Step | Hyp | Ref
| Expression |
1 | | psrplusg.s |
. . . . 5
⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
2 | | eqid 2738 |
. . . . 5
⊢
(Base‘𝑅) =
(Base‘𝑅) |
3 | | psrplusg.a |
. . . . 5
⊢ + =
(+g‘𝑅) |
4 | | eqid 2738 |
. . . . 5
⊢
(.r‘𝑅) = (.r‘𝑅) |
5 | | eqid 2738 |
. . . . 5
⊢
(TopOpen‘𝑅) =
(TopOpen‘𝑅) |
6 | | eqid 2738 |
. . . . 5
⊢ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} |
7 | | psrplusg.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝑆) |
8 | | simpl 486 |
. . . . . 6
⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐼 ∈ V) |
9 | 1, 2, 6, 7, 8 | psrbas 20757 |
. . . . 5
⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈
Fin})) |
10 | | eqid 2738 |
. . . . 5
⊢ (
∘f + ↾ (𝐵 × 𝐵)) = ( ∘f + ↾
(𝐵 × 𝐵)) |
11 | | eqid 2738 |
. . . . 5
⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg
(𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘f − 𝑥))))))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg
(𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘f − 𝑥))))))) |
12 | | eqid 2738 |
. . . . 5
⊢ (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑓)) = (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑓)) |
13 | | eqidd 2739 |
. . . . 5
⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) →
(∏t‘({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ×
{(TopOpen‘𝑅)})) =
(∏t‘({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ×
{(TopOpen‘𝑅)}))) |
14 | | simpr 488 |
. . . . 5
⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑅 ∈ V) |
15 | 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 8, 14 | psrval 20728 |
. . . 4
⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ({〈(Base‘ndx),
𝐵〉,
〈(+g‘ndx), ( ∘f + ↾ (𝐵 × 𝐵))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg
(𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘f − 𝑥)))))))〉} ∪
{〈(Scalar‘ndx), 𝑅〉, 〈(
·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑓))〉, 〈(TopSet‘ndx),
(∏t‘({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ×
{(TopOpen‘𝑅)}))〉})) |
16 | 15 | fveq2d 6678 |
. . 3
⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) →
(+g‘𝑆) =
(+g‘({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (
∘f + ↾ (𝐵 × 𝐵))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg
(𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘f − 𝑥)))))))〉} ∪
{〈(Scalar‘ndx), 𝑅〉, 〈(
·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑓))〉, 〈(TopSet‘ndx),
(∏t‘({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ×
{(TopOpen‘𝑅)}))〉}))) |
17 | | psrplusg.p |
. . 3
⊢ ✚ =
(+g‘𝑆) |
18 | 7 | fvexi 6688 |
. . . . 5
⊢ 𝐵 ∈ V |
19 | 18, 18 | xpex 7494 |
. . . 4
⊢ (𝐵 × 𝐵) ∈ V |
20 | | ofexg 7429 |
. . . 4
⊢ ((𝐵 × 𝐵) ∈ V → ( ∘f
+ ↾
(𝐵 × 𝐵)) ∈ V) |
21 | | psrvalstr 20729 |
. . . . 5
⊢
({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (
∘f + ↾ (𝐵 × 𝐵))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg
(𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘f − 𝑥)))))))〉} ∪
{〈(Scalar‘ndx), 𝑅〉, 〈(
·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑓))〉, 〈(TopSet‘ndx),
(∏t‘({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ×
{(TopOpen‘𝑅)}))〉}) Struct 〈1,
9〉 |
22 | | plusgid 16699 |
. . . . 5
⊢
+g = Slot (+g‘ndx) |
23 | | snsstp2 4705 |
. . . . . 6
⊢
{〈(+g‘ndx), ( ∘f + ↾
(𝐵 × 𝐵))〉} ⊆
{〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (
∘f + ↾ (𝐵 × 𝐵))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg
(𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘f − 𝑥)))))))〉} |
24 | | ssun1 4062 |
. . . . . 6
⊢
{〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (
∘f + ↾ (𝐵 × 𝐵))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg
(𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘f − 𝑥)))))))〉} ⊆
({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (
∘f + ↾ (𝐵 × 𝐵))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg
(𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘f − 𝑥)))))))〉} ∪
{〈(Scalar‘ndx), 𝑅〉, 〈(
·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑓))〉, 〈(TopSet‘ndx),
(∏t‘({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ×
{(TopOpen‘𝑅)}))〉}) |
25 | 23, 24 | sstri 3886 |
. . . . 5
⊢
{〈(+g‘ndx), ( ∘f + ↾
(𝐵 × 𝐵))〉} ⊆
({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (
∘f + ↾ (𝐵 × 𝐵))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg
(𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘f − 𝑥)))))))〉} ∪
{〈(Scalar‘ndx), 𝑅〉, 〈(
·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑓))〉, 〈(TopSet‘ndx),
(∏t‘({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ×
{(TopOpen‘𝑅)}))〉}) |
26 | 21, 22, 25 | strfv 16634 |
. . . 4
⊢ ((
∘f + ↾ (𝐵 × 𝐵)) ∈ V → ( ∘f
+ ↾
(𝐵 × 𝐵)) =
(+g‘({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (
∘f + ↾ (𝐵 × 𝐵))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg
(𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘f − 𝑥)))))))〉} ∪
{〈(Scalar‘ndx), 𝑅〉, 〈(
·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑓))〉, 〈(TopSet‘ndx),
(∏t‘({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ×
{(TopOpen‘𝑅)}))〉}))) |
27 | 19, 20, 26 | mp2b 10 |
. . 3
⊢ (
∘f + ↾ (𝐵 × 𝐵)) =
(+g‘({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (
∘f + ↾ (𝐵 × 𝐵))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝑅 Σg
(𝑥 ∈ {𝑦 ∈ {ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ∣ 𝑦 ∘r ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑅)(𝑔‘(𝑘 ∘f − 𝑥)))))))〉} ∪
{〈(Scalar‘ndx), 𝑅〉, 〈(
·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓 ∈ 𝐵 ↦ (({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} × {𝑥}) ∘f
(.r‘𝑅)𝑓))〉, 〈(TopSet‘ndx),
(∏t‘({ℎ ∈ (ℕ0
↑m 𝐼)
∣ (◡ℎ “ ℕ) ∈ Fin} ×
{(TopOpen‘𝑅)}))〉})) |
28 | 16, 17, 27 | 3eqtr4g 2798 |
. 2
⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → ✚ = (
∘f + ↾ (𝐵 × 𝐵))) |
29 | | reldmpsr 20727 |
. . . . . . 7
⊢ Rel dom
mPwSer |
30 | 29 | ovprc 7208 |
. . . . . 6
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅) |
31 | 1, 30 | syl5eq 2785 |
. . . . 5
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ∅) |
32 | 31 | fveq2d 6678 |
. . . 4
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) →
(+g‘𝑆) =
(+g‘∅)) |
33 | 22 | str0 16638 |
. . . 4
⊢ ∅ =
(+g‘∅) |
34 | 32, 17, 33 | 3eqtr4g 2798 |
. . 3
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → ✚ =
∅) |
35 | 31 | fveq2d 6678 |
. . . . . . . 8
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) →
(Base‘𝑆) =
(Base‘∅)) |
36 | | base0 16639 |
. . . . . . . 8
⊢ ∅ =
(Base‘∅) |
37 | 35, 7, 36 | 3eqtr4g 2798 |
. . . . . . 7
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅) |
38 | 37 | xpeq2d 5555 |
. . . . . 6
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐵 × 𝐵) = (𝐵 × ∅)) |
39 | | xp0 5990 |
. . . . . 6
⊢ (𝐵 × ∅) =
∅ |
40 | 38, 39 | eqtrdi 2789 |
. . . . 5
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐵 × 𝐵) = ∅) |
41 | 40 | reseq2d 5825 |
. . . 4
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → (
∘f + ↾ (𝐵 × 𝐵)) = ( ∘f + ↾
∅)) |
42 | | res0 5829 |
. . . 4
⊢ (
∘f + ↾ ∅) =
∅ |
43 | 41, 42 | eqtrdi 2789 |
. . 3
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → (
∘f + ↾ (𝐵 × 𝐵)) = ∅) |
44 | 34, 43 | eqtr4d 2776 |
. 2
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → ✚ = (
∘f + ↾ (𝐵 × 𝐵))) |
45 | 28, 44 | pm2.61i 185 |
1
⊢ ✚ = (
∘f + ↾ (𝐵 × 𝐵)) |