| 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 2737 | . . . . 5
⊢
((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊) | 
| 5 |  | dvhfvadd.u | . . . . 5
⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | 
| 6 | 1, 2, 3, 4, 5 | dvhset 41083 | . . . 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 6910 | . . 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 41084 | . . . . . . . . . . 11
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊)) | 
| 11 | 10 | fveq2d 6910 | . . . . . . . . . 10
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝐷) =
(+g‘((EDRing‘𝐾)‘𝑊))) | 
| 12 | 8, 11 | eqtrid 2789 | . . . . . . . . 9
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ⨣ =
(+g‘((EDRing‘𝐾)‘𝑊))) | 
| 13 | 12 | oveqd 7448 | . . . . . . . 8
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((2nd ‘𝑓) ⨣ (2nd
‘𝑔)) =
((2nd ‘𝑓)(+g‘((EDRing‘𝐾)‘𝑊))(2nd ‘𝑔))) | 
| 14 | 13 | 3ad2ant1 1134 | . . . . . . 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 8047 | . . . . . . . . . 10
⊢ (𝑓 ∈ (𝑇 × 𝐸) → (2nd ‘𝑓) ∈ 𝐸) | 
| 16 |  | xp2nd 8047 | . . . . . . . . . 10
⊢ (𝑔 ∈ (𝑇 × 𝐸) → (2nd ‘𝑔) ∈ 𝐸) | 
| 17 | 15, 16 | anim12i 613 | . . . . . . . . 9
⊢ ((𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((2nd ‘𝑓) ∈ 𝐸 ∧ (2nd ‘𝑔) ∈ 𝐸)) | 
| 18 |  | eqid 2737 | . . . . . . . . . 10
⊢
(+g‘((EDRing‘𝐾)‘𝑊)) =
(+g‘((EDRing‘𝐾)‘𝑊)) | 
| 19 | 1, 2, 3, 4, 18 | erngplus 40805 | . . . . . . . . 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 593 | . . . . . . . 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 1115 | . . . . . . 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 2777 | . . . . . 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 4880 | . . . . 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 7510 | . . . 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 6920 | . . . . . . 7
⊢ 𝑇 ∈ V | 
| 26 | 3 | fvexi 6920 | . . . . . . 7
⊢ 𝐸 ∈ V | 
| 27 | 25, 26 | xpex 7773 | . . . . . 6
⊢ (𝑇 × 𝐸) ∈ V | 
| 28 | 27, 27 | mpoex 8104 | . . . . 5
⊢ (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
(ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉) ∈ V | 
| 29 |  | eqid 2737 | . . . . . 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 17371 | . . . . 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 2794 | . . 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 2777 | . 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 2802 | 1
⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → + = ✚ ) |