Step | Hyp | Ref
| Expression |
1 | | lkr0f.d |
. . . . . . 7
β’ π· = (Scalarβπ) |
2 | | eqid 2724 |
. . . . . . 7
β’
(Baseβπ·) =
(Baseβπ·) |
3 | | lkr0f.v |
. . . . . . 7
β’ π = (Baseβπ) |
4 | | lkr0f.f |
. . . . . . 7
β’ πΉ = (LFnlβπ) |
5 | 1, 2, 3, 4 | lflf 38436 |
. . . . . 6
β’ ((π β LMod β§ πΊ β πΉ) β πΊ:πβΆ(Baseβπ·)) |
6 | 5 | ffnd 6709 |
. . . . 5
β’ ((π β LMod β§ πΊ β πΉ) β πΊ Fn π) |
7 | 6 | adantr 480 |
. . . 4
β’ (((π β LMod β§ πΊ β πΉ) β§ (πΎβπΊ) = π) β πΊ Fn π) |
8 | | lkr0f.o |
. . . . . . 7
β’ 0 =
(0gβπ·) |
9 | | lkr0f.k |
. . . . . . 7
β’ πΎ = (LKerβπ) |
10 | 1, 8, 4, 9 | lkrval 38461 |
. . . . . 6
β’ ((π β LMod β§ πΊ β πΉ) β (πΎβπΊ) = (β‘πΊ β { 0 })) |
11 | 10 | eqeq1d 2726 |
. . . . 5
β’ ((π β LMod β§ πΊ β πΉ) β ((πΎβπΊ) = π β (β‘πΊ β { 0 }) = π)) |
12 | 11 | biimpa 476 |
. . . 4
β’ (((π β LMod β§ πΊ β πΉ) β§ (πΎβπΊ) = π) β (β‘πΊ β { 0 }) = π) |
13 | 8 | fvexi 6896 |
. . . . . 6
β’ 0 β
V |
14 | 13 | fconst2 7199 |
. . . . 5
β’ (πΊ:πβΆ{ 0 } β πΊ = (π Γ { 0 })) |
15 | | fconst4 7208 |
. . . . 5
β’ (πΊ:πβΆ{ 0 } β (πΊ Fn π β§ (β‘πΊ β { 0 }) = π)) |
16 | 14, 15 | bitr3i 277 |
. . . 4
β’ (πΊ = (π Γ { 0 }) β (πΊ Fn π β§ (β‘πΊ β { 0 }) = π)) |
17 | 7, 12, 16 | sylanbrc 582 |
. . 3
β’ (((π β LMod β§ πΊ β πΉ) β§ (πΎβπΊ) = π) β πΊ = (π Γ { 0 })) |
18 | 17 | ex 412 |
. 2
β’ ((π β LMod β§ πΊ β πΉ) β ((πΎβπΊ) = π β πΊ = (π Γ { 0 }))) |
19 | 16 | biimpi 215 |
. . . . . 6
β’ (πΊ = (π Γ { 0 }) β (πΊ Fn π β§ (β‘πΊ β { 0 }) = π)) |
20 | 19 | adantl 481 |
. . . . 5
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β (πΊ Fn π β§ (β‘πΊ β { 0 }) = π)) |
21 | | simpr 484 |
. . . . . . . . 9
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β πΊ = (π Γ { 0 })) |
22 | | eqid 2724 |
. . . . . . . . . . 11
β’
(LFnlβπ) =
(LFnlβπ) |
23 | 1, 8, 3, 22 | lfl0f 38442 |
. . . . . . . . . 10
β’ (π β LMod β (π Γ { 0 }) β
(LFnlβπ)) |
24 | 23 | adantr 480 |
. . . . . . . . 9
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β (π Γ { 0 }) β
(LFnlβπ)) |
25 | 21, 24 | eqeltrd 2825 |
. . . . . . . 8
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β πΊ β (LFnlβπ)) |
26 | 1, 8, 22, 9 | lkrval 38461 |
. . . . . . . 8
β’ ((π β LMod β§ πΊ β (LFnlβπ)) β (πΎβπΊ) = (β‘πΊ β { 0 })) |
27 | 25, 26 | syldan 590 |
. . . . . . 7
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β (πΎβπΊ) = (β‘πΊ β { 0 })) |
28 | 27 | eqeq1d 2726 |
. . . . . 6
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β ((πΎβπΊ) = π β (β‘πΊ β { 0 }) = π)) |
29 | | ffn 6708 |
. . . . . . . . 9
β’ (πΊ:πβΆ{ 0 } β πΊ Fn π) |
30 | 14, 29 | sylbir 234 |
. . . . . . . 8
β’ (πΊ = (π Γ { 0 }) β πΊ Fn π) |
31 | 30 | adantl 481 |
. . . . . . 7
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β πΊ Fn π) |
32 | 31 | biantrurd 532 |
. . . . . 6
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β ((β‘πΊ β { 0 }) = π β (πΊ Fn π β§ (β‘πΊ β { 0 }) = π))) |
33 | 28, 32 | bitrd 279 |
. . . . 5
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β ((πΎβπΊ) = π β (πΊ Fn π β§ (β‘πΊ β { 0 }) = π))) |
34 | 20, 33 | mpbird 257 |
. . . 4
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β (πΎβπΊ) = π) |
35 | 34 | ex 412 |
. . 3
β’ (π β LMod β (πΊ = (π Γ { 0 }) β (πΎβπΊ) = π)) |
36 | 35 | adantr 480 |
. 2
β’ ((π β LMod β§ πΊ β πΉ) β (πΊ = (π Γ { 0 }) β (πΎβπΊ) = π)) |
37 | 18, 36 | impbid 211 |
1
β’ ((π β LMod β§ πΊ β πΉ) β ((πΎβπΊ) = π β πΊ = (π Γ { 0 }))) |