Step | Hyp | Ref
| Expression |
1 | | prdsval.p |
. 2
⊢ 𝑃 = (𝑆Xs𝑅) |
2 | | df-prds 16550 |
. . . 4
⊢ Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ ⦋X𝑥 ∈
dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({〈(Base‘ndx),
𝑣〉,
〈(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))〉} ∪ {〈(Scalar‘ndx),
𝑠〉, 〈(
·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠
‘(𝑟‘𝑥))(𝑔‘𝑥))))〉,
〈(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))〉}) ∪ ({〈(TopSet‘ndx),
(∏t‘(TopOpen ∘ 𝑟))〉, 〈(le‘ndx), {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}〉, 〈(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, <
))〉} ∪ {〈(Hom ‘ndx), ℎ〉, 〈(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉}))) |
3 | 2 | a1i 11 |
. . 3
⊢ (𝜑 → Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ ⦋X𝑥 ∈
dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({〈(Base‘ndx),
𝑣〉,
〈(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))〉} ∪ {〈(Scalar‘ndx),
𝑠〉, 〈(
·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠
‘(𝑟‘𝑥))(𝑔‘𝑥))))〉,
〈(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))〉}) ∪ ({〈(TopSet‘ndx),
(∏t‘(TopOpen ∘ 𝑟))〉, 〈(le‘ndx), {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}〉, 〈(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, <
))〉} ∪ {〈(Hom ‘ndx), ℎ〉, 〈(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉})))) |
4 | | vex 3440 |
. . . . . . . . . . . 12
⊢ 𝑟 ∈ V |
5 | 4 | rnex 7473 |
. . . . . . . . . . 11
⊢ ran 𝑟 ∈ V |
6 | 5 | uniex 7323 |
. . . . . . . . . 10
⊢ ∪ ran 𝑟 ∈ V |
7 | 6 | rnex 7473 |
. . . . . . . . 9
⊢ ran ∪ ran 𝑟 ∈ V |
8 | 7 | uniex 7323 |
. . . . . . . 8
⊢ ∪ ran ∪ ran 𝑟 ∈ V |
9 | | baseid 16372 |
. . . . . . . . . . . 12
⊢ Base =
Slot (Base‘ndx) |
10 | 9 | strfvss 16335 |
. . . . . . . . . . 11
⊢
(Base‘(𝑟‘𝑥)) ⊆ ∪ ran
(𝑟‘𝑥) |
11 | | fvssunirn 6567 |
. . . . . . . . . . . 12
⊢ (𝑟‘𝑥) ⊆ ∪ ran
𝑟 |
12 | | rnss 5691 |
. . . . . . . . . . . 12
⊢ ((𝑟‘𝑥) ⊆ ∪ ran
𝑟 → ran (𝑟‘𝑥) ⊆ ran ∪
ran 𝑟) |
13 | | uniss 4766 |
. . . . . . . . . . . 12
⊢ (ran
(𝑟‘𝑥) ⊆ ran ∪
ran 𝑟 → ∪ ran (𝑟‘𝑥) ⊆ ∪ ran
∪ ran 𝑟) |
14 | 11, 12, 13 | mp2b 10 |
. . . . . . . . . . 11
⊢ ∪ ran (𝑟‘𝑥) ⊆ ∪ ran
∪ ran 𝑟 |
15 | 10, 14 | sstri 3898 |
. . . . . . . . . 10
⊢
(Base‘(𝑟‘𝑥)) ⊆ ∪ ran
∪ ran 𝑟 |
16 | 15 | rgenw 3117 |
. . . . . . . . 9
⊢
∀𝑥 ∈ dom
𝑟(Base‘(𝑟‘𝑥)) ⊆ ∪ ran
∪ ran 𝑟 |
17 | | iunss 4868 |
. . . . . . . . 9
⊢ (∪ 𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ⊆ ∪ ran
∪ ran 𝑟 ↔ ∀𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ⊆ ∪ ran
∪ ran 𝑟) |
18 | 16, 17 | mpbir 232 |
. . . . . . . 8
⊢ ∪ 𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ⊆ ∪ ran
∪ ran 𝑟 |
19 | 8, 18 | ssexi 5117 |
. . . . . . 7
⊢ ∪ 𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∈ V |
20 | | ixpssmap2g 8339 |
. . . . . . 7
⊢ (∪ 𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∈ V → X𝑥 ∈
dom 𝑟(Base‘(𝑟‘𝑥)) ⊆ (∪ 𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ↑𝑚 dom 𝑟)) |
21 | 19, 20 | ax-mp 5 |
. . . . . 6
⊢ X𝑥 ∈
dom 𝑟(Base‘(𝑟‘𝑥)) ⊆ (∪ 𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ↑𝑚 dom 𝑟) |
22 | | ovex 7048 |
. . . . . . 7
⊢ (∪ 𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ↑𝑚 dom 𝑟) ∈ V |
23 | 22 | ssex 5116 |
. . . . . 6
⊢ (X𝑥 ∈
dom 𝑟(Base‘(𝑟‘𝑥)) ⊆ (∪ 𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ↑𝑚 dom 𝑟) → X𝑥 ∈
dom 𝑟(Base‘(𝑟‘𝑥)) ∈ V) |
24 | 21, 23 | mp1i 13 |
. . . . 5
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∈ V) |
25 | | simpr 485 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅) |
26 | 25 | fveq1d 6540 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑟‘𝑥) = (𝑅‘𝑥)) |
27 | 26 | fveq2d 6542 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (Base‘(𝑟‘𝑥)) = (Base‘(𝑅‘𝑥))) |
28 | 27 | ixpeq2dv 8326 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥 ∈ 𝐼 (Base‘(𝑟‘𝑥)) = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) |
29 | 25 | dmeqd 5660 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → dom 𝑟 = dom 𝑅) |
30 | | prdsval.i |
. . . . . . . . 9
⊢ (𝜑 → dom 𝑅 = 𝐼) |
31 | 30 | ad2antrr 722 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → dom 𝑅 = 𝐼) |
32 | 29, 31 | eqtrd 2831 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → dom 𝑟 = 𝐼) |
