| Step | Hyp | Ref
| Expression |
|---|
| 1 | | lkr0f.d |
. . . . . . 7
β’ π· = (Scalarβπ) |
| 2 | | eqid 2732 |
. . . . . . 7
β’
(Baseβπ·) =
(Baseβπ·) |
| 3 | | lkr0f.v |
. . . . . . 7
β’ π = (Baseβπ) |
| 4 | | lkr0f.f |
. . . . . . 7
β’ πΉ = (LFnlβπ) |
| 5 | 1, 2, 3, 4 | lflf 37921 |
. . . . . 6
β’ ((π β LMod β§ πΊ β πΉ) β πΊ:πβΆ(Baseβπ·)) |
| 6 | 5 | ffnd 6715 |
. . . . 5
β’ ((π β LMod β§ πΊ β πΉ) β πΊ Fn π) |
| 7 | 6 | adantr 481 |
. . . 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 37946 |
. . . . . 6
β’ ((π β LMod β§ πΊ β πΉ) β (πΎβπΊ) = (β‘πΊ β { 0 })) |
| 11 | 10 | eqeq1d 2734 |
. . . . 5
β’ ((π β LMod β§ πΊ β πΉ) β ((πΎβπΊ) = π β (β‘πΊ β { 0 }) = π)) |
| 12 | 11 | biimpa 477 |
. . . 4
β’ (((π β LMod β§ πΊ β πΉ) β§ (πΎβπΊ) = π) β (β‘πΊ β { 0 }) = π) |
| 13 | 8 | fvexi 6902 |
. . . . . 6
β’ 0 β
V |
| 14 | 13 | fconst2 7202 |
. . . . 5
β’ (πΊ:πβΆ{ 0 } β πΊ = (π Γ { 0 })) |
| 15 | | fconst4 7212 |
. . . . 5
β’ (πΊ:πβΆ{ 0 } β (πΊ Fn π β§ (β‘πΊ β { 0 }) = π)) |
| 16 | 14, 15 | bitr3i 276 |
. . . 4
β’ (πΊ = (π Γ { 0 }) β (πΊ Fn π β§ (β‘πΊ β { 0 }) = π)) |
| 17 | 7, 12, 16 | sylanbrc 583 |
. . 3
β’ (((π β LMod β§ πΊ β πΉ) β§ (πΎβπΊ) = π) β πΊ = (π Γ { 0 })) |
| 18 | 17 | ex 413 |
. 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 482 |
. . . . 5
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β (πΊ Fn π β§ (β‘πΊ β { 0 }) = π)) |
| 21 | | simpr 485 |
. . . . . . . . 9
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β πΊ = (π Γ { 0 })) |
| 22 | | eqid 2732 |
. . . . . . . . . . 11
β’
(LFnlβπ) =
(LFnlβπ) |
| 23 | 1, 8, 3, 22 | lfl0f 37927 |
. . . . . . . . . 10
β’ (π β LMod β (π Γ { 0 }) β
(LFnlβπ)) |
| 24 | 23 | adantr 481 |
. . . . . . . . 9
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β (π Γ { 0 }) β
(LFnlβπ)) |
| 25 | 21, 24 | eqeltrd 2833 |
. . . . . . . 8
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β πΊ β (LFnlβπ)) |
| 26 | 1, 8, 22, 9 | lkrval 37946 |
. . . . . . . 8
β’ ((π β LMod β§ πΊ β (LFnlβπ)) β (πΎβπΊ) = (β‘πΊ β { 0 })) |
| 27 | 25, 26 | syldan 591 |
. . . . . . 7
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β (πΎβπΊ) = (β‘πΊ β { 0 })) |
| 28 | 27 | eqeq1d 2734 |
. . . . . 6
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β ((πΎβπΊ) = π β (β‘πΊ β { 0 }) = π)) |
| 29 | | ffn 6714 |
. . . . . . . . 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 482 |
. . . . . . 7
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β πΊ Fn π) |
| 32 | 31 | biantrurd 533 |
. . . . . 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 278 |
. . . . 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 256 |
. . . 4
β’ ((π β LMod β§ πΊ = (π Γ { 0 })) β (πΎβπΊ) = π) |
| 35 | 34 | ex 413 |
. . 3
β’ (π β LMod β (πΊ = (π Γ { 0 }) β (πΎβπΊ) = π)) |
| 36 | 35 | adantr 481 |
. 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 }))) |
