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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 13638 | . . . 4 β’ ((π β LMod β§ π β π β§ π β π) β π β π) |
| 9 | 2, 4, 5, 8 | syl3anc 1249 | . . 3 β’ ((π β§ π β π) β π β π) |
| 10 | lspsnel5.n | . . . . 5 β’ π = (LSpanβπ) | |
| 11 | 7, 10 | lspsnss 13681 | . . . 4 β’ ((π β LMod β§ π β π β§ π β π) β (πβ{π}) β π) |
| 12 | 2, 4, 5, 11 | syl3anc 1249 | . . 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 13684 | . . . . 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 3164 | . . . 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 1364 β wcel 2160 β wss 3144 {csn 3607 βcfv 5231 Basecbs 12480 LModclmod 13564 LSubSpclss 13629 LSpanclspn 13663 |
| 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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-cnex 7920 ax-resscn 7921 ax-1re 7923 ax-addrcl 7926 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-inn 8938 df-2 8996 df-3 8997 df-4 8998 df-5 8999 df-6 9000 df-ndx 12483 df-slot 12484 df-base 12486 df-plusg 12568 df-mulr 12569 df-sca 12571 df-vsca 12572 df-0g 12729 df-mgm 12798 df-sgrp 12831 df-mnd 12844 df-grp 12914 df-lmod 13566 df-lssm 13630 df-lsp 13664 |
| This theorem is referenced by: lspsnel5 13686 |
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