| Step | Hyp | Ref
| Expression |
| 1 | | dihval.i |
. . 3
⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
| 2 | | dihval.b |
. . . . 5
⊢ 𝐵 = (Base‘𝐾) |
| 3 | | dihval.l |
. . . . 5
⊢ ≤ =
(le‘𝐾) |
| 4 | | dihval.j |
. . . . 5
⊢ ∨ =
(join‘𝐾) |
| 5 | | dihval.m |
. . . . 5
⊢ ∧ =
(meet‘𝐾) |
| 6 | | dihval.a |
. . . . 5
⊢ 𝐴 = (Atoms‘𝐾) |
| 7 | | dihval.h |
. . . . 5
⊢ 𝐻 = (LHyp‘𝐾) |
| 8 | 2, 3, 4, 5, 6, 7 | dihffval 41232 |
. . . 4
⊢ (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))))))))) |
| 9 | 8 | fveq1d 6908 |
. . 3
⊢ (𝐾 ∈ 𝑉 → ((DIsoH‘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))))))))‘𝑊)) |
| 10 | 1, 9 | eqtrid 2789 |
. 2
⊢ (𝐾 ∈ 𝑉 → 𝐼 = ((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))))))))‘𝑊)) |
| 11 | | breq2 5147 |
. . . . 5
⊢ (𝑤 = 𝑊 → (𝑥 ≤ 𝑤 ↔ 𝑥 ≤ 𝑊)) |
| 12 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑤 = 𝑊 → ((DIsoB‘𝐾)‘𝑤) = ((DIsoB‘𝐾)‘𝑊)) |
| 13 | | dihval.d |
. . . . . . 7
⊢ 𝐷 = ((DIsoB‘𝐾)‘𝑊) |
| 14 | 12, 13 | eqtr4di 2795 |
. . . . . 6
⊢ (𝑤 = 𝑊 → ((DIsoB‘𝐾)‘𝑤) = 𝐷) |
| 15 | 14 | fveq1d 6908 |
. . . . 5
⊢ (𝑤 = 𝑊 → (((DIsoB‘𝐾)‘𝑤)‘𝑥) = (𝐷‘𝑥)) |
| 16 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊)) |
| 17 | | dihval.u |
. . . . . . . . 9
⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
| 18 | 16, 17 | eqtr4di 2795 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈) |
| 19 | 18 | fveq2d 6910 |
. . . . . . 7
⊢ (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = (LSubSp‘𝑈)) |
| 20 | | dihval.s |
. . . . . . 7
⊢ 𝑆 = (LSubSp‘𝑈) |
| 21 | 19, 20 | eqtr4di 2795 |
. . . . . 6
⊢ (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = 𝑆) |
| 22 | | breq2 5147 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → (𝑞 ≤ 𝑤 ↔ 𝑞 ≤ 𝑊)) |
| 23 | 22 | notbid 318 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → (¬ 𝑞 ≤ 𝑤 ↔ ¬ 𝑞 ≤ 𝑊)) |
| 24 | | oveq2 7439 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑊 → (𝑥 ∧ 𝑤) = (𝑥 ∧ 𝑊)) |
| 25 | 24 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → (𝑞 ∨ (𝑥 ∧ 𝑤)) = (𝑞 ∨ (𝑥 ∧ 𝑊))) |
| 26 | 25 | eqeq1d 2739 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → ((𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥 ↔ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥)) |
| 27 | 23, 26 | anbi12d 632 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) ↔ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥))) |
| 28 | 18 | fveq2d 6910 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑊 → (LSSum‘((DVecH‘𝐾)‘𝑤)) = (LSSum‘𝑈)) |
| 29 | | dihval.p |
. . . . . . . . . . 11
⊢ ⊕ =
(LSSum‘𝑈) |
| 30 | 28, 29 | eqtr4di 2795 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → (LSSum‘((DVecH‘𝐾)‘𝑤)) = ⊕ ) |
| 31 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑊 → ((DIsoC‘𝐾)‘𝑤) = ((DIsoC‘𝐾)‘𝑊)) |
| 32 | | dihval.c |
. . . . . . . . . . . 12
⊢ 𝐶 = ((DIsoC‘𝐾)‘𝑊) |
| 33 | 31, 32 | eqtr4di 2795 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑊 → ((DIsoC‘𝐾)‘𝑤) = 𝐶) |
| 34 | 33 | fveq1d 6908 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → (((DIsoC‘𝐾)‘𝑤)‘𝑞) = (𝐶‘𝑞)) |
| 35 | 14, 24 | fveq12d 6913 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑊 → (((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)) = (𝐷‘(𝑥 ∧ 𝑊))) |
| 36 | 30, 34, 35 | oveq123d 7452 |
. . . . . . . . 9
⊢ (𝑤 = 𝑊 → ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤))) = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊)))) |
| 37 | 36 | eqeq2d 2748 |
. . . . . . . 8
⊢ (𝑤 = 𝑊 → (𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤))) ↔ 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊))))) |
| 38 | 27, 37 | imbi12d 344 |
. . . . . . 7
⊢ (𝑤 = 𝑊 → (((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))) ↔ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊)))))) |
| 39 | 38 | ralbidv 3178 |
. . . . . 6
⊢ (𝑤 = 𝑊 → (∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑤 ∧ (𝑞 ∨ (𝑥 ∧ 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 ∧ 𝑤)))) ↔ ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊)))))) |
| 40 | 21, 39 | riotaeqbidv 7391 |
. . . . 5
⊢ (𝑤 = 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))))) = (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊)))))) |
| 41 | 11, 15, 40 | ifbieq12d 4554 |
. . . 4
⊢ (𝑤 = 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)))))) = if(𝑥 ≤ 𝑊, (𝐷‘𝑥), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊))))))) |
| 42 | 41 | mpteq2dv 5244 |
. . 3
⊢ (𝑤 = 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))))))) = (𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑊, (𝐷‘𝑥), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊)))))))) |
| 43 | | eqid 2737 |
. . 3
⊢ (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)))))))) = (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)))))))) |
| 44 | 42, 43, 2 | mptfvmpt 7248 |
. 2
⊢ (𝑊 ∈ 𝐻 → ((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))))))))‘𝑊) = (𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑊, (𝐷‘𝑥), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊)))))))) |
| 45 | 10, 44 | sylan9eq 2797 |
1
⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ 𝐵 ↦ if(𝑥 ≤ 𝑊, (𝐷‘𝑥), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑥 ∧ 𝑊)))))))) |