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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 32281 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 2726 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2726 | . . . . 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 38691 | . . . 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 231 | . 2 ⊢ (𝜑 → ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣})) |
10 | eldifsn 4795 | . . . . 5 ⊢ (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ↔ (𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 )) | |
11 | 3, 5, 4 | lspsneq0 20991 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑊)) → (((LSpan‘𝑊)‘{𝑣}) = { 0 } ↔ 𝑣 = 0 )) |
12 | 2, 11 | sylan 578 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → (((LSpan‘𝑊)‘{𝑣}) = { 0 } ↔ 𝑣 = 0 )) |
13 | 12 | biimpd 228 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → (((LSpan‘𝑊)‘{𝑣}) = { 0 } → 𝑣 = 0 )) |
14 | 13 | necon3d 2951 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑣 ≠ 0 → ((LSpan‘𝑊)‘{𝑣}) ≠ { 0 })) |
15 | 14 | expimpd 452 | . . . . 5 ⊢ (𝜑 → ((𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 ) → ((LSpan‘𝑊)‘{𝑣}) ≠ { 0 })) |
16 | 10, 15 | biimtrid 241 | . . . 4 ⊢ (𝜑 → (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) → ((LSpan‘𝑊)‘{𝑣}) ≠ { 0 })) |
17 | neeq1 2993 | . . . . 5 ⊢ (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → (𝑈 ≠ { 0 } ↔ ((LSpan‘𝑊)‘{𝑣}) ≠ { 0 })) | |
18 | 17 | biimprcd 249 | . . . 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 3143 | . 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 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∃wrex 3060 ∖ cdif 3944 {csn 4633 ‘cfv 6556 Basecbs 17215 0gc0g 17456 LModclmod 20838 LSpanclspn 20950 LSAtomsclsa 38674 |
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-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
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 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-int 4957 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-2 12329 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17216 df-plusg 17281 df-0g 17458 df-mgm 18635 df-sgrp 18714 df-mnd 18730 df-grp 18933 df-minusg 18934 df-cmn 19782 df-abl 19783 df-mgp 20120 df-rng 20138 df-ur 20167 df-ring 20220 df-lmod 20840 df-lss 20911 df-lsp 20951 df-lsatoms 38676 |
This theorem is referenced by: lsatspn0 38700 lsatssn0 38702 lsatcmp 38703 lsatcv0 38731 dochsat 41084 dochsatshp 41152 dochshpsat 41155 dochexmidlem1 41161 |
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