| Step | Hyp | Ref
| Expression |
| 1 | | elex 3501 |
. 2
⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V) |
| 2 | | fveq2 6906 |
. . . . 5
⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾)) |
| 3 | | dihval.h |
. . . . 5
⊢ 𝐻 = (LHyp‘𝐾) |
| 4 | 2, 3 | eqtr4di 2795 |
. . . 4
⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻) |
| 5 | | fveq2 6906 |
. . . . . 6
⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾)) |
| 6 | | dihval.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝐾) |
| 7 | 5, 6 | eqtr4di 2795 |
. . . . 5
⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵) |
| 8 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾)) |
| 9 | | dihval.l |
. . . . . . . 8
⊢ ≤ =
(le‘𝐾) |
| 10 | 8, 9 | eqtr4di 2795 |
. . . . . . 7
⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ ) |
| 11 | 10 | breqd 5154 |
. . . . . 6
⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑤 ↔ 𝑥 ≤ 𝑤)) |
| 12 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑘 = 𝐾 → (DIsoB‘𝑘) = (DIsoB‘𝐾)) |
| 13 | 12 | fveq1d 6908 |
. . . . . . 7
⊢ (𝑘 = 𝐾 → ((DIsoB‘𝑘)‘𝑤) = ((DIsoB‘𝐾)‘𝑤)) |
| 14 | 13 | fveq1d 6908 |
. . . . . 6
⊢ (𝑘 = 𝐾 → (((DIsoB‘𝑘)‘𝑤)‘𝑥) = (((DIsoB‘𝐾)‘𝑤)‘𝑥)) |
| 15 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾)) |
| 16 | 15 | fveq1d 6908 |
. . . . . . . 8
⊢ (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤)) |
| 17 | 16 | fveq2d 6910 |
. . . . . . 7
⊢ (𝑘 = 𝐾 → (LSubSp‘((DVecH‘𝑘)‘𝑤)) = (LSubSp‘((DVecH‘𝐾)‘𝑤))) |
| 18 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾)) |
| 19 | | dihval.a |
. . . . . . . . 9
⊢ 𝐴 = (Atoms‘𝐾) |
| 20 | 18, 19 | eqtr4di 2795 |
. . . . . . . 8
⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴) |
| 21 | 10 | breqd 5154 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝐾 → (𝑞(le‘𝑘)𝑤 ↔ 𝑞 ≤ 𝑤)) |
| 22 | 21 | notbid 318 |
. . . . . . . . . 10
⊢ (𝑘 = 𝐾 → (¬ 𝑞(le‘𝑘)𝑤 ↔ ¬ 𝑞 ≤ 𝑤)) |
| 23 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾)) |
| 24 | | dihval.j |
. . . . . . . . . . . . 13
⊢ ∨ =
(join‘𝐾) |
| 25 | 23, 24 | eqtr4di 2795 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ ) |
| 26 | | eqidd 2738 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝐾 → 𝑞 = 𝑞) |
| 27 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾)) |
| 28 | | dihval.m |
. . . . . . . . . . . . . 14
⊢ ∧ =
(meet‘𝐾) |
| 29 | 27, 28 | eqtr4di 2795 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝐾 → (meet‘𝑘) = ∧ ) |
| 30 | 29 | oveqd 7448 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝐾 → (𝑥(meet‘𝑘)𝑤) = (𝑥 ∧ 𝑤)) |
| 31 | 25, 26, 30 | oveq123d 7452 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝐾 → (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = (𝑞 ∨ (𝑥 ∧ 𝑤))) |
| 32 | 31 | eqeq1d 2739 |
. . . . . . . . . 10
⊢ (𝑘 = 𝐾 → ((𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥 ↔ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥)) |
| 33 | 22, 32 | anbi12d 632 |
. . . . . . . . 9
⊢ (𝑘 = 𝐾 → ((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) ↔ (¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥))) |
| 34 | 16 | fveq2d 6910 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝐾 → (LSSum‘((DVecH‘𝑘)‘𝑤)) = (LSSum‘((DVecH‘𝐾)‘𝑤))) |
| 35 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝐾 → (DIsoC‘𝑘) = (DIsoC‘𝐾)) |
| 36 | 35 | fveq1d 6908 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝐾 → ((DIsoC‘𝑘)‘𝑤) = ((DIsoC‘𝐾)‘𝑤)) |
| 37 | 36 | fveq1d 6908 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝐾 → (((DIsoC‘𝑘)‘𝑤)‘𝑞) = (((DIsoC‘𝐾)‘𝑤)‘𝑞)) |
| 38 | 13, 30 | fveq12d 6913 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝐾 → (((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)) = (((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤))) |
| 39 | 34, 37, 38 | oveq123d 7452 |
. . . . . . . . . 10
⊢ (𝑘 = 𝐾 → ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))) = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))) |
| 40 | 39 | eqeq2d 2748 |
. . . . . . . . 9
⊢ (𝑘 = 𝐾 → (𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))) ↔ 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤))))) |
| 41 | 33, 40 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑘 = 𝐾 → (((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))) ↔ ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))))) |
| 42 | 20, 41 | raleqbidv 3346 |
. . . . . . 7
⊢ (𝑘 = 𝐾 → (∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))) ↔ ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))))) |
| 43 | 17, 42 | riotaeqbidv 7391 |
. . . . . 6
⊢ (𝑘 = 𝐾 → (℩𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))))) = (℩𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))))) |
| 44 | 11, 14, 43 | ifbieq12d 4554 |
. . . . 5
⊢ (𝑘 = 𝐾 → if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))))) = if(𝑥 ≤ 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤))))))) |
| 45 | 7, 44 | mpteq12dv 5233 |
. . . 4
⊢ (𝑘 = 𝐾 → (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))))))) = (𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))))))) |
| 46 | 4, 45 | mpteq12dv 5233 |
. . 3
⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))))))) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤))))))))) |
| 47 | | df-dih 41231 |
. . 3
⊢ DIsoH =
(𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))))))))) |
| 48 | 46, 47, 3 | mptfvmpt 7248 |
. 2
⊢ (𝐾 ∈ V →
(DIsoH‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤))))))))) |
| 49 | 1, 48 | syl 17 |
1
⊢ (𝐾 ∈ 𝑉 → (DIsoH‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤))))))))) |