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Mirrors > Home > MPE Home > Th. List > lssneln0 | Structured version Visualization version GIF version |
Description: A vector 𝑋 which doesn't belong to a subspace 𝑈 is nonzero. (Contributed by NM, 14-May-2015.) (Revised by AV, 17-Jul-2022.) (Proof shortened by AV, 19-Jul-2022.) |
Ref | Expression |
---|---|
lssneln0.o | ⊢ 0 = (0g‘𝑊) |
lssneln0.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lssneln0.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lssneln0.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lssneln0.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lssneln0.n | ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) |
Ref | Expression |
---|---|
lssneln0 | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lssneln0.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
2 | lssneln0.o | . . 3 ⊢ 0 = (0g‘𝑊) | |
3 | lssneln0.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
4 | lssneln0.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
5 | lssneln0.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
6 | lssneln0.n | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) | |
7 | 2, 3, 4, 5, 6 | lssvneln0 20923 | . 2 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
8 | eldifsn 4786 | . 2 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) | |
9 | 1, 7, 8 | sylanbrc 581 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∖ cdif 3944 {csn 4624 ‘cfv 6544 0gc0g 17447 LModclmod 20830 LSubSpclss 20902 |
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 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-2 12319 df-sets 17159 df-slot 17177 df-ndx 17189 df-base 17207 df-plusg 17272 df-0g 17449 df-mgm 18626 df-sgrp 18705 df-mnd 18721 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mgp 20112 df-ur 20159 df-ring 20212 df-lmod 20832 df-lss 20903 |
This theorem is referenced by: lspexchn1 21105 baerlem5amN 41426 baerlem5bmN 41427 baerlem5abmN 41428 mapdh6iN 41454 hdmaplem3 41483 mapdh8ad 41489 mapdh8e 41494 mapdh9a 41499 mapdh9aOLDN 41500 hdmap1l6i 41528 hdmap1eulem 41532 hdmap1eulemOLDN 41533 hdmapval3lemN 41547 hdmap10lem 41549 hdmap11lem1 41551 hdmaprnlem3N 41560 hdmap14lem11 41588 |
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