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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 2760 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2760 | . . . . . 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 34799 | . . . . 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 222 | . . 3 ⊢ (𝜑 → ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣})) |
10 | eldifi 3875 | . . . . . 6 ⊢ (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) → 𝑣 ∈ (Base‘𝑊)) | |
11 | 3, 4 | lspsnid 19215 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑣 ∈ ((LSpan‘𝑊)‘{𝑣})) |
12 | 2, 10, 11 | syl2an 495 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑣 ∈ ((LSpan‘𝑊)‘{𝑣})) |
13 | eleq2 2828 | . . . . 5 ⊢ (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → (𝑣 ∈ 𝑈 ↔ 𝑣 ∈ ((LSpan‘𝑊)‘{𝑣}))) | |
14 | 12, 13 | syl5ibrcom 237 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Base‘𝑊) ∖ { 0 })) → (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → 𝑣 ∈ 𝑈)) |
15 | 14 | reximdva 3155 | . . 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 4462 | . . . . . . 7 ⊢ (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ↔ (𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 )) | |
18 | 17 | anbi1i 733 | . . . . . 6 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) ↔ ((𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 ) ∧ 𝑣 ∈ 𝑈)) |
19 | anass 684 | . . . . . 6 ⊢ (((𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 ) ∧ 𝑣 ∈ 𝑈) ↔ (𝑣 ∈ (Base‘𝑊) ∧ (𝑣 ≠ 0 ∧ 𝑣 ∈ 𝑈))) | |
20 | 18, 19 | bitri 264 | . . . . 5 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) ↔ (𝑣 ∈ (Base‘𝑊) ∧ (𝑣 ≠ 0 ∧ 𝑣 ∈ 𝑈))) |
21 | 20 | simprbi 483 | . . . 4 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) → (𝑣 ≠ 0 ∧ 𝑣 ∈ 𝑈)) |
22 | 21 | ancomd 466 | . . 3 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) → (𝑣 ∈ 𝑈 ∧ 𝑣 ≠ 0 )) |
23 | 22 | reximi2 3148 | . 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 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∃wrex 3051 ∖ cdif 3712 {csn 4321 ‘cfv 6049 Basecbs 16079 0gc0g 16322 LModclmod 19085 LSpanclspn 19193 LSAtomsclsa 34782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-0g 16324 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-grp 17646 df-lmod 19087 df-lss 19155 df-lsp 19194 df-lsatoms 34784 |
This theorem is referenced by: dvh1dim 37251 dochkr1 37287 dochkr1OLDN 37288 lcfrlem40 37391 |
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