33 | 32 | ixpeq1d 8322 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) = X𝑥 ∈ 𝐼 (Base‘(𝑟‘𝑥))) |
34 | | prdsval.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) |
35 | 34 | ad2antrr 722 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) |
36 | 28, 33, 35 | 3eqtr4d 2841 |
. . . . 5
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) = 𝐵) |
37 | | ovex 7048 |
. . . . . . . . . . 11
⊢ (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ∈ V |
38 | | ovssunirn 7051 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran
(Hom ‘(𝑟‘𝑥)) |
39 | | df-hom 16418 |
. . . . . . . . . . . . . . . . . 18
⊢ Hom =
Slot ;14 |
40 | 39 | strfvss 16335 |
. . . . . . . . . . . . . . . . 17
⊢ (Hom
‘(𝑟‘𝑥)) ⊆ ∪ ran (𝑟‘𝑥) |
41 | 40, 14 | sstri 3898 |
. . . . . . . . . . . . . . . 16
⊢ (Hom
‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟 |
42 | | rnss 5691 |
. . . . . . . . . . . . . . . 16
⊢ ((Hom
‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran 𝑟 → ran (Hom ‘(𝑟‘𝑥)) ⊆ ran ∪
ran ∪ ran 𝑟) |
43 | | uniss 4766 |
. . . . . . . . . . . . . . . 16
⊢ (ran (Hom
‘(𝑟‘𝑥)) ⊆ ran ∪ ran ∪ ran 𝑟 → ∪ ran (Hom
‘(𝑟‘𝑥)) ⊆ ∪ ran ∪ ran ∪ ran 𝑟) |
44 | 41, 42, 43 | mp2b 10 |
. . . . . . . . . . . . . . 15
⊢ ∪ ran (Hom ‘(𝑟‘𝑥)) ⊆ ∪ ran
∪ ran ∪ ran 𝑟 |
45 | 38, 44 | sstri 3898 |
. . . . . . . . . . . . . 14
⊢ ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran
∪ ran ∪ ran 𝑟 |
46 | 45 | rgenw 3117 |
. . . . . . . . . . . . 13
⊢
∀𝑥 ∈ dom
𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran
∪ ran ∪ ran 𝑟 |
47 | | ss2ixp 8323 |
. . . . . . . . . . . . 13
⊢
(∀𝑥 ∈
dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ ∪ ran
∪ ran ∪ ran 𝑟 → X𝑥 ∈
dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ X𝑥 ∈ dom 𝑟∪ ran ∪ ran ∪ ran 𝑟) |
48 | 46, 47 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ X𝑥 ∈
dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ X𝑥 ∈ dom 𝑟∪ ran ∪ ran ∪ ran 𝑟 |
49 | 4 | dmex 7472 |
. . . . . . . . . . . . 13
⊢ dom 𝑟 ∈ V |
50 | 8 | rnex 7473 |
. . . . . . . . . . . . . 14
⊢ ran ∪ ran ∪ ran 𝑟 ∈ V |
51 | 50 | uniex 7323 |
. . . . . . . . . . . . 13
⊢ ∪ ran ∪ ran ∪ ran 𝑟 ∈ V |
52 | 49, 51 | ixpconst 8320 |
. . . . . . . . . . . 12
⊢ X𝑥 ∈
dom 𝑟∪ ran ∪ ran ∪ ran 𝑟 = (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) |
53 | 48, 52 | sseqtri 3924 |
. . . . . . . . . . 11
⊢ X𝑥 ∈
dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ⊆ (∪ ran
∪ ran ∪ ran 𝑟 ↑𝑚 dom
𝑟) |
54 | 37, 53 | elpwi2 5140 |
. . . . . . . . . 10
⊢ X𝑥 ∈
dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) |
55 | 54 | rgen2w 3118 |
. . . . . . . . 9
⊢
∀𝑓 ∈
𝑣 ∀𝑔 ∈ 𝑣 X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) |
56 | | eqid 2795 |
. . . . . . . . . 10
⊢ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) |
57 | 56 | fmpo 7622 |
. . . . . . . . 9
⊢
(∀𝑓 ∈
𝑣 ∀𝑔 ∈ 𝑣 X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) ∈ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ↔ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))):(𝑣 × 𝑣)⟶𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟)) |
58 | 55, 57 | mpbi 231 |
. . . . . . . 8
⊢ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))):(𝑣 × 𝑣)⟶𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) |
59 | | vex 3440 |
. . . . . . . . 9
⊢ 𝑣 ∈ V |
60 | 59, 59 | xpex 7333 |
. . . . . . . 8
⊢ (𝑣 × 𝑣) ∈ V |
61 | 37 | pwex 5172 |
. . . . . . . 8
⊢ 𝒫
(∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ∈ V |
62 | | fex2 7494 |
. . . . . . . 8
⊢ (((𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))):(𝑣 × 𝑣)⟶𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ∧ (𝑣 × 𝑣) ∈ V ∧ 𝒫 (∪ ran ∪ ran ∪ ran 𝑟 ↑𝑚 dom 𝑟) ∈ V) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V) |
63 | 58, 60, 61, 62 | mp3an 1453 |
. . . . . . 7
⊢ (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V |
64 | 63 | a1i 11 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V) |
65 | | simpr 485 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → 𝑣 = 𝐵) |
66 | 32 | adantr 481 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → dom 𝑟 = 𝐼) |
67 | 66 | ixpeq1d 8322 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) = X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) |
