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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatcveq0 | Structured version Visualization version GIF version |
Description: A subspace covered by an atom must be the zero subspace. (atcveq0 29724 analog.) (Contributed by NM, 7-Jan-2015.) |
Ref | Expression |
---|---|
lsatcveq0.o | ⊢ 0 = (0g‘𝑊) |
lsatcveq0.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lsatcveq0.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatcveq0.c | ⊢ 𝐶 = ( ⋖L ‘𝑊) |
lsatcveq0.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lsatcveq0.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lsatcveq0.q | ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
Ref | Expression |
---|---|
lsatcveq0 | ⊢ (𝜑 → (𝑈𝐶𝑄 ↔ 𝑈 = { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatcveq0.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
2 | lsatcveq0.c | . . . . 5 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
3 | lsatcveq0.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
4 | 3 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑈𝐶𝑄) → 𝑊 ∈ LVec) |
5 | lsatcveq0.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
6 | 5 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑈𝐶𝑄) → 𝑈 ∈ 𝑆) |
7 | lsatcveq0.a | . . . . . . 7 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
8 | lveclmod 19424 | . . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
9 | 3, 8 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod) |
10 | lsatcveq0.q | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ 𝐴) | |
11 | 1, 7, 9, 10 | lsatlssel 35010 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
12 | 11 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑈𝐶𝑄) → 𝑄 ∈ 𝑆) |
13 | simpr 478 | . . . . 5 ⊢ ((𝜑 ∧ 𝑈𝐶𝑄) → 𝑈𝐶𝑄) | |
14 | 1, 2, 4, 6, 12, 13 | lcvpss 35037 | . . . 4 ⊢ ((𝜑 ∧ 𝑈𝐶𝑄) → 𝑈 ⊊ 𝑄) |
15 | 14 | ex 402 | . . 3 ⊢ (𝜑 → (𝑈𝐶𝑄 → 𝑈 ⊊ 𝑄)) |
16 | lsatcveq0.o | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
17 | 16, 7, 2, 3, 10 | lsatcv0 35044 | . . . 4 ⊢ (𝜑 → { 0 }𝐶𝑄) |
18 | 3 | 3ad2ant1 1164 | . . . . . 6 ⊢ ((𝜑 ∧ { 0 }𝐶𝑄 ∧ 𝑈 ⊊ 𝑄) → 𝑊 ∈ LVec) |
19 | 16, 1 | lsssn0 19263 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → { 0 } ∈ 𝑆) |
20 | 9, 19 | syl 17 | . . . . . . 7 ⊢ (𝜑 → { 0 } ∈ 𝑆) |
21 | 20 | 3ad2ant1 1164 | . . . . . 6 ⊢ ((𝜑 ∧ { 0 }𝐶𝑄 ∧ 𝑈 ⊊ 𝑄) → { 0 } ∈ 𝑆) |
22 | 11 | 3ad2ant1 1164 | . . . . . 6 ⊢ ((𝜑 ∧ { 0 }𝐶𝑄 ∧ 𝑈 ⊊ 𝑄) → 𝑄 ∈ 𝑆) |
23 | 5 | 3ad2ant1 1164 | . . . . . 6 ⊢ ((𝜑 ∧ { 0 }𝐶𝑄 ∧ 𝑈 ⊊ 𝑄) → 𝑈 ∈ 𝑆) |
24 | simp2 1168 | . . . . . 6 ⊢ ((𝜑 ∧ { 0 }𝐶𝑄 ∧ 𝑈 ⊊ 𝑄) → { 0 }𝐶𝑄) | |
25 | 16, 1 | lss0ss 19264 | . . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → { 0 } ⊆ 𝑈) |
26 | 9, 5, 25 | syl2anc 580 | . . . . . . 7 ⊢ (𝜑 → { 0 } ⊆ 𝑈) |
27 | 26 | 3ad2ant1 1164 | . . . . . 6 ⊢ ((𝜑 ∧ { 0 }𝐶𝑄 ∧ 𝑈 ⊊ 𝑄) → { 0 } ⊆ 𝑈) |
28 | simp3 1169 | . . . . . 6 ⊢ ((𝜑 ∧ { 0 }𝐶𝑄 ∧ 𝑈 ⊊ 𝑄) → 𝑈 ⊊ 𝑄) | |
29 | 1, 2, 18, 21, 22, 23, 24, 27, 28 | lcvnbtwn3 35041 | . . . . 5 ⊢ ((𝜑 ∧ { 0 }𝐶𝑄 ∧ 𝑈 ⊊ 𝑄) → 𝑈 = { 0 }) |
30 | 29 | 3exp 1149 | . . . 4 ⊢ (𝜑 → ({ 0 }𝐶𝑄 → (𝑈 ⊊ 𝑄 → 𝑈 = { 0 }))) |
31 | 17, 30 | mpd 15 | . . 3 ⊢ (𝜑 → (𝑈 ⊊ 𝑄 → 𝑈 = { 0 })) |
32 | 15, 31 | syld 47 | . 2 ⊢ (𝜑 → (𝑈𝐶𝑄 → 𝑈 = { 0 })) |
33 | breq1 4844 | . . 3 ⊢ (𝑈 = { 0 } → (𝑈𝐶𝑄 ↔ { 0 }𝐶𝑄)) | |
34 | 17, 33 | syl5ibrcom 239 | . 2 ⊢ (𝜑 → (𝑈 = { 0 } → 𝑈𝐶𝑄)) |
35 | 32, 34 | impbid 204 | 1 ⊢ (𝜑 → (𝑈𝐶𝑄 ↔ 𝑈 = { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ⊆ wss 3767 ⊊ wpss 3768 {csn 4366 class class class wbr 4841 ‘cfv 6099 0gc0g 16412 LModclmod 19178 LSubSpclss 19247 LVecclvec 19420 LSAtomsclsa 34987 ⋖L clcv 35031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-tpos 7588 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-2 11372 df-3 11373 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-0g 16414 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-grp 17738 df-minusg 17739 df-sbg 17740 df-cmn 18507 df-abl 18508 df-mgp 18803 df-ur 18815 df-ring 18862 df-oppr 18936 df-dvdsr 18954 df-unit 18955 df-invr 18985 df-drng 19064 df-lmod 19180 df-lss 19248 df-lsp 19290 df-lvec 19421 df-lsatoms 34989 df-lcv 35032 |
This theorem is referenced by: lcvp 35053 lsatcv1 35061 |
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