Step | Hyp | Ref
| Expression |
1 | | psrvsca.s |
. . . . 5
⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
2 | | psrvsca.k |
. . . . 5
⊢ 𝐾 = (Base‘𝑅) |
3 | | eqid 2825 |
. . . . 5
⊢
(+g‘𝑅) = (+g‘𝑅) |
4 | | psrvsca.m |
. . . . 5
⊢ · =
(.r‘𝑅) |
5 | | eqid 2825 |
. . . . 5
⊢
(TopOpen‘𝑅) =
(TopOpen‘𝑅) |
6 | | psrvsca.d |
. . . . 5
⊢ 𝐷 = {ℎ ∈ (ℕ0
↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
7 | | psrvsca.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝑆) |
8 | | simpl 476 |
. . . . . 6
⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐼 ∈ V) |
9 | 1, 2, 6, 7, 8 | psrbas 19739 |
. . . . 5
⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = (𝐾 ↑𝑚 𝐷)) |
10 | | eqid 2825 |
. . . . . 6
⊢
(+g‘𝑆) = (+g‘𝑆) |
11 | 1, 7, 3, 10 | psrplusg 19742 |
. . . . 5
⊢
(+g‘𝑆) = ( ∘𝑓
(+g‘𝑅)
↾ (𝐵 × 𝐵)) |
12 | | eqid 2825 |
. . . . . 6
⊢
(.r‘𝑆) = (.r‘𝑆) |
13 | 1, 7, 4, 12, 6 | psrmulr 19745 |
. . . . 5
⊢
(.r‘𝑆) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘𝑓 − 𝑥))))))) |
14 | | eqid 2825 |
. . . . 5
⊢ (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)) = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)) |
15 | | eqidd 2826 |
. . . . 5
⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) →
(∏t‘(𝐷 × {(TopOpen‘𝑅)})) = (∏t‘(𝐷 × {(TopOpen‘𝑅)}))) |
16 | | simpr 479 |
. . . . 5
⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑅 ∈ V) |
17 | 1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 15, 8, 16 | psrval 19723 |
. . . 4
⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ({〈(Base‘ndx),
𝐵〉,
〈(+g‘ndx), (+g‘𝑆)〉, 〈(.r‘ndx),
(.r‘𝑆)〉} ∪ {〈(Scalar‘ndx),
𝑅〉, 〈(
·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))〉,
〈(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))〉})) |
18 | 17 | fveq2d 6437 |
. . 3
⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (
·𝑠 ‘𝑆) = ( ·𝑠
‘({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx),
(+g‘𝑆)〉, 〈(.r‘ndx),
(.r‘𝑆)〉} ∪ {〈(Scalar‘ndx),
𝑅〉, 〈(
·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))〉,
〈(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))〉}))) |
19 | | psrvsca.n |
. . 3
⊢ ∙ = (
·𝑠 ‘𝑆) |
20 | 2 | fvexi 6447 |
. . . . 5
⊢ 𝐾 ∈ V |
21 | 7 | fvexi 6447 |
. . . . 5
⊢ 𝐵 ∈ V |
22 | 20, 21 | mpt2ex 7510 |
. . . 4
⊢ (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)) ∈ V |
23 | | psrvalstr 19724 |
. . . . 5
⊢
({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx),
(+g‘𝑆)〉, 〈(.r‘ndx),
(.r‘𝑆)〉} ∪ {〈(Scalar‘ndx),
𝑅〉, 〈(
·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))〉,
〈(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))〉}) Struct 〈1,
9〉 |
24 | | vscaid 16375 |
. . . . 5
⊢
·𝑠 = Slot (
·𝑠 ‘ndx) |
25 | | snsstp2 4566 |
. . . . . 6
⊢ {〈(
·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))〉} ⊆
{〈(Scalar‘ndx), 𝑅〉, 〈(
·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))〉,
〈(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))〉} |
26 | | ssun2 4004 |
. . . . . 6
⊢
{〈(Scalar‘ndx), 𝑅〉, 〈(
·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))〉,
〈(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))〉} ⊆
({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx),
(+g‘𝑆)〉, 〈(.r‘ndx),
(.r‘𝑆)〉} ∪ {〈(Scalar‘ndx),
𝑅〉, 〈(
·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))〉,
〈(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))〉}) |
27 | 25, 26 | sstri 3836 |
. . . . 5
⊢ {〈(
·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))〉} ⊆
({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx),
(+g‘𝑆)〉, 〈(.r‘ndx),
(.r‘𝑆)〉} ∪ {〈(Scalar‘ndx),
𝑅〉, 〈(
·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))〉,
