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| Mirrors > Home > MPE Home > Th. List > lspsnss | Structured version Visualization version GIF version | ||
| Description: The span of the singleton of a subspace member is included in the subspace. (spansnss 31329 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.) |
| Ref | Expression |
|---|---|
| lspsnss.s | β’ π = (LSubSpβπ) |
| lspsnss.n | β’ π = (LSpanβπ) |
| Ref | Expression |
|---|---|
| lspsnss | β’ ((π β LMod β§ π β π β§ π β π) β (πβ{π}) β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | snssi 4806 | . 2 β’ (π β π β {π} β π) | |
| 2 | lspsnss.s | . . 3 β’ π = (LSubSpβπ) | |
| 3 | lspsnss.n | . . 3 β’ π = (LSpanβπ) | |
| 4 | 2, 3 | lspssp 20833 | . 2 β’ ((π β LMod β§ π β π β§ {π} β π) β (πβ{π}) β π) |
| 5 | 1, 4 | syl3an3 1162 | 1 β’ ((π β LMod 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§ w3a 1084 = wceq 1533 β wcel 2098 β wss 3943 {csn 4623 βcfv 6536 LModclmod 20704 LSubSpclss 20776 LSpanclspn 20816 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-lmod 20706 df-lss 20777 df-lsp 20817 |
| This theorem is referenced by: lspsnel3 20836 lspsnel6 20839 |
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