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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 31399 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 4814 | . 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 20877 | . 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 3947 {csn 4630 βcfv 6551 LModclmod 20748 LSubSpclss 20820 LSpanclspn 20860 |
| 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 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-0g 17428 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-grp 18898 df-lmod 20750 df-lss 20821 df-lsp 20861 |
| This theorem is referenced by: lspsnel3 20880 lspsnel6 20883 |
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