Step | Hyp | Ref
| Expression |
1 | | islbs2.v |
. . . . 5
⊢ 𝑉 = (Base‘𝑊) |
2 | | islbs2.j |
. . . . 5
⊢ 𝐽 = (LBasis‘𝑊) |
3 | 1, 2 | lbsss 19980 |
. . . 4
⊢ (𝐵 ∈ 𝐽 → 𝐵 ⊆ 𝑉) |
4 | 3 | adantl 485 |
. . 3
⊢ ((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) → 𝐵 ⊆ 𝑉) |
5 | | islbs2.n |
. . . . 5
⊢ 𝑁 = (LSpan‘𝑊) |
6 | 1, 2, 5 | lbssp 19982 |
. . . 4
⊢ (𝐵 ∈ 𝐽 → (𝑁‘𝐵) = 𝑉) |
7 | 6 | adantl 485 |
. . 3
⊢ ((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) → (𝑁‘𝐵) = 𝑉) |
8 | | lveclmod 20009 |
. . . . . . 7
⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) |
9 | | eqid 2739 |
. . . . . . . . 9
⊢
(Scalar‘𝑊) =
(Scalar‘𝑊) |
10 | 9 | lvecdrng 20008 |
. . . . . . . 8
⊢ (𝑊 ∈ LVec →
(Scalar‘𝑊) ∈
DivRing) |
11 | | eqid 2739 |
. . . . . . . . 9
⊢
(0g‘(Scalar‘𝑊)) =
(0g‘(Scalar‘𝑊)) |
12 | | eqid 2739 |
. . . . . . . . 9
⊢
(1r‘(Scalar‘𝑊)) =
(1r‘(Scalar‘𝑊)) |
13 | 11, 12 | drngunz 19648 |
. . . . . . . 8
⊢
((Scalar‘𝑊)
∈ DivRing → (1r‘(Scalar‘𝑊)) ≠
(0g‘(Scalar‘𝑊))) |
14 | 10, 13 | syl 17 |
. . . . . . 7
⊢ (𝑊 ∈ LVec →
(1r‘(Scalar‘𝑊)) ≠
(0g‘(Scalar‘𝑊))) |
15 | 8, 14 | jca 515 |
. . . . . 6
⊢ (𝑊 ∈ LVec → (𝑊 ∈ LMod ∧
(1r‘(Scalar‘𝑊)) ≠
(0g‘(Scalar‘𝑊)))) |
16 | 2, 5, 9, 12, 11 | lbsind2 19984 |
. . . . . 6
⊢ (((𝑊 ∈ LMod ∧
(1r‘(Scalar‘𝑊)) ≠
(0g‘(Scalar‘𝑊))) ∧ 𝐵 ∈ 𝐽 ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥}))) |
17 | 15, 16 | syl3an1 1164 |
. . . . 5
⊢ ((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽 ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥}))) |
18 | 17 | 3expa 1119 |
. . . 4
⊢ (((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥}))) |
19 | 18 | ralrimiva 3097 |
. . 3
⊢ ((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) → ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥}))) |
20 | 4, 7, 19 | 3jca 1129 |
. 2
⊢ ((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽) → (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) |
21 | | simpr1 1195 |
. . 3
⊢ ((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) → 𝐵 ⊆ 𝑉) |
22 | | simpr2 1196 |
. . 3
⊢ ((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) → (𝑁‘𝐵) = 𝑉) |
23 | | id 22 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) |
24 | | sneq 4536 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) |
25 | 24 | difeq2d 4023 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑦})) |
26 | 25 | fveq2d 6690 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (𝑁‘(𝐵 ∖ {𝑥})) = (𝑁‘(𝐵 ∖ {𝑦}))) |
27 | 23, 26 | eleq12d 2828 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})) ↔ 𝑦 ∈ (𝑁‘(𝐵 ∖ {𝑦})))) |
28 | 27 | notbid 321 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})) ↔ ¬ 𝑦 ∈ (𝑁‘(𝐵 ∖ {𝑦})))) |
29 | | simplr3 1218 |
. . . . . 6
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥}))) |
30 | | simprl 771 |
. . . . . 6
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → 𝑦 ∈ 𝐵) |
31 | 28, 29, 30 | rspcdva 3531 |
. . . . 5
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → ¬ 𝑦 ∈ (𝑁‘(𝐵 ∖ {𝑦}))) |
32 | | simpll 767 |
. . . . . . . 8
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → 𝑊 ∈ LVec) |
33 | | simprr 773 |
. . . . . . . . 9
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))})) |
34 | | eldifsn 4685 |
. . . . . . . . 9
⊢ (𝑧 ∈
((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}) ↔ (𝑧 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑧 ≠
(0g‘(Scalar‘𝑊)))) |
35 | 33, 34 | sylib 221 |
. . . . . . . 8
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → (𝑧 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑧 ≠
(0g‘(Scalar‘𝑊)))) |
36 | 21 | adantr 484 |
. . . . . . . . 9
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → 𝐵 ⊆ 𝑉) |
37 | 36, 30 | sseldd 3888 |
. . . . . . . 8
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → 𝑦 ∈ 𝑉) |
38 | | eqid 2739 |
. . . . . . . . 9
⊢ (
·𝑠 ‘𝑊) = ( ·𝑠
‘𝑊) |
39 | | eqid 2739 |
. . . . . . . . 9
⊢
(Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) |
40 | 1, 9, 38, 39, 11, 5 | lspsnvs 20017 |
. . . . . . . 8
⊢ ((𝑊 ∈ LVec ∧ (𝑧 ∈
(Base‘(Scalar‘𝑊)) ∧ 𝑧 ≠
(0g‘(Scalar‘𝑊))) ∧ 𝑦 ∈ 𝑉) → (𝑁‘{(𝑧( ·𝑠
‘𝑊)𝑦)}) = (𝑁‘{𝑦})) |
41 | 32, 35, 37, 40 | syl3anc 1372 |
. . . . . . 7
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → (𝑁‘{(𝑧( ·𝑠
‘𝑊)𝑦)}) = (𝑁‘{𝑦})) |
42 | 41 | sseq1d 3918 |
. . . . . 6
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → ((𝑁‘{(𝑧( ·𝑠
‘𝑊)𝑦)}) ⊆ (𝑁‘(𝐵 ∖ {𝑦})) ↔ (𝑁‘{𝑦}) ⊆ (𝑁‘(𝐵 ∖ {𝑦})))) |
43 | | eqid 2739 |
. . . . . . 7
⊢
(LSubSp‘𝑊) =
(LSubSp‘𝑊) |
44 | 8 | adantr 484 |
. . . . . . . 8
⊢ ((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) → 𝑊 ∈ LMod) |
45 | 44 | adantr 484 |
. . . . . . 7
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → 𝑊 ∈ LMod) |
46 | 36 | ssdifssd 4043 |
. . . . . . . 8
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → (𝐵 ∖ {𝑦}) ⊆ 𝑉) |
47 | 1, 43, 5 | lspcl 19879 |
. . . . . . . 8
⊢ ((𝑊 ∈ LMod ∧ (𝐵 ∖ {𝑦}) ⊆ 𝑉) → (𝑁‘(𝐵 ∖ {𝑦})) ∈ (LSubSp‘𝑊)) |
48 | 45, 46, 47 | syl2anc 587 |
. . . . . . 7
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → (𝑁‘(𝐵 ∖ {𝑦})) ∈ (LSubSp‘𝑊)) |
49 | 35 | simpld 498 |
. . . . . . . 8
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → 𝑧 ∈ (Base‘(Scalar‘𝑊))) |
50 | 1, 9, 38, 39 | lmodvscl 19782 |
. . . . . . . 8
⊢ ((𝑊 ∈ LMod ∧ 𝑧 ∈
(Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑉) → (𝑧( ·𝑠
‘𝑊)𝑦) ∈ 𝑉) |
51 | 45, 49, 37, 50 | syl3anc 1372 |
. . . . . . 7
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → (𝑧( ·𝑠
‘𝑊)𝑦) ∈ 𝑉) |
52 | 1, 43, 5, 45, 48, 51 | lspsnel5 19898 |
. . . . . 6
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → ((𝑧( ·𝑠
‘𝑊)𝑦) ∈ (𝑁‘(𝐵 ∖ {𝑦})) ↔ (𝑁‘{(𝑧( ·𝑠
‘𝑊)𝑦)}) ⊆ (𝑁‘(𝐵 ∖ {𝑦})))) |
53 | 1, 43, 5, 45, 48, 37 | lspsnel5 19898 |
. . . . . 6
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → (𝑦 ∈ (𝑁‘(𝐵 ∖ {𝑦})) ↔ (𝑁‘{𝑦}) ⊆ (𝑁‘(𝐵 ∖ {𝑦})))) |
54 | 42, 52, 53 | 3bitr4d 314 |
. . . . 5
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → ((𝑧( ·𝑠
‘𝑊)𝑦) ∈ (𝑁‘(𝐵 ∖ {𝑦})) ↔ 𝑦 ∈ (𝑁‘(𝐵 ∖ {𝑦})))) |
55 | 31, 54 | mtbird 328 |
. . . 4
⊢ (((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → ¬ (𝑧( ·𝑠
‘𝑊)𝑦) ∈ (𝑁‘(𝐵 ∖ {𝑦}))) |
56 | 55 | ralrimivva 3104 |
. . 3
⊢ ((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}) ¬ (𝑧( ·𝑠
‘𝑊)𝑦) ∈ (𝑁‘(𝐵 ∖ {𝑦}))) |
57 | 1, 9, 38, 39, 2, 5, 11 | islbs 19979 |
. . . 4
⊢ (𝑊 ∈ LVec → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}) ¬ (𝑧( ·𝑠
‘𝑊)𝑦) ∈ (𝑁‘(𝐵 ∖ {𝑦}))))) |
58 | 57 | adantr 484 |
. . 3
⊢ ((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}) ¬ (𝑧( ·𝑠
‘𝑊)𝑦) ∈ (𝑁‘(𝐵 ∖ {𝑦}))))) |
59 | 21, 22, 56, 58 | mpbir3and 1343 |
. 2
⊢ ((𝑊 ∈ LVec ∧ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))) → 𝐵 ∈ 𝐽) |
60 | 20, 59 | impbida 801 |
1
⊢ (𝑊 ∈ LVec → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥}))))) |