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| Mirrors > Home > ILE Home > Th. List > lspsnel6 | 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.w | . . . . 5 β’ (π β π β LMod) | |
| 2 | 1 | adantr 276 | . . . 4 β’ ((π β§ π β π) β π β LMod) |
| 3 | lspsnel5.a | . . . . 5 β’ (π β π β π) | |
| 4 | 3 | adantr 276 | . . . 4 β’ ((π β§ π β π) β π β π) |
| 5 | simpr 110 | . . . 4 β’ ((π β§ π β π) β π β π) | |
| 6 | lspsnel5.v | . . . . 5 β’ π = (Baseβπ) | |
| 7 | lspsnel5.s | . . . . 5 β’ π = (LSubSpβπ) | |
| 8 | 6, 7 | lsselg 13514 | . . . 4 β’ ((π β LMod β§ π β π β§ π β π) β π β π) |
| 9 | 2, 4, 5, 8 | syl3anc 1248 | . . 3 β’ ((π β§ π β π) β π β π) |
| 10 | lspsnel5.n | . . . . 5 β’ π = (LSpanβπ) | |
| 11 | 7, 10 | lspsnss 13557 | . . . 4 β’ ((π β LMod β§ π β π β§ π β π) β (πβ{π}) β π) |
| 12 | 2, 4, 5, 11 | syl3anc 1248 | . . 3 β’ ((π β§ π β π) β (πβ{π}) β π) |
| 13 | 9, 12 | jca 306 | . 2 β’ ((π β§ π β π) β (π β π β§ (πβ{π}) β π)) |
| 14 | 6, 10 | lspsnid 13560 | . . . . 5 β’ ((π β LMod β§ π β π) β π β (πβ{π})) |
| 15 | 1, 14 | sylan 283 | . . . 4 β’ ((π β§ π β π) β π β (πβ{π})) |
| 16 | ssel 3161 | . . . 4 β’ ((πβ{π}) β π β (π β (πβ{π}) β π β π)) | |
| 17 | 15, 16 | syl5com 29 | . . 3 β’ ((π β§ π β π) β ((πβ{π}) β π β π β π)) |
| 18 | 17 | impr 379 | . 2 β’ ((π β§ (π β π β§ (πβ{π}) β π)) β π β π) |
| 19 | 13, 18 | impbida 596 | 1 β’ (π β (π β π β (π β π β§ (πβ{π}) β π))) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ wa 104 β wb 105 = wceq 1363 β wcel 2158 β wss 3141 {csn 3604 βcfv 5228 Basecbs 12475 LModclmod 13440 LSubSpclss 13505 LSpanclspn 13539 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-cnex 7915 ax-resscn 7916 ax-1re 7918 ax-addrcl 7921 |
| This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-5 8994 df-6 8995 df-ndx 12478 df-slot 12479 df-base 12481 df-plusg 12563 df-mulr 12564 df-sca 12566 df-vsca 12567 df-0g 12724 df-mgm 12793 df-sgrp 12826 df-mnd 12837 df-grp 12899 df-lmod 13442 df-lssm 13506 df-lsp 13540 |
| This theorem is referenced by: lspsnel5 13562 |
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