Step | Hyp | Ref
| Expression |
1 | | lfl0f.o |
. . . . 5
⊢ 0 =
(0g‘𝐷) |
2 | 1 | fvexi 6460 |
. . . 4
⊢ 0 ∈
V |
3 | 2 | fconst 6341 |
. . 3
⊢ (𝑉 × { 0 }):𝑉⟶{ 0 } |
4 | | lfl0f.d |
. . . . 5
⊢ 𝐷 = (Scalar‘𝑊) |
5 | | eqid 2777 |
. . . . 5
⊢
(Base‘𝐷) =
(Base‘𝐷) |
6 | 4, 5, 1 | lmod0cl 19281 |
. . . 4
⊢ (𝑊 ∈ LMod → 0 ∈
(Base‘𝐷)) |
7 | 6 | snssd 4571 |
. . 3
⊢ (𝑊 ∈ LMod → { 0 } ⊆
(Base‘𝐷)) |
8 | | fss 6304 |
. . 3
⊢ (((𝑉 × { 0 }):𝑉⟶{ 0 } ∧ { 0 } ⊆
(Base‘𝐷)) →
(𝑉 × { 0 }):𝑉⟶(Base‘𝐷)) |
9 | 3, 7, 8 | sylancr 581 |
. 2
⊢ (𝑊 ∈ LMod → (𝑉 × { 0 }):𝑉⟶(Base‘𝐷)) |
10 | 4 | lmodring 19263 |
. . . . . . . . 9
⊢ (𝑊 ∈ LMod → 𝐷 ∈ Ring) |
11 | 10 | ad2antrr 716 |
. . . . . . . 8
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → 𝐷 ∈ Ring) |
12 | | simplrl 767 |
. . . . . . . 8
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → 𝑟 ∈ (Base‘𝐷)) |
13 | | eqid 2777 |
. . . . . . . . 9
⊢
(.r‘𝐷) = (.r‘𝐷) |
14 | 5, 13, 1 | ringrz 18975 |
. . . . . . . 8
⊢ ((𝐷 ∈ Ring ∧ 𝑟 ∈ (Base‘𝐷)) → (𝑟(.r‘𝐷) 0 ) = 0 ) |
15 | 11, 12, 14 | syl2anc 579 |
. . . . . . 7
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → (𝑟(.r‘𝐷) 0 ) = 0 ) |
16 | 15 | oveq1d 6937 |
. . . . . 6
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → ((𝑟(.r‘𝐷) 0
)(+g‘𝐷)
0 ) = (
0
(+g‘𝐷)
0
)) |
17 | | ringgrp 18939 |
. . . . . . . 8
⊢ (𝐷 ∈ Ring → 𝐷 ∈ Grp) |
18 | 11, 17 | syl 17 |
. . . . . . 7
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → 𝐷 ∈ Grp) |
19 | 5, 1 | grpidcl 17837 |
. . . . . . . 8
⊢ (𝐷 ∈ Grp → 0 ∈
(Base‘𝐷)) |
20 | 18, 19 | syl 17 |
. . . . . . 7
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → 0 ∈ (Base‘𝐷)) |
21 | | eqid 2777 |
. . . . . . . 8
⊢
(+g‘𝐷) = (+g‘𝐷) |
22 | 5, 21, 1 | grplid 17839 |
. . . . . . 7
⊢ ((𝐷 ∈ Grp ∧ 0 ∈
(Base‘𝐷)) → (
0
(+g‘𝐷)
0 ) =
0
) |
23 | 18, 20, 22 | syl2anc 579 |
. . . . . 6
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → ( 0 (+g‘𝐷) 0 ) = 0 ) |
24 | 16, 23 | eqtrd 2813 |
. . . . 5
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → ((𝑟(.r‘𝐷) 0
)(+g‘𝐷)
0 ) =
0
) |
25 | | simplrr 768 |
. . . . . . . 8
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ 𝑉) |
26 | 2 | fvconst2 6741 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝑉 → ((𝑉 × { 0 })‘𝑥) = 0 ) |
27 | 25, 26 | syl 17 |
. . . . . . 7
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → ((𝑉 × { 0 })‘𝑥) = 0 ) |
28 | 27 | oveq2d 6938 |
. . . . . 6
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → (𝑟(.r‘𝐷)((𝑉 × { 0 })‘𝑥)) = (𝑟(.r‘𝐷) 0 )) |
29 | 2 | fvconst2 6741 |
. . . . . . 7
⊢ (𝑦 ∈ 𝑉 → ((𝑉 × { 0 })‘𝑦) = 0 ) |
30 | 29 | adantl 475 |
. . . . . 6
