Step | Hyp | Ref
| Expression |
1 | | dvhfvadd.h |
. . . . 5
⊢ 𝐻 = (LHyp‘𝐾) |
2 | | dvhfvadd.t |
. . . . 5
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
3 | | dvhfvadd.e |
. . . . 5
⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
4 | | eqid 2739 |
. . . . 5
⊢
((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊) |
5 | | dvhfvadd.u |
. . . . 5
⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
6 | 1, 2, 3, 4, 5 | dvhset 39074 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 = ({〈(Base‘ndx), (𝑇 × 𝐸)〉, 〈(+g‘ndx),
(𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
(ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉)〉, 〈(Scalar‘ndx),
((EDRing‘𝐾)‘𝑊)〉} ∪ {〈(
·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉)〉})) |
7 | 6 | fveq2d 6772 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝑈) =
(+g‘({〈(Base‘ndx), (𝑇 × 𝐸)〉, 〈(+g‘ndx),
(𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
(ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉)〉, 〈(Scalar‘ndx),
((EDRing‘𝐾)‘𝑊)〉} ∪ {〈(
·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉)〉}))) |
8 | | dvhfvadd.p |
. . . . . . . . . 10
⊢ ⨣ =
(+g‘𝐷) |
9 | | dvhfvadd.f |
. . . . . . . . . . . 12
⊢ 𝐷 = (Scalar‘𝑈) |
10 | 1, 4, 5, 9 | dvhsca 39075 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊)) |
11 | 10 | fveq2d 6772 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝐷) =
(+g‘((EDRing‘𝐾)‘𝑊))) |
12 | 8, 11 | eqtrid 2791 |
. . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ⨣ =
(+g‘((EDRing‘𝐾)‘𝑊))) |
13 | 12 | oveqd 7285 |
. . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((2nd ‘𝑓) ⨣ (2nd
‘𝑔)) =
((2nd ‘𝑓)(+g‘((EDRing‘𝐾)‘𝑊))(2nd ‘𝑔))) |
14 | 13 | 3ad2ant1 1131 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((2nd ‘𝑓) ⨣ (2nd
‘𝑔)) =
((2nd ‘𝑓)(+g‘((EDRing‘𝐾)‘𝑊))(2nd ‘𝑔))) |
15 | | xp2nd 7850 |
. . . . . . . . . 10
⊢ (𝑓 ∈ (𝑇 × 𝐸) → (2nd ‘𝑓) ∈ 𝐸) |
16 | | xp2nd 7850 |
. . . . . . . . . 10
⊢ (𝑔 ∈ (𝑇 × 𝐸) → (2nd ‘𝑔) ∈ 𝐸) |
17 | 15, 16 | anim12i 612 |
. . . . . . . . 9
⊢ ((𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((2nd ‘𝑓) ∈ 𝐸 ∧ (2nd ‘𝑔) ∈ 𝐸)) |
18 | | eqid 2739 |
. . . . . . . . . 10
⊢
(+g‘((EDRing‘𝐾)‘𝑊)) =
(+g‘((EDRing‘𝐾)‘𝑊)) |
19 | 1, 2, 3, 4, 18 | erngplus 38796 |
. . . . . . . . 9
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2nd ‘𝑓) ∈ 𝐸 ∧ (2nd ‘𝑔) ∈ 𝐸)) → ((2nd ‘𝑓)(+g‘((EDRing‘𝐾)‘𝑊))(2nd ‘𝑔)) = (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))) |
20 | 17, 19 | sylan2 592 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸))) → ((2nd ‘𝑓)(+g‘((EDRing‘𝐾)‘𝑊))(2nd ‘𝑔)) = (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))) |
21 | 20 | 3impb 1113 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((2nd ‘𝑓)(+g‘((EDRing‘𝐾)‘𝑊))(2nd ‘𝑔)) = (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))) |
22 | 14, 21 | eqtrd 2779 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((2nd ‘𝑓) ⨣ (2nd
‘𝑔)) = (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))) |
23 | 22 | opeq2d 4816 |
. . . . 5
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
((2nd ‘𝑓)
⨣
(2nd ‘𝑔))〉 = 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
(ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉) |
24 | 23 | mpoeq3dva 7343 |
. . . 4
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
((2nd ‘𝑓)
⨣
(2nd ‘𝑔))〉) = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
(ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉)) |
25 | 2 | fvexi 6782 |
. . . . . . 7
⊢ 𝑇 ∈ V |
26 | 3 | fvexi 6782 |
. . . . . . 7
⊢ 𝐸 ∈ V |
27 | 25, 26 | xpex 7594 |
. . . . . 6
⊢ (𝑇 × 𝐸) ∈ V |
28 | 27, 27 | mpoex 7906 |
. . . . 5
⊢ (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
(ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉) ∈ V |
29 | | eqid 2739 |
. . . . . 6
⊢
({〈(Base‘ndx), (𝑇 × 𝐸)〉, 〈(+g‘ndx),
(𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
(ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉)〉, 〈(Scalar‘ndx),
((EDRing‘𝐾)‘𝑊)〉} ∪ {〈(
·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉)〉}) =
({〈(Base‘ndx), (𝑇 × 𝐸)〉, 〈(+g‘ndx),
(𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
(ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉)〉, 〈(Scalar‘ndx),
((EDRing‘𝐾)‘𝑊)〉} ∪ {〈(
·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉)〉}) |
30 | 29 | lmodplusg 17018 |
. . . . 5
⊢ ((𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
(ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉) ∈ V → (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
(ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉) =
(+g‘({〈(Base‘ndx), (𝑇 × 𝐸)〉, 〈(+g‘ndx),
(𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
(ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉)〉, 〈(Scalar‘ndx),
((EDRing‘𝐾)‘𝑊)〉} ∪ {〈(
·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉)〉}))) |
31 | 28, 30 | ax-mp 5 |
. . . 4
⊢ (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
(ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉) =
(+g‘({〈(Base‘ndx), (𝑇 × 𝐸)〉, 〈(+g‘ndx),
(𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
(ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉)〉, 〈(Scalar‘ndx),
((EDRing‘𝐾)‘𝑊)〉} ∪ {〈(
·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉)〉})) |
32 | 24, 31 | eqtr2di 2796 |
. . 3
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) →
(+g‘({〈(Base‘ndx), (𝑇 × 𝐸)〉, 〈(+g‘ndx),
(𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
(ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉)〉, 〈(Scalar‘ndx),
((EDRing‘𝐾)‘𝑊)〉} ∪ {〈(
·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉)〉})) = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
((2nd ‘𝑓)
⨣
(2nd ‘𝑔))〉)) |
33 | 7, 32 | eqtrd 2779 |
. 2
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝑈) = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
((2nd ‘𝑓)
⨣
(2nd ‘𝑔))〉)) |
34 | | dvhfvadd.s |
. 2
⊢ + =
(+g‘𝑈) |
35 | | dvhfvadd.a |
. 2
⊢ ✚ =
(𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
((2nd ‘𝑓)
⨣
(2nd ‘𝑔))〉) |
36 | 33, 34, 35 | 3eqtr4g 2804 |
1
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → + = ✚ ) |