Proof of Theorem islssd
| Step | Hyp | Ref
| Expression |
| 1 | | islssd.u |
. . . 4
⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
| 2 | | islssd.v |
. . . 4
⊢ (𝜑 → 𝑉 = (Base‘𝑊)) |
| 3 | 1, 2 | sseqtrd 4020 |
. . 3
⊢ (𝜑 → 𝑈 ⊆ (Base‘𝑊)) |
| 4 | | islssd.z |
. . 3
⊢ (𝜑 → 𝑈 ≠ ∅) |
| 5 | | islssd.c |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈) |
| 6 | 5 | 3exp2 1355 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐵 → (𝑎 ∈ 𝑈 → (𝑏 ∈ 𝑈 → ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈)))) |
| 7 | 6 | imp43 427 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈) |
| 8 | 7 | ralrimivva 3202 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈) |
| 9 | 8 | ex 412 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝐵 → ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈)) |
| 10 | | islssd.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 = (Base‘𝐹)) |
| 11 | | islssd.f |
. . . . . . . 8
⊢ (𝜑 → 𝐹 = (Scalar‘𝑊)) |
| 12 | 11 | fveq2d 6910 |
. . . . . . 7
⊢ (𝜑 → (Base‘𝐹) =
(Base‘(Scalar‘𝑊))) |
| 13 | 10, 12 | eqtrd 2777 |
. . . . . 6
⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘𝑊))) |
| 14 | 13 | eleq2d 2827 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘(Scalar‘𝑊)))) |
| 15 | | islssd.p |
. . . . . . . . 9
⊢ (𝜑 → + =
(+g‘𝑊)) |
| 16 | 15 | oveqd 7448 |
. . . . . . . 8
⊢ (𝜑 → ((𝑥 · 𝑎) + 𝑏) = ((𝑥 · 𝑎)(+g‘𝑊)𝑏)) |
| 17 | | islssd.t |
. . . . . . . . . 10
⊢ (𝜑 → · = (
·𝑠 ‘𝑊)) |
| 18 | 17 | oveqd 7448 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 · 𝑎) = (𝑥( ·𝑠
‘𝑊)𝑎)) |
| 19 | 18 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝜑 → ((𝑥 · 𝑎)(+g‘𝑊)𝑏) = ((𝑥( ·𝑠
‘𝑊)𝑎)(+g‘𝑊)𝑏)) |
| 20 | 16, 19 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 · 𝑎) + 𝑏) = ((𝑥( ·𝑠
‘𝑊)𝑎)(+g‘𝑊)𝑏)) |
| 21 | 20 | eleq1d 2826 |
. . . . . 6
⊢ (𝜑 → (((𝑥 · 𝑎) + 𝑏) ∈ 𝑈 ↔ ((𝑥( ·𝑠
‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ 𝑈)) |
| 22 | 21 | 2ralbidv 3221 |
. . . . 5
⊢ (𝜑 → (∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈 ↔ ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥( ·𝑠
‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ 𝑈)) |
| 23 | 9, 14, 22 | 3imtr3d 293 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ (Base‘(Scalar‘𝑊)) → ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥( ·𝑠
‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ 𝑈)) |
| 24 | 23 | ralrimiv 3145 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥( ·𝑠
‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ 𝑈) |
| 25 | | eqid 2737 |
. . . 4
⊢
(Scalar‘𝑊) =
(Scalar‘𝑊) |
| 26 | | eqid 2737 |
. . . 4
⊢
(Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) |
| 27 | | eqid 2737 |
. . . 4
⊢
(Base‘𝑊) =
(Base‘𝑊) |
| 28 | | eqid 2737 |
. . . 4
⊢
(+g‘𝑊) = (+g‘𝑊) |
| 29 | | eqid 2737 |
. . . 4
⊢ (
·𝑠 ‘𝑊) = ( ·𝑠
‘𝑊) |
| 30 | | eqid 2737 |
. . . 4
⊢
(LSubSp‘𝑊) =
(LSubSp‘𝑊) |
| 31 | 25, 26, 27, 28, 29, 30 | islss 20932 |
. . 3
⊢ (𝑈 ∈ (LSubSp‘𝑊) ↔ (𝑈 ⊆ (Base‘𝑊) ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈
(Base‘(Scalar‘𝑊))∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥( ·𝑠
‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ 𝑈)) |
| 32 | 3, 4, 24, 31 | syl3anbrc 1344 |
. 2
⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝑊)) |
| 33 | | islssd.s |
. 2
⊢ (𝜑 → 𝑆 = (LSubSp‘𝑊)) |
| 34 | 32, 33 | eleqtrrd 2844 |
1
⊢ (𝜑 → 𝑈 ∈ 𝑆) |