Step | Hyp | Ref
| Expression |
1 | | eqid 2733 |
. . . . 5
β’
(LSubSpβπ) =
(LSubSpβπ) |
2 | | lspabs2.n |
. . . . 5
β’ π = (LSpanβπ) |
3 | | lspabs2.w |
. . . . . 6
β’ (π β π β LVec) |
4 | | lveclmod 20582 |
. . . . . 6
β’ (π β LVec β π β LMod) |
5 | 3, 4 | syl 17 |
. . . . 5
β’ (π β π β LMod) |
6 | | lspabs2.x |
. . . . . . 7
β’ (π β π β π) |
7 | | lspabs2.v |
. . . . . . . 8
β’ π = (Baseβπ) |
8 | 7, 1, 2 | lspsncl 20453 |
. . . . . . 7
β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
9 | 5, 6, 8 | syl2anc 585 |
. . . . . 6
β’ (π β (πβ{π}) β (LSubSpβπ)) |
10 | | lspabs3.y |
. . . . . . 7
β’ (π β π β π) |
11 | 7, 1, 2 | lspsncl 20453 |
. . . . . . 7
β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
12 | 5, 10, 11 | syl2anc 585 |
. . . . . 6
β’ (π β (πβ{π}) β (LSubSpβπ)) |
13 | | eqid 2733 |
. . . . . . 7
β’
(LSSumβπ) =
(LSSumβπ) |
14 | 1, 13 | lsmcl 20559 |
. . . . . 6
β’ ((π β LMod β§ (πβ{π}) β (LSubSpβπ) β§ (πβ{π}) β (LSubSpβπ)) β ((πβ{π})(LSSumβπ)(πβ{π})) β (LSubSpβπ)) |
15 | 5, 9, 12, 14 | syl3anc 1372 |
. . . . 5
β’ (π β ((πβ{π})(LSSumβπ)(πβ{π})) β (LSubSpβπ)) |
16 | 7, 2 | lspsnsubg 20456 |
. . . . . . 7
β’ ((π β LMod β§ π β π) β (πβ{π}) β (SubGrpβπ)) |
17 | 5, 6, 16 | syl2anc 585 |
. . . . . 6
β’ (π β (πβ{π}) β (SubGrpβπ)) |
18 | | lspabs3.e |
. . . . . . 7
β’ (π β (πβ{π}) = (πβ{π})) |
19 | 18, 17 | eqeltrrd 2835 |
. . . . . 6
β’ (π β (πβ{π}) β (SubGrpβπ)) |
20 | 7, 2 | lspsnid 20469 |
. . . . . . 7
β’ ((π β LMod β§ π β π) β π β (πβ{π})) |
21 | 5, 6, 20 | syl2anc 585 |
. . . . . 6
β’ (π β π β (πβ{π})) |
22 | 7, 2 | lspsnid 20469 |
. . . . . . 7
β’ ((π β LMod β§ π β π) β π β (πβ{π})) |
23 | 5, 10, 22 | syl2anc 585 |
. . . . . 6
β’ (π β π β (πβ{π})) |
24 | | lspabs2.p |
. . . . . . 7
β’ + =
(+gβπ) |
25 | 24, 13 | lsmelvali 19437 |
. . . . . 6
β’ ((((πβ{π}) β (SubGrpβπ) β§ (πβ{π}) β (SubGrpβπ)) β§ (π β (πβ{π}) β§ π β (πβ{π}))) β (π + π) β ((πβ{π})(LSSumβπ)(πβ{π}))) |
26 | 17, 19, 21, 23, 25 | syl22anc 838 |
. . . . 5
β’ (π β (π + π) β ((πβ{π})(LSSumβπ)(πβ{π}))) |
27 | 1, 2, 5, 15, 26 | lspsnel5a 20472 |
. . . 4
β’ (π β (πβ{(π + π)}) β ((πβ{π})(LSSumβπ)(πβ{π}))) |
28 | 18 | oveq2d 7374 |
. . . . 5
β’ (π β ((πβ{π})(LSSumβπ)(πβ{π})) = ((πβ{π})(LSSumβπ)(πβ{π}))) |
29 | 13 | lsmidm 19450 |
. . . . . 6
β’ ((πβ{π}) β (SubGrpβπ) β ((πβ{π})(LSSumβπ)(πβ{π})) = (πβ{π})) |
30 | 17, 29 | syl 17 |
. . . . 5
β’ (π β ((πβ{π})(LSSumβπ)(πβ{π})) = (πβ{π})) |
31 | 28, 30 | eqtr3d 2775 |
. . . 4
β’ (π β ((πβ{π})(LSSumβπ)(πβ{π})) = (πβ{π})) |
32 | 27, 31 | sseqtrd 3985 |
. . 3
β’ (π β (πβ{(π + π)}) β (πβ{π})) |
33 | | lspabs2.o |
. . . 4
β’ 0 =
(0gβπ) |
34 | 7, 24 | lmodvacl 20351 |
. . . . . 6
β’ ((π β LMod β§ π β π β§ π β π) β (π + π) β π) |
35 | 5, 6, 10, 34 | syl3anc 1372 |
. . . . 5
β’ (π β (π + π) β π) |
36 | | lspabs3.xy |
. . . . 5
β’ (π β (π + π) β 0 ) |
37 | | eldifsn 4748 |
. . . . 5
β’ ((π + π) β (π β { 0 }) β ((π + π) β π β§ (π + π) β 0 )) |
38 | 35, 36, 37 | sylanbrc 584 |
. . . 4
β’ (π β (π + π) β (π β { 0 })) |
39 | 7, 33, 2, 3, 38, 6 | lspsncmp 20593 |
. . 3
β’ (π β ((πβ{(π + π)}) β (πβ{π}) β (πβ{(π + π)}) = (πβ{π}))) |
40 | 32, 39 | mpbid 231 |
. 2
β’ (π β (πβ{(π + π)}) = (πβ{π})) |
41 | 40 | eqcomd 2739 |
1
β’ (π β (πβ{π}) = (πβ{(π + π)})) |