Detailed syntax breakdown of Definition df-lindf
Step | Hyp | Ref
| Expression |
1 | | clindf 20766 |
. 2
class
LIndF |
2 | | vf |
. . . . . . 7
setvar 𝑓 |
3 | 2 | cv 1542 |
. . . . . 6
class 𝑓 |
4 | 3 | cdm 5551 |
. . . . 5
class dom 𝑓 |
5 | | vw |
. . . . . . 7
setvar 𝑤 |
6 | 5 | cv 1542 |
. . . . . 6
class 𝑤 |
7 | | cbs 16760 |
. . . . . 6
class
Base |
8 | 6, 7 | cfv 6380 |
. . . . 5
class
(Base‘𝑤) |
9 | 4, 8, 3 | wf 6376 |
. . . 4
wff 𝑓:dom 𝑓⟶(Base‘𝑤) |
10 | | vk |
. . . . . . . . . . 11
setvar 𝑘 |
11 | 10 | cv 1542 |
. . . . . . . . . 10
class 𝑘 |
12 | | vx |
. . . . . . . . . . . 12
setvar 𝑥 |
13 | 12 | cv 1542 |
. . . . . . . . . . 11
class 𝑥 |
14 | 13, 3 | cfv 6380 |
. . . . . . . . . 10
class (𝑓‘𝑥) |
15 | | cvsca 16806 |
. . . . . . . . . . 11
class
·𝑠 |
16 | 6, 15 | cfv 6380 |
. . . . . . . . . 10
class (
·𝑠 ‘𝑤) |
17 | 11, 14, 16 | co 7213 |
. . . . . . . . 9
class (𝑘(
·𝑠 ‘𝑤)(𝑓‘𝑥)) |
18 | 13 | csn 4541 |
. . . . . . . . . . . 12
class {𝑥} |
19 | 4, 18 | cdif 3863 |
. . . . . . . . . . 11
class (dom
𝑓 ∖ {𝑥}) |
20 | 3, 19 | cima 5554 |
. . . . . . . . . 10
class (𝑓 “ (dom 𝑓 ∖ {𝑥})) |
21 | | clspn 20008 |
. . . . . . . . . . 11
class
LSpan |
22 | 6, 21 | cfv 6380 |
. . . . . . . . . 10
class
(LSpan‘𝑤) |
23 | 20, 22 | cfv 6380 |
. . . . . . . . 9
class
((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) |
24 | 17, 23 | wcel 2110 |
. . . . . . . 8
wff (𝑘(
·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) |
25 | 24 | wn 3 |
. . . . . . 7
wff ¬
(𝑘(
·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) |
26 | | vs |
. . . . . . . . . 10
setvar 𝑠 |
27 | 26 | cv 1542 |
. . . . . . . . 9
class 𝑠 |
28 | 27, 7 | cfv 6380 |
. . . . . . . 8
class
(Base‘𝑠) |
29 | | c0g 16944 |
. . . . . . . . . 10
class
0g |
30 | 27, 29 | cfv 6380 |
. . . . . . . . 9
class
(0g‘𝑠) |
31 | 30 | csn 4541 |
. . . . . . . 8
class
{(0g‘𝑠)} |
32 | 28, 31 | cdif 3863 |
. . . . . . 7
class
((Base‘𝑠)
∖ {(0g‘𝑠)}) |
33 | 25, 10, 32 | wral 3061 |
. . . . . 6
wff
∀𝑘 ∈
((Base‘𝑠) ∖
{(0g‘𝑠)})
¬ (𝑘(
·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) |
34 | 33, 12, 4 | wral 3061 |
. . . . 5
wff
∀𝑥 ∈ dom
𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠
‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) |
35 | | csca 16805 |
. . . . . 6
class
Scalar |
36 | 6, 35 | cfv 6380 |
. . . . 5
class
(Scalar‘𝑤) |
37 | 34, 26, 36 | wsbc 3694 |
. . . 4
wff
[(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠
‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))) |
38 | 9, 37 | wa 399 |
. . 3
wff (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠
‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥})))) |
39 | 38, 2, 5 | copab 5115 |
. 2
class
{〈𝑓, 𝑤〉 ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠
‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))} |
40 | 1, 39 | wceq 1543 |
1
wff LIndF =
{〈𝑓, 𝑤〉 ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠
‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))} |