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Mirrors > Home > MPE Home > Th. List > lspsnel6 | Structured version Visualization version GIF version |
Description: Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.) (Revised by Mario Carneiro, 8-Jan-2015.) |
Ref | Expression |
---|---|
lspsnel5.v | β’ π = (Baseβπ) |
lspsnel5.s | β’ π = (LSubSpβπ) |
lspsnel5.n | β’ π = (LSpanβπ) |
lspsnel5.w | β’ (π β π β LMod) |
lspsnel5.a | β’ (π β π β π) |
Ref | Expression |
---|---|
lspsnel6 | β’ (π β (π β π β (π β π β§ (πβ{π}) β π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsnel5.a | . . . 4 β’ (π β π β π) | |
2 | lspsnel5.v | . . . . 5 β’ π = (Baseβπ) | |
3 | lspsnel5.s | . . . . 5 β’ π = (LSubSpβπ) | |
4 | 2, 3 | lssel 20414 | . . . 4 β’ ((π β π β§ π β π) β π β π) |
5 | 1, 4 | sylan 581 | . . 3 β’ ((π β§ π β π) β π β π) |
6 | lspsnel5.w | . . . . 5 β’ (π β π β LMod) | |
7 | 6 | adantr 482 | . . . 4 β’ ((π β§ π β π) β π β LMod) |
8 | 1 | adantr 482 | . . . 4 β’ ((π β§ π β π) β π β π) |
9 | simpr 486 | . . . 4 β’ ((π β§ π β π) β π β π) | |
10 | lspsnel5.n | . . . . 5 β’ π = (LSpanβπ) | |
11 | 3, 10 | lspsnss 20467 | . . . 4 β’ ((π β LMod β§ π β π β§ π β π) β (πβ{π}) β π) |
12 | 7, 8, 9, 11 | syl3anc 1372 | . . 3 β’ ((π β§ π β π) β (πβ{π}) β π) |
13 | 5, 12 | jca 513 | . 2 β’ ((π β§ π β π) β (π β π β§ (πβ{π}) β π)) |
14 | 2, 10 | lspsnid 20470 | . . . . 5 β’ ((π β LMod β§ π β π) β π β (πβ{π})) |
15 | 6, 14 | sylan 581 | . . . 4 β’ ((π β§ π β π) β π β (πβ{π})) |
16 | ssel 3942 | . . . 4 β’ ((πβ{π}) β π β (π β (πβ{π}) β π β π)) | |
17 | 15, 16 | syl5com 31 | . . 3 β’ ((π β§ π β π) β ((πβ{π}) β π β π β π)) |
18 | 17 | impr 456 | . 2 β’ ((π β§ (π β π β§ (πβ{π}) β π)) β π β π) |
19 | 13, 18 | impbida 800 | 1 β’ (π β (π β π β (π β π β§ (πβ{π}) β π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β wss 3915 {csn 4591 βcfv 6501 Basecbs 17090 LModclmod 20338 LSubSpclss 20408 LSpanclspn 20448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-0g 17330 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-grp 18758 df-lmod 20340 df-lss 20409 df-lsp 20449 |
This theorem is referenced by: lspsnel5 20472 lsmelval2 20562 dihjat1lem 39920 |
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