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| Mirrors > Home > MPE Home > Th. List > lss0v | Structured version Visualization version GIF version | ||
| Description: The zero vector in a submodule equals the zero vector in the including module. (Contributed by NM, 15-Mar-2015.) |
| Ref | Expression |
|---|---|
| lss0v.x | β’ π = (π βΎs π) |
| lss0v.o | β’ 0 = (0gβπ) |
| lss0v.z | β’ π = (0gβπ) |
| lss0v.l | β’ πΏ = (LSubSpβπ) |
| Ref | Expression |
|---|---|
| lss0v | β’ ((π β LMod β§ π β πΏ) β π = 0 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0ss 4397 | . . . . 5 β’ β β π | |
| 2 | lss0v.x | . . . . . 6 β’ π = (π βΎs π) | |
| 3 | eqid 2728 | . . . . . 6 β’ (LSpanβπ) = (LSpanβπ) | |
| 4 | eqid 2728 | . . . . . 6 β’ (LSpanβπ) = (LSpanβπ) | |
| 5 | lss0v.l | . . . . . 6 β’ πΏ = (LSubSpβπ) | |
| 6 | 2, 3, 4, 5 | lsslsp 20898 | . . . . 5 β’ ((π β LMod β§ π β πΏ β§ β β π) β ((LSpanβπ)ββ ) = ((LSpanβπ)ββ )) |
| 7 | 1, 6 | mp3an3 1447 | . . . 4 β’ ((π β LMod β§ π β πΏ) β ((LSpanβπ)ββ ) = ((LSpanβπ)ββ )) |
| 8 | 2, 5 | lsslmod 20843 | . . . . 5 β’ ((π β LMod β§ π β πΏ) β π β LMod) |
| 9 | lss0v.z | . . . . . 6 β’ π = (0gβπ) | |
| 10 | 9, 4 | lsp0 20892 | . . . . 5 β’ (π β LMod β ((LSpanβπ)ββ ) = {π}) |
| 11 | 8, 10 | syl 17 | . . . 4 β’ ((π β LMod β§ π β πΏ) β ((LSpanβπ)ββ ) = {π}) |
| 12 | lss0v.o | . . . . . 6 β’ 0 = (0gβπ) | |
| 13 | 12, 3 | lsp0 20892 | . . . . 5 β’ (π β LMod β ((LSpanβπ)ββ ) = { 0 }) |
| 14 | 13 | adantr 480 | . . . 4 β’ ((π β LMod β§ π β πΏ) β ((LSpanβπ)ββ ) = { 0 }) |
| 15 | 7, 11, 14 | 3eqtr3d 2776 | . . 3 β’ ((π β LMod β§ π β πΏ) β {π} = { 0 }) |
| 16 | 15 | unieqd 4921 | . 2 β’ ((π β LMod β§ π β πΏ) β βͺ {π} = βͺ { 0 }) |
| 17 | 9 | fvexi 6911 | . . 3 β’ π β V |
| 18 | 17 | unisn 4929 | . 2 β’ βͺ {π} = π |
| 19 | 12 | fvexi 6911 | . . 3 β’ 0 β V |
| 20 | 19 | unisn 4929 | . 2 β’ βͺ { 0 } = 0 |
| 21 | 16, 18, 20 | 3eqtr3g 2791 | 1 β’ ((π β LMod β§ π β πΏ) β π = 0 ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wss 3947 β c0 4323 {csn 4629 βͺ cuni 4908 βcfv 6548 (class class class)co 7420 βΎs cress 17208 0gc0g 17420 LModclmod 20742 LSubSpclss 20814 LSpanclspn 20854 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 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 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-sca 17248 df-vsca 17249 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-lmod 20744 df-lss 20815 df-lsp 20855 |
| This theorem is referenced by: phlssphl 21590 lcd0v 41084 |
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