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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 4389 | . . . . 5 β’ β β π | |
| 2 | lss0v.x | . . . . . 6 β’ π = (π βΎs π) | |
| 3 | eqid 2724 | . . . . . 6 β’ (LSpanβπ) = (LSpanβπ) | |
| 4 | eqid 2724 | . . . . . 6 β’ (LSpanβπ) = (LSpanβπ) | |
| 5 | lss0v.l | . . . . . 6 β’ πΏ = (LSubSpβπ) | |
| 6 | 2, 3, 4, 5 | lsslsp 20858 | . . . . 5 β’ ((π β LMod β§ π β πΏ β§ β β π) β ((LSpanβπ)ββ ) = ((LSpanβπ)ββ )) |
| 7 | 1, 6 | mp3an3 1446 | . . . 4 β’ ((π β LMod β§ π β πΏ) β ((LSpanβπ)ββ ) = ((LSpanβπ)ββ )) |
| 8 | 2, 5 | lsslmod 20803 | . . . . 5 β’ ((π β LMod β§ π β πΏ) β π β LMod) |
| 9 | lss0v.z | . . . . . 6 β’ π = (0gβπ) | |
| 10 | 9, 4 | lsp0 20852 | . . . . 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 20852 | . . . . 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 2772 | . . 3 β’ ((π β LMod β§ π β πΏ) β {π} = { 0 }) |
| 16 | 15 | unieqd 4913 | . 2 β’ ((π β LMod β§ π β πΏ) β βͺ {π} = βͺ { 0 }) |
| 17 | 9 | fvexi 6896 | . . 3 β’ π β V |
| 18 | 17 | unisn 4921 | . 2 β’ βͺ {π} = π |
| 19 | 12 | fvexi 6896 | . . 3 β’ 0 β V |
| 20 | 19 | unisn 4921 | . 2 β’ βͺ { 0 } = 0 |
| 21 | 16, 18, 20 | 3eqtr3g 2787 | 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 1533 β wcel 2098 β wss 3941 β c0 4315 {csn 4621 βͺ cuni 4900 βcfv 6534 (class class class)co 7402 βΎs cress 17178 0gc0g 17390 LModclmod 20702 LSubSpclss 20774 LSpanclspn 20814 |
| 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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-sca 17218 df-vsca 17219 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-minusg 18863 df-sbg 18864 df-subg 19046 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-lmod 20704 df-lss 20775 df-lsp 20815 |
| This theorem is referenced by: phlssphl 21541 lcd0v 40985 |
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