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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lsateln0 | Structured version Visualization version GIF version | ||
| Description: A 1-dim subspace (atom) (of a left module or left vector space) contains a nonzero vector. (Contributed by NM, 2-Jan-2015.) | 
| Ref | Expression | 
|---|---|
| lsateln0.z | ⊢ 0 = (0g‘𝑊) | 
| lsateln0.a | ⊢ 𝐴 = (LSAtoms‘𝑊) | 
| lsateln0.w | ⊢ (𝜑 → 𝑊 ∈ LMod) | 
| lsateln0.u | ⊢ (𝜑 → 𝑈 ∈ 𝐴) | 
| Ref | Expression | 
|---|---|
| lsateln0 | ⊢ (𝜑 → ∃𝑣 ∈ 𝑈 𝑣 ≠ 0 ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | lsateln0.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐴) | |
| 2 | lsateln0.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
| 3 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 4 | eqid 2736 | . . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
| 5 | lsateln0.z | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
| 6 | lsateln0.a | . . . . . 6 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
| 7 | 3, 4, 5, 6 | islsat 38993 | . . . . 5 ⊢ (𝑊 ∈ LMod → (𝑈 ∈ 𝐴 ↔ ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣}))) | 
| 8 | 2, 7 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ↔ ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣}))) | 
| 9 | 1, 8 | mpbid 232 | . . 3 ⊢ (𝜑 → ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣})) | 
| 10 | eldifi 4130 | . . . . . 6 ⊢ (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) → 𝑣 ∈ (Base‘𝑊)) | |
| 11 | 3, 4 | lspsnid 20992 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑣 ∈ ((LSpan‘𝑊)‘{𝑣})) | 
| 12 | 2, 10, 11 | syl2an 596 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑣 ∈ ((LSpan‘𝑊)‘{𝑣})) | 
| 13 | eleq2 2829 | . . . . 5 ⊢ (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → (𝑣 ∈ 𝑈 ↔ 𝑣 ∈ ((LSpan‘𝑊)‘{𝑣}))) | |
| 14 | 12, 13 | syl5ibrcom 247 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Base‘𝑊) ∖ { 0 })) → (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → 𝑣 ∈ 𝑈)) | 
| 15 | 14 | reximdva 3167 | . . 3 ⊢ (𝜑 → (∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣}) → ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑣 ∈ 𝑈)) | 
| 16 | 9, 15 | mpd 15 | . 2 ⊢ (𝜑 → ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑣 ∈ 𝑈) | 
| 17 | eldifsn 4785 | . . . . . . 7 ⊢ (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ↔ (𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 )) | |
| 18 | 17 | anbi1i 624 | . . . . . 6 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) ↔ ((𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 ) ∧ 𝑣 ∈ 𝑈)) | 
| 19 | anass 468 | . . . . . 6 ⊢ (((𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 ) ∧ 𝑣 ∈ 𝑈) ↔ (𝑣 ∈ (Base‘𝑊) ∧ (𝑣 ≠ 0 ∧ 𝑣 ∈ 𝑈))) | |
| 20 | 18, 19 | bitri 275 | . . . . 5 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) ↔ (𝑣 ∈ (Base‘𝑊) ∧ (𝑣 ≠ 0 ∧ 𝑣 ∈ 𝑈))) | 
| 21 | 20 | simprbi 496 | . . . 4 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) → (𝑣 ≠ 0 ∧ 𝑣 ∈ 𝑈)) | 
| 22 | 21 | ancomd 461 | . . 3 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) → (𝑣 ∈ 𝑈 ∧ 𝑣 ≠ 0 )) | 
| 23 | 22 | reximi2 3078 | . 2 ⊢ (∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑣 ∈ 𝑈 → ∃𝑣 ∈ 𝑈 𝑣 ≠ 0 ) | 
| 24 | 16, 23 | syl 17 | 1 ⊢ (𝜑 → ∃𝑣 ∈ 𝑈 𝑣 ≠ 0 ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∃wrex 3069 ∖ cdif 3947 {csn 4625 ‘cfv 6560 Basecbs 17248 0gc0g 17485 LModclmod 20859 LSpanclspn 20970 LSAtomsclsa 38976 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 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-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-lmod 20861 df-lss 20931 df-lsp 20971 df-lsatoms 38978 | 
| This theorem is referenced by: dvh1dim 41445 dochkr1 41481 dochkr1OLDN 41482 lcfrlem40 41585 | 
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