68 | 26 | fveq2d 6542 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (Hom ‘(𝑟‘𝑥)) = (Hom ‘(𝑅‘𝑥))) |
69 | 68 | oveqd 7033 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) = ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))) |
70 | 69 | ixpeq2dv 8326 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) = X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))) |
71 | 70 | adantr 481 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) = X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))) |
72 | 67, 71 | eqtrd 2831 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥)) = X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))) |
73 | 65, 65, 72 | mpoeq123dv 7087 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))) |
74 | | prdsval.h |
. . . . . . . 8
⊢ (𝜑 → 𝐻 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))) |
75 | 74 | ad3antrrr 726 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → 𝐻 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))) |
76 | 73, 75 | eqtr4d 2834 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) = 𝐻) |
77 | | simplr 765 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 𝑣 = 𝐵) |
78 | 77 | opeq2d 4717 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 〈(Base‘ndx), 𝑣〉 = 〈(Base‘ndx),
𝐵〉) |
79 | 26 | fveq2d 6542 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (+g‘(𝑟‘𝑥)) = (+g‘(𝑅‘𝑥))) |
80 | 79 | oveqd 7033 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥)) = ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))) |
81 | 32, 80 | mpteq12dv 5045 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))) |
82 | 81 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))) |
83 | 65, 65, 82 | mpoeq123dv 7087 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))) |
84 | 83 | adantr 481 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))) |
85 | | prdsval.a |
. . . . . . . . . . . 12
⊢ (𝜑 → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))) |
86 | 85 | ad4antr 728 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))) |
87 | 84, 86 | eqtr4d 2834 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥)))) = + ) |
88 | 87 | opeq2d 4717 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 〈(+g‘ndx),
(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))〉 = 〈(+g‘ndx),
+
〉) |
89 | 26 | fveq2d 6542 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (.r‘(𝑟‘𝑥)) = (.r‘(𝑅‘𝑥))) |
90 | 89 | oveqd 7033 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥)) = ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))) |
91 | 32, 90 | mpteq12dv 5045 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))) |
92 | 91 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))) |
93 | 65, 65, 92 | mpoeq123dv 7087 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))) |
94 | 93 | adantr 481 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))) |
95 | | prdsval.t |
. . . . . . . . . . . 12
⊢ (𝜑 → × = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))) |
96 | 95 | ad4antr 728 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → × = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))) |
97 | 94, 96 | eqtr4d 2834 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥)))) = × ) |
98 | 97 | opeq2d 4717 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 〈(.r‘ndx),
(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))〉 = 〈(.r‘ndx),
×
〉) |
99 | 78, 88, 98 | tpeq123d 4591 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → {〈(Base‘ndx), 𝑣〉,
〈(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))〉} = {〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), ×
〉}) |
100 | | simp-4r 780 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 𝑠 = 𝑆) |
101 | 100 | opeq2d 4717 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 〈(Scalar‘ndx), 𝑠〉 =
〈(Scalar‘ndx), 𝑆〉) |
102 | | simpllr 772 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → 𝑠 = 𝑆) |
103 | 102 | fveq2d 6542 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (Base‘𝑠) = (Base‘𝑆)) |
104 | | prdsval.k |
. . . . . . . . . . . . . 14
⊢ 𝐾 = (Base‘𝑆) |
105 | 103, 104 | syl6eqr 2849 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (Base‘𝑠) = 𝐾) |