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → ((𝑉 × { 0 })‘𝑦) = 0 ) |
31 | 28, 30 | oveq12d 6940 |
. . . . 5
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → ((𝑟(.r‘𝐷)((𝑉 × { 0 })‘𝑥))(+g‘𝐷)((𝑉 × { 0 })‘𝑦)) = ((𝑟(.r‘𝐷) 0
)(+g‘𝐷)
0
)) |
32 | | simpll 757 |
. . . . . . 7
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → 𝑊 ∈ LMod) |
33 | | lfl0f.v |
. . . . . . . . 9
⊢ 𝑉 = (Base‘𝑊) |
34 | | eqid 2777 |
. . . . . . . . 9
⊢ (
·𝑠 ‘𝑊) = ( ·𝑠
‘𝑊) |
35 | 33, 4, 34, 5 | lmodvscl 19272 |
. . . . . . . 8
⊢ ((𝑊 ∈ LMod ∧ 𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉) → (𝑟( ·𝑠
‘𝑊)𝑥) ∈ 𝑉) |
36 | 32, 12, 25, 35 | syl3anc 1439 |
. . . . . . 7
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → (𝑟( ·𝑠
‘𝑊)𝑥) ∈ 𝑉) |
37 | | simpr 479 |
. . . . . . 7
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉) |
38 | | eqid 2777 |
. . . . . . . 8
⊢
(+g‘𝑊) = (+g‘𝑊) |
39 | 33, 38 | lmodvacl 19269 |
. . . . . . 7
⊢ ((𝑊 ∈ LMod ∧ (𝑟(
·𝑠 ‘𝑊)𝑥) ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝑟( ·𝑠
‘𝑊)𝑥)(+g‘𝑊)𝑦) ∈ 𝑉) |
40 | 32, 36, 37, 39 | syl3anc 1439 |
. . . . . 6
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → ((𝑟( ·𝑠
‘𝑊)𝑥)(+g‘𝑊)𝑦) ∈ 𝑉) |
41 | 2 | fvconst2 6741 |
. . . . . 6
⊢ (((𝑟(
·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦) ∈ 𝑉 → ((𝑉 × { 0 })‘((𝑟(
·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = 0 ) |
42 | 40, 41 | syl 17 |
. . . . 5
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → ((𝑉 × { 0 })‘((𝑟(
·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = 0 ) |
43 | 24, 31, 42 | 3eqtr4rd 2824 |
. . . 4
⊢ (((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) ∧ 𝑦 ∈ 𝑉) → ((𝑉 × { 0 })‘((𝑟(
·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑟(.r‘𝐷)((𝑉 × { 0 })‘𝑥))(+g‘𝐷)((𝑉 × { 0 })‘𝑦))) |
44 | 43 | ralrimiva 3147 |
. . 3
⊢ ((𝑊 ∈ LMod ∧ (𝑟 ∈ (Base‘𝐷) ∧ 𝑥 ∈ 𝑉)) → ∀𝑦 ∈ 𝑉 ((𝑉 × { 0 })‘((𝑟(
·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑟(.r‘𝐷)((𝑉 × { 0 })‘𝑥))(+g‘𝐷)((𝑉 × { 0 })‘𝑦))) |
45 | 44 | ralrimivva 3152 |
. 2
⊢ (𝑊 ∈ LMod →
∀𝑟 ∈
(Base‘𝐷)∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ((𝑉 × { 0 })‘((𝑟(
·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑟(.r‘𝐷)((𝑉 × { 0 })‘𝑥))(+g‘𝐷)((𝑉 × { 0 })‘𝑦))) |
46 | | lfl0f.f |
. . 3
⊢ 𝐹 = (LFnl‘𝑊) |
47 | 33, 38, 4, 34, 5, 21, 13, 46 | islfl 35198 |
. 2
⊢ (𝑊 ∈ LMod → ((𝑉 × { 0 }) ∈ 𝐹 ↔ ((𝑉 × { 0 }):𝑉⟶(Base‘𝐷) ∧ ∀𝑟 ∈ (Base‘𝐷)∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ((𝑉 × { 0 })‘((𝑟(
·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑟(.r‘𝐷)((𝑉 × { 0 })‘𝑥))(+g‘𝐷)((𝑉 × { 0 })‘𝑦))))) |
48 | 9, 45, 47 | mpbir2and 703 |
1
⊢ (𝑊 ∈ LMod → (𝑉 × { 0 }) ∈ 𝐹) |