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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 | ⊢ 𝑉 = (Base‘𝑊) |
lspsneq0.z | ⊢ 0 = (0g‘𝑊) |
lspsneq0.n | ⊢ 𝑁 = (LSpan‘𝑊) |
Ref | Expression |
---|---|
lspsneq0 | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspsneq0.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lspsneq0.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
3 | 1, 2 | lspsnid 19499 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) |
4 | eleq2 2856 | . . . 4 ⊢ ((𝑁‘{𝑋}) = { 0 } → (𝑋 ∈ (𝑁‘{𝑋}) ↔ 𝑋 ∈ { 0 })) | |
5 | 3, 4 | syl5ibcom 237 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑁‘{𝑋}) = { 0 } → 𝑋 ∈ { 0 })) |
6 | elsni 4461 | . . 3 ⊢ (𝑋 ∈ { 0 } → 𝑋 = 0 ) | |
7 | 5, 6 | syl6 35 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑁‘{𝑋}) = { 0 } → 𝑋 = 0 )) |
8 | lspsneq0.z | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
9 | 8, 2 | lspsn0 19514 | . . . 4 ⊢ (𝑊 ∈ LMod → (𝑁‘{ 0 }) = { 0 }) |
10 | 9 | adantr 473 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{ 0 }) = { 0 }) |
11 | sneq 4454 | . . . 4 ⊢ (𝑋 = 0 → {𝑋} = { 0 }) | |
12 | 11 | fveqeq2d 6512 | . . 3 ⊢ (𝑋 = 0 → ((𝑁‘{𝑋}) = { 0 } ↔ (𝑁‘{ 0 }) = { 0 })) |
13 | 10, 12 | syl5ibrcom 239 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋 = 0 → (𝑁‘{𝑋}) = { 0 })) |
14 | 7, 13 | impbid 204 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 {csn 4444 ‘cfv 6193 Basecbs 16345 0gc0g 16575 LModclmod 19368 LSpanclspn 19477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-int 4755 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-om 7403 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-er 8095 df-en 8313 df-dom 8314 df-sdom 8315 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-nn 11446 df-2 11509 df-ndx 16348 df-slot 16349 df-base 16351 df-sets 16352 df-plusg 16440 df-0g 16577 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-grp 17906 df-mgp 18975 df-ring 19034 df-lmod 19370 df-lss 19438 df-lsp 19478 |
This theorem is referenced by: lspsneq0b 19519 lsatn0 35620 lsator0sp 35622 lsat0cv 35654 dih0vbN 37903 dihlspsnat 37954 mapdn0 38290 mapdindp1 38341 hdmapeq0 38465 |
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