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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 20548 | . . . 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 20601 | . . . 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 20604 | . . . . 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 3976 | . . . 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 3949 {csn 4629 βcfv 6544 Basecbs 17144 LModclmod 20471 LSubSpclss 20542 LSpanclspn 20582 |
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 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-lmod 20473 df-lss 20543 df-lsp 20583 |
This theorem is referenced by: lspsnel5 20606 lsmelval2 20696 dihjat1lem 40299 |
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