106 | 26 | fveq2d 6542 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (
·𝑠 ‘(𝑟‘𝑥)) = ( ·𝑠
‘(𝑅‘𝑥))) |
107 | 106 | oveqd 7033 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑓( ·𝑠
‘(𝑟‘𝑥))(𝑔‘𝑥)) = (𝑓( ·𝑠
‘(𝑅‘𝑥))(𝑔‘𝑥))) |
108 | 32, 107 | mpteq12dv 5045 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠
‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠
‘(𝑅‘𝑥))(𝑔‘𝑥)))) |
109 | 108 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠
‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠
‘(𝑅‘𝑥))(𝑔‘𝑥)))) |
110 | 105, 65, 109 | mpoeq123dv 7087 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠
‘(𝑟‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠
‘(𝑅‘𝑥))(𝑔‘𝑥))))) |
111 | 110 | adantr 481 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠
‘(𝑟‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠
‘(𝑅‘𝑥))(𝑔‘𝑥))))) |
112 | | prdsval.m |
. . . . . . . . . . . 12
⊢ (𝜑 → · = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠
‘(𝑅‘𝑥))(𝑔‘𝑥))))) |
113 | 112 | ad4antr 728 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → · = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠
‘(𝑅‘𝑥))(𝑔‘𝑥))))) |
114 | 111, 113 | eqtr4d 2834 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠
‘(𝑟‘𝑥))(𝑔‘𝑥)))) = · ) |
115 | 114 | opeq2d 4717 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 〈(
·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠
‘(𝑟‘𝑥))(𝑔‘𝑥))))〉 = 〈(
·𝑠 ‘ndx), ·
〉) |
116 | 26 | fveq2d 6542 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) →
(·𝑖‘(𝑟‘𝑥)) =
(·𝑖‘(𝑅‘𝑥))) |
117 | 116 | oveqd 7033 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)) = ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))) |
118 | 32, 117 | mpteq12dv 5045 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))) |
119 | 118 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))) |
120 | 102, 119 | oveq12d 7034 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))) = (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))) |
121 | 65, 65, 120 | mpoeq123dv 7087 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥))))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))) |
122 | 121 | adantr 481 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥))))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))) |
123 | | prdsval.j |
. . . . . . . . . . . 12
⊢ (𝜑 → , = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))) |
124 | 123 | ad4antr 728 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → , = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))) |
125 | 122, 124 | eqtr4d 2834 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥))))) = , ) |
126 | 125 | opeq2d 4717 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) →
〈(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))〉 =
〈(·𝑖‘ndx), , 〉) |
127 | 101, 115,
126 | tpeq123d 4591 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → {〈(Scalar‘ndx), 𝑠〉, 〈(
·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠
‘(𝑟‘𝑥))(𝑔‘𝑥))))〉,
〈(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))〉} = {〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), , 〉}) |
128 | 99, 127 | uneq12d 4061 |
. . . . . . 7
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ({〈(Base‘ndx), 𝑣〉,
〈(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))〉} ∪ {〈(Scalar‘ndx),
𝑠〉, 〈(
·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠
‘(𝑟‘𝑥))(𝑔‘𝑥))))〉,
〈(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))〉}) = ({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), × 〉} ∪
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), , 〉})) |
129 | | simpllr 772 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 𝑟 = 𝑅) |
130 | 129 | coeq2d 5619 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (TopOpen ∘ 𝑟) = (TopOpen ∘ 𝑅)) |
131 | 130 | fveq2d 6542 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (∏t‘(TopOpen
∘ 𝑟)) =
(∏t‘(TopOpen ∘ 𝑅))) |
132 | | prdsval.o |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑂 = (∏t‘(TopOpen
∘ 𝑅))) |
133 | 132 | ad4antr 728 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 𝑂 = (∏t‘(TopOpen
∘ 𝑅))) |
134 | 131, 133 | eqtr4d 2834 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (∏t‘(TopOpen
