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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lsatn0 | Structured version Visualization version GIF version |
Description: A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 29780 analog.) (Contributed by NM, 25-Aug-2014.) |
Ref | Expression |
---|---|
lsatn0.o | ⊢ 0 = (0g‘𝑊) |
lsatn0.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lsatn0.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lsatn0.u | ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
Ref | Expression |
---|---|
lsatn0 | ⊢ (𝜑 → 𝑈 ≠ { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsatn0.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐴) | |
2 | lsatn0.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | eqid 2778 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2778 | . . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
5 | lsatn0.o | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
6 | lsatn0.a | . . . . 5 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
7 | 3, 4, 5, 6 | islsat 35150 | . . . 4 ⊢ (𝑊 ∈ LMod → (𝑈 ∈ 𝐴 ↔ ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣}))) |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ↔ ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣}))) |
9 | 1, 8 | mpbid 224 | . 2 ⊢ (𝜑 → ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣})) |
10 | eldifsn 4550 | . . . . 5 ⊢ (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ↔ (𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 )) | |
11 | 3, 5, 4 | lspsneq0 19411 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑊)) → (((LSpan‘𝑊)‘{𝑣}) = { 0 } ↔ 𝑣 = 0 )) |
12 | 2, 11 | sylan 575 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → (((LSpan‘𝑊)‘{𝑣}) = { 0 } ↔ 𝑣 = 0 )) |
13 | 12 | biimpd 221 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → (((LSpan‘𝑊)‘{𝑣}) = { 0 } → 𝑣 = 0 )) |
14 | 13 | necon3d 2990 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑣 ≠ 0 → ((LSpan‘𝑊)‘{𝑣}) ≠ { 0 })) |
15 | 14 | expimpd 447 | . . . . 5 ⊢ (𝜑 → ((𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑣}) ≠ { 0 })) |
16 | 10, 15 | syl5bi 234 | . . . 4 ⊢ (𝜑 → (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) → ((LSpan‘𝑊)‘{𝑣}) ≠ { 0 })) |
17 | neeq1 3031 | . . . . 5 ⊢ (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → (𝑈 ≠ { 0 } ↔ ((LSpan‘𝑊)‘{𝑣}) ≠ { 0 })) | |
18 | 17 | biimprcd 242 | . . . 4 ⊢ (((LSpan‘𝑊)‘{𝑣}) ≠ { 0 } → (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → 𝑈 ≠ { 0 })) |
19 | 16, 18 | syl6 35 | . . 3 ⊢ (𝜑 → (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) → (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → 𝑈 ≠ { 0 }))) |
20 | 19 | rexlimdv 3212 | . 2 ⊢ (𝜑 → (∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣}) → 𝑈 ≠ { 0 })) |
21 | 9, 20 | mpd 15 | 1 ⊢ (𝜑 → 𝑈 ≠ { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∃wrex 3091 ∖ cdif 3789 {csn 4398 ‘cfv 6137 Basecbs 16259 0gc0g 16490 LModclmod 19259 LSpanclspn 19370 LSAtomsclsa 35133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-2 11442 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-plusg 16355 df-0g 16492 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-grp 17816 df-mgp 18881 df-ring 18940 df-lmod 19261 df-lss 19329 df-lsp 19371 df-lsatoms 35135 |
This theorem is referenced by: lsatspn0 35159 lsatssn0 35161 lsatcmp 35162 lsatcv0 35190 dochsat 37542 dochsatshp 37610 dochshpsat 37613 dochexmidlem1 37619 |
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