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| Mirrors > Home > MPE Home > Th. List > lspsneq0 | Structured version Visualization version GIF version | ||
| Description: Span of the singleton is the zero subspace iff the vector is zero. (Contributed by NM, 27-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
| Ref | Expression |
|---|---|
| lspsneq0.v | β’ π = (Baseβπ) |
| lspsneq0.z | β’ 0 = (0gβπ) |
| lspsneq0.n | β’ π = (LSpanβπ) |
| Ref | Expression |
|---|---|
| lspsneq0 | β’ ((π β LMod β§ π β π) β ((πβ{π}) = { 0 } β π = 0 )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lspsneq0.v | . . . . 5 β’ π = (Baseβπ) | |
| 2 | lspsneq0.n | . . . . 5 β’ π = (LSpanβπ) | |
| 3 | 1, 2 | lspsnid 20891 | . . . 4 β’ ((π β LMod β§ π β π) β π β (πβ{π})) |
| 4 | eleq2 2818 | . . . 4 β’ ((πβ{π}) = { 0 } β (π β (πβ{π}) β π β { 0 })) | |
| 5 | 3, 4 | syl5ibcom 244 | . . 3 β’ ((π β LMod β§ π β π) β ((πβ{π}) = { 0 } β π β { 0 })) |
| 6 | elsni 4649 | . . 3 β’ (π β { 0 } β π = 0 ) | |
| 7 | 5, 6 | syl6 35 | . 2 β’ ((π β LMod β§ π β π) β ((πβ{π}) = { 0 } β π = 0 )) |
| 8 | lspsneq0.z | . . . . 5 β’ 0 = (0gβπ) | |
| 9 | 8, 2 | lspsn0 20906 | . . . 4 β’ (π β LMod β (πβ{ 0 }) = { 0 }) |
| 10 | 9 | adantr 479 | . . 3 β’ ((π β LMod β§ π β π) β (πβ{ 0 }) = { 0 }) |
| 11 | sneq 4642 | . . . 4 β’ (π = 0 β {π} = { 0 }) | |
| 12 | 11 | fveqeq2d 6910 | . . 3 β’ (π = 0 β ((πβ{π}) = { 0 } β (πβ{ 0 }) = { 0 })) |
| 13 | 10, 12 | syl5ibrcom 246 | . 2 β’ ((π β LMod β§ π β π) β (π = 0 β (πβ{π}) = { 0 })) |
| 14 | 7, 13 | impbid 211 | 1 β’ ((π β LMod β§ π β π) β ((πβ{π}) = { 0 } β π = 0 )) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 {csn 4632 βcfv 6553 Basecbs 17189 0gc0g 17430 LModclmod 20757 LSpanclspn 20869 |
| 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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-plusg 17255 df-0g 17432 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-grp 18907 df-minusg 18908 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-lmod 20759 df-lss 20830 df-lsp 20870 |
| This theorem is referenced by: lspsneq0b 20911 lsatn0 38511 lsator0sp 38513 lsat0cv 38545 dih0vbN 40795 dihlspsnat 40846 mapdn0 41182 mapdindp1 41233 hdmapeq0 41357 |
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