∘ 𝑟)) = 𝑂) |
135 | 134 | opeq2d 4717 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 〈(TopSet‘ndx),
(∏t‘(TopOpen ∘ 𝑟))〉 = 〈(TopSet‘ndx), 𝑂〉) |
136 | 65 | sseq2d 3920 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → ({𝑓, 𝑔} ⊆ 𝑣 ↔ {𝑓, 𝑔} ⊆ 𝐵)) |
137 | 26 | fveq2d 6542 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (le‘(𝑟‘𝑥)) = (le‘(𝑅‘𝑥))) |
138 | 137 | breqd 4973 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥) ↔ (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))) |
139 | 32, 138 | raleqbidv 3361 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥) ↔ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))) |
140 | 139 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥) ↔ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))) |
141 | 136, 140 | anbi12d 630 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥)) ↔ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥)))) |
142 | 141 | opabbidv 5028 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))} = {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}) |
143 | 142 | adantr 481 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))} = {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}) |
144 | | prdsval.l |
. . . . . . . . . . . 12
⊢ (𝜑 → ≤ = {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}) |
145 | 144 | ad4antr 728 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ≤ = {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}) |
146 | 143, 145 | eqtr4d 2834 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))} = ≤ ) |
147 | 146 | opeq2d 4717 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 〈(le‘ndx), {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}〉 = 〈(le‘ndx), ≤
〉) |
148 | 26 | fveq2d 6542 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (dist‘(𝑟‘𝑥)) = (dist‘(𝑅‘𝑥))) |
149 | 148 | oveqd 7033 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥)) = ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) |
150 | 32, 149 | mpteq12dv 5045 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥)))) |
151 | 150 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥)))) |
152 | 151 | rneqd 5690 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) = ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥)))) |
153 | 152 | uneq1d 4059 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}) = (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0})) |
154 | 153 | supeq1d 8756 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ) =
sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, <
)) |
155 | 65, 65, 154 | mpoeq123dv 7087 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))
= (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, <
))) |
156 | 155 | adantr 481 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))
= (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, <
))) |
157 | | prdsval.d |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, <
))) |
158 | 157 | ad4antr 728 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, <
))) |
159 | 156, 158 | eqtr4d 2834 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))
= 𝐷) |
160 | 159 | opeq2d 4717 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 〈(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, <
))〉 = 〈(dist‘ndx), 𝐷〉) |
161 | 135, 147,
160 | tpeq123d 4591 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → {〈(TopSet‘ndx),
(∏t‘(TopOpen ∘ 𝑟))〉, 〈(le‘ndx), {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}〉, 〈(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, <
))〉} = {〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), ≤ 〉,
〈(dist‘ndx), 𝐷〉}) |
162 | | simpr 485 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ℎ = 𝐻) |
163 | 162 | opeq2d 4717 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 〈(Hom ‘ndx), ℎ〉 = 〈(Hom ‘ndx),
𝐻〉) |
164 | 77 | sqxpeqd 5475 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑣 × 𝑣) = (𝐵 × 𝐵)) |
165 | 162 | oveqd 7033 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑐ℎ(2nd ‘𝑎)) = (𝑐𝐻(2nd ‘𝑎))) |
166 | 162 | fveq1d 6540 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (ℎ‘𝑎) = (𝐻‘𝑎)) |
