 |
|
eqidd |
⊢ ( 𝜑 → ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) = ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ) |
17 |
|
eqid |
⊢ ( le ‘ 𝑃 ) = ( le ‘ 𝑃 ) |
18 |
1 2 3 4 5 17
|
prdsle |
⊢ ( 𝜑 → ( le ‘ 𝑃 ) = { 〈 𝑓 , 𝑔 〉 ∣ ( { 𝑓 , 𝑔 } ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝑓 ‘ 𝑥 ) ( le ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) } ) |
19 |
|
eqid |
⊢ ( dist ‘ 𝑃 ) = ( dist ‘ 𝑃 ) |
20 |
1 2 3 4 5 19
|
prdsds |
⊢ ( 𝜑 → ( dist ‘ 𝑃 ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ sup ( ( ran ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( dist ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ∪ { 0 } ) , ℝ* , < ) ) ) |
21 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) |
22 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( 〈 ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) 〉 ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) = ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( 〈 ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) 〉 ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) ) |
23 |
1 7 5 8 10 12 14 15 16 18 20 21 22 2 3
|
prdsval |
⊢ ( 𝜑 → 𝑃 = ( ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( +g ‘ 𝑃 ) 〉 , 〈 ( .r ‘ ndx ) , ( .r ‘ 𝑃 ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑆 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ 𝑃 ) 〉 , 〈 ( ·𝑖 ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) 〉 } ) ∪ ( { 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) 〉 , 〈 ( le ‘ ndx ) , ( le ‘ 𝑃 ) 〉 , 〈 ( dist ‘ ndx ) , ( dist ‘ 𝑃 ) 〉 } ∪ { 〈 ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 〉 , 〈 ( comp ‘ ndx ) , ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( 〈 ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) 〉 ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) 〉 } ) ) ) |
24 |
|
tsetid |
⊢ TopSet = Slot ( TopSet ‘ ndx ) |
25 |
|
fvexd |
⊢ ( 𝜑 → ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ∈ V ) |
26 |
|
snsstp1 |
⊢ { 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) 〉 } ⊆ { 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) 〉 , 〈 ( le ‘ ndx ) , ( le ‘ 𝑃 ) 〉 , 〈 ( dist ‘ ndx ) , ( dist ‘ 𝑃 ) 〉 } |
27 |
|
ssun1 |
⊢ { 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) 〉 , 〈 ( le ‘ ndx ) , ( le ‘ 𝑃 ) 〉 , 〈 ( dist ‘ ndx ) , ( dist ‘ 𝑃 ) 〉 } ⊆ ( { 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) 〉 , 〈 ( le ‘ ndx ) , ( le ‘ 𝑃 ) 〉 , 〈 ( dist ‘ ndx ) , ( dist ‘ 𝑃 ) 〉 } ∪ { 〈 ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 〉 , 〈 ( comp ‘ ndx ) , ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( 〈 ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) 〉 ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) 〉 } ) |
28 |
26 27
|
sstri |
⊢ { 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) 〉 } ⊆ ( { 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) 〉 , 〈 ( le ‘ ndx ) , ( le ‘ 𝑃 ) 〉 , 〈 ( dist ‘ ndx ) , ( dist ‘ 𝑃 ) 〉 } ∪ { 〈 ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 〉 , 〈 ( comp ‘ ndx ) , ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( 〈 ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) 〉 ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) 〉 } ) |
29 |
|
ssun2 |
⊢ ( { 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) 〉 , 〈 ( le ‘ ndx ) , ( le ‘ 𝑃 ) 〉 , 〈 ( dist ‘ ndx ) , ( dist ‘ 𝑃 ) 〉 } ∪ { 〈 ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 〉 , 〈 ( comp ‘ ndx ) , ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( 〈 ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) 〉 ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) 〉 } ) ⊆ ( ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( +g ‘ 𝑃 ) 〉 , 〈 ( .r ‘ ndx ) , ( .r ‘ 𝑃 ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑆 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ 𝑃 ) 〉 , 〈 ( ·𝑖 ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) 〉 } ) ∪ ( { 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) 〉 , 〈 ( le ‘ ndx ) , ( le ‘ 𝑃 ) 〉 , 〈 ( dist ‘ ndx ) , ( dist ‘ 𝑃 ) 〉 } ∪ { 〈 ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 〉 , 〈 ( comp ‘ ndx ) , ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( 〈 ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) 〉 ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) 〉 } ) ) |
30 |
28 29
|
sstri |
⊢ { 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) 〉 } ⊆ ( ( { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , ( +g ‘ 𝑃 ) 〉 , 〈 ( .r ‘ ndx ) , ( .r ‘ 𝑃 ) 〉 } ∪ { 〈 ( Scalar ‘ ndx ) , 𝑆 〉 , 〈 ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ 𝑃 ) 〉 , 〈 ( ·𝑖 ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑆 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑓 ‘ 𝑥 ) ( ·𝑖 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ) ) 〉 } ) ∪ ( { 〈 ( TopSet ‘ ndx ) , ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) 〉 , 〈 ( le ‘ ndx ) , ( le ‘ 𝑃 ) 〉 , 〈 ( dist ‘ ndx ) , ( dist ‘ 𝑃 ) 〉 } ∪ { 〈 ( Hom ‘ ndx ) , ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 〉 , 〈 ( comp ‘ ndx ) , ( 𝑎 ∈ ( 𝐵 × 𝐵 ) , 𝑐 ∈ 𝐵 ↦ ( 𝑑 ∈ ( ( 2nd ‘ 𝑎 ) ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) 𝑐 ) , 𝑒 ∈ ( ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ X 𝑥 ∈ 𝐼 ( ( 𝑓 ‘ 𝑥 ) ( Hom ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑔 ‘ 𝑥 ) ) ) ‘ 𝑎 ) ↦ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑑 ‘ 𝑥 ) ( 〈 ( ( 1st ‘ 𝑎 ) ‘ 𝑥 ) , ( ( 2nd ‘ 𝑎 ) ‘ 𝑥 ) 〉 ( comp ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑐 ‘ 𝑥 ) ) ( 𝑒 ‘ 𝑥 ) ) ) ) ) 〉 } ) ) |
31 |
23 6 24 25 30
|
prdsbaslem |
⊢ ( 𝜑 → 𝑂 = ( ∏t ‘ ( TopOpen ∘ 𝑅 ) ) ) |