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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 2821 | . . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2821 | . . . . . 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 36142 | . . . . 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 234 | . . 3 ⊢ (𝜑 → ∃𝑣 ∈ ((Base‘𝑊) ∖ { 0 })𝑈 = ((LSpan‘𝑊)‘{𝑣})) |
10 | eldifi 4103 | . . . . . 6 ⊢ (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) → 𝑣 ∈ (Base‘𝑊)) | |
11 | 3, 4 | lspsnid 19765 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑊)) → 𝑣 ∈ ((LSpan‘𝑊)‘{𝑣})) |
12 | 2, 10, 11 | syl2an 597 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Base‘𝑊) ∖ { 0 })) → 𝑣 ∈ ((LSpan‘𝑊)‘{𝑣})) |
13 | eleq2 2901 | . . . . 5 ⊢ (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → (𝑣 ∈ 𝑈 ↔ 𝑣 ∈ ((LSpan‘𝑊)‘{𝑣}))) | |
14 | 12, 13 | syl5ibrcom 249 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ ((Base‘𝑊) ∖ { 0 })) → (𝑈 = ((LSpan‘𝑊)‘{𝑣}) → 𝑣 ∈ 𝑈)) |
15 | 14 | reximdva 3274 | . . 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 4719 | . . . . . . 7 ⊢ (𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ↔ (𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 )) | |
18 | 17 | anbi1i 625 | . . . . . 6 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) ↔ ((𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 ) ∧ 𝑣 ∈ 𝑈)) |
19 | anass 471 | . . . . . 6 ⊢ (((𝑣 ∈ (Base‘𝑊) ∧ 𝑣 ≠ 0 ) ∧ 𝑣 ∈ 𝑈) ↔ (𝑣 ∈ (Base‘𝑊) ∧ (𝑣 ≠ 0 ∧ 𝑣 ∈ 𝑈))) | |
20 | 18, 19 | bitri 277 | . . . . 5 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) ↔ (𝑣 ∈ (Base‘𝑊) ∧ (𝑣 ≠ 0 ∧ 𝑣 ∈ 𝑈))) |
21 | 20 | simprbi 499 | . . . 4 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) → (𝑣 ≠ 0 ∧ 𝑣 ∈ 𝑈)) |
22 | 21 | ancomd 464 | . . 3 ⊢ ((𝑣 ∈ ((Base‘𝑊) ∖ { 0 }) ∧ 𝑣 ∈ 𝑈) → (𝑣 ∈ 𝑈 ∧ 𝑣 ≠ 0 )) |
23 | 22 | reximi2 3244 | . 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 ∖ cdif 3933 {csn 4567 ‘cfv 6355 Basecbs 16483 0gc0g 16713 LModclmod 19634 LSpanclspn 19743 LSAtomsclsa 36125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lsatoms 36127 |
This theorem is referenced by: dvh1dim 38593 dochkr1 38629 dochkr1OLDN 38630 lcfrlem40 38733 |
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