167 | 26 | fveq2d 6542 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (comp‘(𝑟‘𝑥)) = (comp‘(𝑅‘𝑥))) |
168 | 167 | oveqd 7033 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (〈((1st
‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥)) = (〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))) |
169 | 168 | oveqd 7033 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)) = ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))) |
170 | 32, 169 | mpteq12dv 5045 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))) |
171 | 170 | ad2antrr 722 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))) |
172 | 165, 166,
171 | mpoeq123dv 7087 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))) = (𝑑 ∈ (𝑐𝐻(2nd ‘𝑎)), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))) |
173 | 164, 77, 172 | mpoeq123dv 7087 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))) = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ (𝑐𝐻(2nd ‘𝑎)), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))) |
174 | | prdsval.x |
. . . . . . . . . . . 12
⊢ (𝜑 → ∙ = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ (𝑐𝐻(2nd ‘𝑎)), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))) |
175 | 174 | ad4antr 728 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ∙ = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ (𝑐𝐻(2nd ‘𝑎)), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))) |
176 | 173, 175 | eqtr4d 2834 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))) = ∙ ) |
177 | 176 | opeq2d 4717 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → 〈(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉 = 〈(comp‘ndx), ∙
〉) |
178 | 163, 177 | preq12d 4584 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → {〈(Hom ‘ndx), ℎ〉, 〈(comp‘ndx),
(𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉} = {〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx),
∙
〉}) |
179 | 161, 178 | uneq12d 4061 |
. . . . . . 7
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → ({〈(TopSet‘ndx),
(∏t‘(TopOpen ∘ 𝑟))〉, 〈(le‘ndx), {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}〉, 〈(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, <
))〉} ∪ {〈(Hom ‘ndx), ℎ〉, 〈(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉}) = ({〈(TopSet‘ndx),
𝑂〉,
〈(le‘ndx), ≤ 〉,
〈(dist‘ndx), 𝐷〉} ∪ {〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx),
∙
〉})) |
180 | 128, 179 | uneq12d 4061 |
. . . . . 6
⊢
(((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) ∧ ℎ = 𝐻) → (({〈(Base‘ndx), 𝑣〉,
〈(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))〉} ∪ {〈(Scalar‘ndx),
𝑠〉, 〈(
·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠
‘(𝑟‘𝑥))(𝑔‘𝑥))))〉,
〈(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))〉}) ∪ ({〈(TopSet‘ndx),
(∏t‘(TopOpen ∘ 𝑟))〉, 〈(le‘ndx), {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}〉, 〈(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, <
))〉} ∪ {〈(Hom ‘ndx), ℎ〉, 〈(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉})) = (({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), × 〉} ∪
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), , 〉}) ∪
({〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), ≤ 〉,
〈(dist‘ndx), 𝐷〉}
∪ {〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), ∙
〉}))) |
181 | 64, 76, 180 | csbied2 3845 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) ∧ 𝑣 = 𝐵) → ⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({〈(Base‘ndx),
𝑣〉,
〈(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))〉} ∪ {〈(Scalar‘ndx),
𝑠〉, 〈(
·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠
‘(𝑟‘𝑥))(𝑔‘𝑥))))〉,
〈(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))〉}) ∪ ({〈(TopSet‘ndx),
(∏t‘(TopOpen ∘ 𝑟))〉, 〈(le‘ndx), {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}〉, 〈(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, <
))〉} ∪ {〈(Hom ‘ndx), ℎ〉, 〈(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉})) = (({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), × 〉} ∪
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), , 〉}) ∪
({〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), ≤ 〉,
〈(dist‘ndx), 𝐷〉}
