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| Mirrors > Home > MPE Home > Th. List > lssne0 | Structured version Visualization version GIF version | ||
| Description: A nonzero subspace has a nonzero vector. (shne0i 31467 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.) |
| Ref | