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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatn0 | Structured version Visualization version GIF version |
Description: A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 32173 analog.) (Contributed by NM, 25-Aug-2014.) |
Ref | Expression |
---|---|
lsatn0.o | β’ 0 = (0gβπ) |
lsatn0.a | β’ π΄ = (LSAtomsβπ) |
lsatn0.w | β’ (π β π β LMod) |
lsatn0.u | β’ (π β π β π΄) |
Ref | Expression |
---|---|
lsatn0 | β’ (π β π β { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatn0.u | . . 3 β’ (π β π β π΄) | |
2 | lsatn0.w | . . . 4 β’ (π β π β LMod) | |
3 | eqid 2727 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
4 | eqid 2727 | . . . . 5 β’ (LSpanβπ) = (LSpanβπ) | |
5 | lsatn0.o | . . . . 5 β’ 0 = (0gβπ) | |
6 | lsatn0.a | . . . . 5 β’ π΄ = (LSAtomsβπ) | |
7 | 3, 4, 5, 6 | islsat 38467 | . . . 4 β’ (π β LMod β (π β π΄ β βπ£ β ((Baseβπ) β { 0 })π = ((LSpanβπ)β{π£}))) |
8 | 2, 7 | syl 17 | . . 3 β’ (π β (π β π΄ β βπ£ β ((Baseβπ) β { 0 })π = ((LSpanβπ)β{π£}))) |
9 | 1, 8 | mpbid 231 | . 2 β’ (π β βπ£ β ((Baseβπ) β { 0 })π = ((LSpanβπ)β{π£})) |
10 | eldifsn 4793 | . . . . 5 β’ (π£ β ((Baseβπ) β { 0 }) β (π£ β (Baseβπ) β§ π£ β 0 )) | |
11 | 3, 5, 4 | lspsneq0 20901 | . . . . . . . . 9 β’ ((π β LMod β§ π£ β (Baseβπ)) β (((LSpanβπ)β{π£}) = { 0 } β π£ = 0 )) |
12 | 2, 11 | sylan 578 | . . . . . . . 8 β’ ((π β§ π£ β (Baseβπ)) β (((LSpanβπ)β{π£}) = { 0 } β π£ = 0 )) |
13 | 12 | biimpd 228 | . . . . . . 7 β’ ((π β§ π£ β (Baseβπ)) β (((LSpanβπ)β{π£}) = { 0 } β π£ = 0 )) |
14 | 13 | necon3d 2957 | . . . . . 6 β’ ((π β§ π£ β (Baseβπ)) β (π£ β 0 β ((LSpanβπ)β{π£}) β { 0 })) |
15 | 14 | expimpd 452 | . . . . 5 β’ (π β ((π£ β (Baseβπ) β§ π£ β 0 ) β ((LSpanβπ)β{π£}) β { 0 })) |
16 | 10, 15 | biimtrid 241 | . . . 4 β’ (π β (π£ β ((Baseβπ) β { 0 }) β ((LSpanβπ)β{π£}) β { 0 })) |
17 | neeq1 2999 | . . . . 5 β’ (π = ((LSpanβπ)β{π£}) β (π β { 0 } β ((LSpanβπ)β{π£}) β { 0 })) | |
18 | 17 | biimprcd 249 | . . . 4 β’ (((LSpanβπ)β{π£}) β { 0 } β (π = ((LSpanβπ)β{π£}) β π β { 0 })) |
19 | 16, 18 | syl6 35 | . . 3 β’ (π β (π£ β ((Baseβπ) β { 0 }) β (π = ((LSpanβπ)β{π£}) β π β { 0 }))) |
20 | 19 | rexlimdv 3149 | . 2 β’ (π β (βπ£ β ((Baseβπ) β { 0 })π = ((LSpanβπ)β{π£}) β π β { 0 })) |
21 | 9, 20 | mpd 15 | 1 β’ (π β π β { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2936 βwrex 3066 β cdif 3944 {csn 4630 βcfv 6551 Basecbs 17185 0gc0g 17426 LModclmod 20748 LSpanclspn 20860 LSAtomsclsa 38450 |
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 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
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 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 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-pss 3966 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-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 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-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 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-oprab 7428 df-mpo 7429 df-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-plusg 17251 df-0g 17428 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-grp 18898 df-minusg 18899 df-cmn 19742 df-abl 19743 df-mgp 20080 df-rng 20098 df-ur 20127 df-ring 20180 df-lmod 20750 df-lss 20821 df-lsp 20861 df-lsatoms 38452 |
This theorem is referenced by: lsatspn0 38476 lsatssn0 38478 lsatcmp 38479 lsatcv0 38507 dochsat 40860 dochsatshp 40928 dochshpsat 40931 dochexmidlem1 40937 |
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