∪ {〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), ∙
〉}))) |
182 | 24, 36, 181 | csbied2 3845 |
. . . 4
⊢ (((𝜑 ∧ 𝑠 = 𝑆) ∧ 𝑟 = 𝑅) → ⦋X𝑥 ∈
dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({〈(Base‘ndx),
𝑣〉,
〈(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))〉} ∪ {〈(Scalar‘ndx),
𝑠〉, 〈(
·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠
‘(𝑟‘𝑥))(𝑔‘𝑥))))〉,
〈(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))〉}) ∪ ({〈(TopSet‘ndx),
(∏t‘(TopOpen ∘ 𝑟))〉, 〈(le‘ndx), {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}〉, 〈(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, <
))〉} ∪ {〈(Hom ‘ndx), ℎ〉, 〈(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉})) = (({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), × 〉} ∪
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), , 〉}) ∪
({〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), ≤ 〉,
〈(dist‘ndx), 𝐷〉}
∪ {〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), ∙
〉}))) |
183 | 182 | anasss 467 |
. . 3
⊢ ((𝜑 ∧ (𝑠 = 𝑆 ∧ 𝑟 = 𝑅)) → ⦋X𝑥 ∈
dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({〈(Base‘ndx),
𝑣〉,
〈(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))〉} ∪ {〈(Scalar‘ndx),
𝑠〉, 〈(
·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠
‘(𝑟‘𝑥))(𝑔‘𝑥))))〉,
〈(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))〉}) ∪ ({〈(TopSet‘ndx),
(∏t‘(TopOpen ∘ 𝑟))〉, 〈(le‘ndx), {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}〉, 〈(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, <
))〉} ∪ {〈(Hom ‘ndx), ℎ〉, 〈(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉})) = (({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), × 〉} ∪
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), , 〉}) ∪
({〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), ≤ 〉,
〈(dist‘ndx), 𝐷〉}
∪ {〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), ∙
〉}))) |
184 | | prdsval.s |
. . . 4
⊢ (𝜑 → 𝑆 ∈ 𝑊) |
185 | 184 | elexd 3457 |
. . 3
⊢ (𝜑 → 𝑆 ∈ V) |
186 | | prdsval.r |
. . . 4
⊢ (𝜑 → 𝑅 ∈ 𝑍) |
187 | 186 | elexd 3457 |
. . 3
⊢ (𝜑 → 𝑅 ∈ V) |
188 | | tpex 7327 |
. . . . . 6
⊢
{〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx),
+ 〉,
〈(.r‘ndx), × 〉} ∈
V |
189 | | tpex 7327 |
. . . . . 6
⊢
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), , 〉} ∈
V |
190 | 188, 189 | unex 7326 |
. . . . 5
⊢
({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx),
+ 〉,
〈(.r‘ndx), × 〉} ∪
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), , 〉}) ∈
V |
191 | | tpex 7327 |
. . . . . 6
⊢
{〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), ≤ 〉,
〈(dist‘ndx), 𝐷〉} ∈ V |
192 | | prex 5224 |
. . . . . 6
⊢
{〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), ∙
〉} ∈ V |
193 | 191, 192 | unex 7326 |
. . . . 5
⊢
({〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), ≤ 〉,
〈(dist‘ndx), 𝐷〉} ∪ {〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx),
∙
〉}) ∈ V |
194 | 190, 193 | unex 7326 |
. . . 4
⊢
(({〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx),
+ 〉,
〈(.r‘ndx), × 〉} ∪
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), , 〉}) ∪
({〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), ≤ 〉,
〈(dist‘ndx), 𝐷〉} ∪ {〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx),
∙
〉})) ∈ V |
195 | 194 | a1i 11 |
. . 3
⊢ (𝜑 → (({〈(Base‘ndx),
𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), × 〉} ∪
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), , 〉}) ∪
({〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), ≤ 〉,
〈(dist‘ndx), 𝐷〉} ∪ {〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx),
∙
〉})) ∈ V) |
196 | 3, 183, 185, 187, 195 | ovmpod 7158 |
. 2
⊢ (𝜑 → (𝑆Xs𝑅) = (({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), × 〉} ∪
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), , 〉}) ∪
({〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), ≤ 〉,
〈(dist‘ndx), 𝐷〉} ∪ {〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx),
∙
〉}))) |
197 | 1, 196 | syl5eq 2843 |
1
⊢ (𝜑 → 𝑃 = (({〈(Base‘ndx), 𝐵〉,
〈(+g‘ndx), + 〉,
〈(.r‘ndx), × 〉} ∪
{〈(Scalar‘ndx), 𝑆〉, 〈(
·𝑠 ‘ndx), · 〉,
〈(·𝑖‘ndx), , 〉}) ∪
({〈(TopSet‘ndx), 𝑂〉, 〈(le‘ndx), ≤ 〉,
〈(dist‘ndx), 𝐷〉} ∪ {〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx),
∙
〉}))) |