Step | Hyp | Ref
| Expression |
1 | | l1cvpat.q |
. . 3
β’ (π β π β π΄) |
2 | | l1cvpat.w |
. . . 4
β’ (π β π β LVec) |
3 | | l1cvpat.v |
. . . . 5
β’ π = (Baseβπ) |
4 | | eqid 2737 |
. . . . 5
β’
(LSpanβπ) =
(LSpanβπ) |
5 | | eqid 2737 |
. . . . 5
β’
(0gβπ) = (0gβπ) |
6 | | l1cvpat.a |
. . . . 5
β’ π΄ = (LSAtomsβπ) |
7 | 3, 4, 5, 6 | islsat 37456 |
. . . 4
β’ (π β LVec β (π β π΄ β βπ£ β (π β {(0gβπ)})π = ((LSpanβπ)β{π£}))) |
8 | 2, 7 | syl 17 |
. . 3
β’ (π β (π β π΄ β βπ£ β (π β {(0gβπ)})π = ((LSpanβπ)β{π£}))) |
9 | 1, 8 | mpbid 231 |
. 2
β’ (π β βπ£ β (π β {(0gβπ)})π = ((LSpanβπ)β{π£})) |
10 | | l1cvpat.m |
. 2
β’ (π β Β¬ π β π) |
11 | | eldifi 4087 |
. . . 4
β’ (π£ β (π β {(0gβπ)}) β π£ β π) |
12 | | l1cvpat.s |
. . . . . . . . 9
β’ π = (LSubSpβπ) |
13 | | lveclmod 20570 |
. . . . . . . . . . 11
β’ (π β LVec β π β LMod) |
14 | 2, 13 | syl 17 |
. . . . . . . . . 10
β’ (π β π β LMod) |
15 | 14 | 3ad2ant1 1134 |
. . . . . . . . 9
β’ ((π β§ π£ β π β§ π = ((LSpanβπ)β{π£})) β π β LMod) |
16 | | l1cvpat.u |
. . . . . . . . . 10
β’ (π β π β π) |
17 | 16 | 3ad2ant1 1134 |
. . . . . . . . 9
β’ ((π β§ π£ β π β§ π = ((LSpanβπ)β{π£})) β π β π) |
18 | | simp2 1138 |
. . . . . . . . 9
β’ ((π β§ π£ β π β§ π = ((LSpanβπ)β{π£})) β π£ β π) |
19 | 3, 12, 4, 15, 17, 18 | lspsnel5 20459 |
. . . . . . . 8
β’ ((π β§ π£ β π β§ π = ((LSpanβπ)β{π£})) β (π£ β π β ((LSpanβπ)β{π£}) β π)) |
20 | 19 | notbid 318 |
. . . . . . 7
β’ ((π β§ π£ β π β§ π = ((LSpanβπ)β{π£})) β (Β¬ π£ β π β Β¬ ((LSpanβπ)β{π£}) β π)) |
21 | | l1cvpat.p |
. . . . . . . . 9
β’ β =
(LSSumβπ) |
22 | | eqid 2737 |
. . . . . . . . 9
β’
(LSHypβπ) =
(LSHypβπ) |
23 | 2 | 3ad2ant1 1134 |
. . . . . . . . 9
β’ ((π β§ π£ β π β§ π = ((LSpanβπ)β{π£})) β π β LVec) |
24 | | l1cvpat.l |
. . . . . . . . . . 11
β’ (π β ππΆπ) |
25 | | l1cvpat.c |
. . . . . . . . . . . 12
β’ πΆ = ( βL
βπ) |
26 | 3, 12, 22, 25, 2 | islshpcv 37518 |
. . . . . . . . . . 11
β’ (π β (π β (LSHypβπ) β (π β π β§ ππΆπ))) |
27 | 16, 24, 26 | mpbir2and 712 |
. . . . . . . . . 10
β’ (π β π β (LSHypβπ)) |
28 | 27 | 3ad2ant1 1134 |
. . . . . . . . 9
β’ ((π β§ π£ β π β§ π = ((LSpanβπ)β{π£})) β π β (LSHypβπ)) |
29 | 3, 4, 21, 22, 23, 28, 18 | lshpnelb 37449 |
. . . . . . . 8
β’ ((π β§ π£ β π β§ π = ((LSpanβπ)β{π£})) β (Β¬ π£ β π β (π β ((LSpanβπ)β{π£})) = π)) |
30 | 29 | biimpd 228 |
. . . . . . 7
β’ ((π β§ π£ β π β§ π = ((LSpanβπ)β{π£})) β (Β¬ π£ β π β (π β ((LSpanβπ)β{π£})) = π)) |
31 | 20, 30 | sylbird 260 |
. . . . . 6
β’ ((π β§ π£ β π β§ π = ((LSpanβπ)β{π£})) β (Β¬ ((LSpanβπ)β{π£}) β π β (π β ((LSpanβπ)β{π£})) = π)) |
32 | | sseq1 3970 |
. . . . . . . . 9
β’ (π = ((LSpanβπ)β{π£}) β (π β π β ((LSpanβπ)β{π£}) β π)) |
33 | 32 | notbid 318 |
. . . . . . . 8
β’ (π = ((LSpanβπ)β{π£}) β (Β¬ π β π β Β¬ ((LSpanβπ)β{π£}) β π)) |
34 | | oveq2 7366 |
. . . . . . . . 9
β’ (π = ((LSpanβπ)β{π£}) β (π β π) = (π β ((LSpanβπ)β{π£}))) |
35 | 34 | eqeq1d 2739 |
. . . . . . . 8
β’ (π = ((LSpanβπ)β{π£}) β ((π β π) = π β (π β ((LSpanβπ)β{π£})) = π)) |
36 | 33, 35 | imbi12d 345 |
. . . . . . 7
β’ (π = ((LSpanβπ)β{π£}) β ((Β¬ π β π β (π β π) = π) β (Β¬ ((LSpanβπ)β{π£}) β π β (π β ((LSpanβπ)β{π£})) = π))) |
37 | 36 | 3ad2ant3 1136 |
. . . . . 6
β’ ((π β§ π£ β π β§ π = ((LSpanβπ)β{π£})) β ((Β¬ π β π β (π β π) = π) β (Β¬ ((LSpanβπ)β{π£}) β π β (π β ((LSpanβπ)β{π£})) = π))) |
38 | 31, 37 | mpbird 257 |
. . . . 5
β’ ((π β§ π£ β π β§ π = ((LSpanβπ)β{π£})) β (Β¬ π β π β (π β π) = π)) |
39 | 38 | 3exp 1120 |
. . . 4
β’ (π β (π£ β π β (π = ((LSpanβπ)β{π£}) β (Β¬ π β π β (π β π) = π)))) |
40 | 11, 39 | syl5 34 |
. . 3
β’ (π β (π£ β (π β {(0gβπ)}) β (π = ((LSpanβπ)β{π£}) β (Β¬ π β π β (π β π) = π)))) |
41 | 40 | rexlimdv 3151 |
. 2
β’ (π β (βπ£ β (π β {(0gβπ)})π = ((LSpanβπ)β{π£}) β (Β¬ π β π β (π β π) = π))) |
42 | 9, 10, 41 | mp2d 49 |
1
β’ (π β (π β π) = π) |