〈(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))〉}) |
28 | 23, 24, 27 | strfv 16270 |
. . . 4
⊢ ((𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)) ∈ V → (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)) = (
·𝑠 ‘({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), (+g‘𝑆)〉, 〈(.r‘ndx),
(.r‘𝑆)〉} ∪ {〈(Scalar‘ndx),
𝑅〉, 〈(
·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))〉,
〈(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))〉}))) |
29 | 22, 28 | ax-mp 5 |
. . 3
⊢ (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)) = (
·𝑠 ‘({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), (+g‘𝑆)〉, 〈(.r‘ndx),
(.r‘𝑆)〉} ∪ {〈(Scalar‘ndx),
𝑅〉, 〈(
·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))〉,
〈(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))〉})) |
30 | 18, 19, 29 | 3eqtr4g 2886 |
. 2
⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → ∙ =
(𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))) |
31 | | eqid 2825 |
. . . . . 6
⊢ ∅ =
∅ |
32 | | fn0 6244 |
. . . . . 6
⊢ (∅
Fn ∅ ↔ ∅ = ∅) |
33 | 31, 32 | mpbir 223 |
. . . . 5
⊢ ∅
Fn ∅ |
34 | | reldmpsr 19722 |
. . . . . . . . . 10
⊢ Rel dom
mPwSer |
35 | 34 | ovprc 6942 |
. . . . . . . . 9
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅) |
36 | 1, 35 | syl5eq 2873 |
. . . . . . . 8
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ∅) |
37 | 36 | fveq2d 6437 |
. . . . . . 7
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → (
·𝑠 ‘𝑆) = ( ·𝑠
‘∅)) |
38 | | df-vsca 16322 |
. . . . . . . 8
⊢
·𝑠 = Slot 6 |
39 | 38 | str0 16274 |
. . . . . . 7
⊢ ∅ =
( ·𝑠 ‘∅) |
40 | 37, 19, 39 | 3eqtr4g 2886 |
. . . . . 6
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → ∙ =
∅) |
41 | 34, 1, 7 | elbasov 16284 |
. . . . . . . . . 10
⊢ (𝑓 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
42 | 41 | con3i 152 |
. . . . . . . . 9
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → ¬ 𝑓 ∈ 𝐵) |
43 | 42 | eq0rdv 4204 |
. . . . . . . 8
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅) |
44 | 43 | xpeq2d 5372 |
. . . . . . 7
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐾 × 𝐵) = (𝐾 × ∅)) |
45 | | xp0 5793 |
. . . . . . 7
⊢ (𝐾 × ∅) =
∅ |
46 | 44, 45 | syl6eq 2877 |
. . . . . 6
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐾 × 𝐵) = ∅) |
47 | 40, 46 | fneq12d 6216 |
. . . . 5
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → ( ∙ Fn
(𝐾 × 𝐵) ↔ ∅ Fn
∅)) |
48 | 33, 47 | mpbiri 250 |
. . . 4
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → ∙ Fn
(𝐾 × 𝐵)) |
49 | | fnov 7028 |
. . . 4
⊢ ( ∙ Fn
(𝐾 × 𝐵) ↔ ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ (𝑥 ∙ 𝑓))) |
50 | 48, 49 | sylib 210 |
. . 3
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → ∙ =
(𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ (𝑥 ∙ 𝑓))) |
51 | 42 | pm2.21d 119 |
. . . . . 6
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑓 ∈ 𝐵 → ((𝐷 × {𝑥}) ∘𝑓 · 𝑓) = (𝑥 ∙ 𝑓))) |
52 | 51 | a1d 25 |
. . . . 5
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑥 ∈ 𝐾 → (𝑓 ∈ 𝐵 → ((𝐷 × {𝑥}) ∘𝑓 · 𝑓) = (𝑥 ∙ 𝑓)))) |
53 | 52 | 3imp 1143 |
. . . 4
⊢ ((¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑥 ∈ 𝐾 ∧ 𝑓 ∈ 𝐵) → ((𝐷 × {𝑥}) ∘𝑓 · 𝑓) = (𝑥 ∙ 𝑓)) |
54 | 53 | mpt2eq3dva 6979 |
. . 3
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)) = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ (𝑥 ∙ 𝑓))) |
55 | 50, 54 | eqtr4d 2864 |
. 2
⊢ (¬
(𝐼 ∈ V ∧ 𝑅 ∈ V) → ∙ =
(𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))) |
56 | 30, 55 | pm2.61i 177 |
1
⊢ ∙ =
(𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)) |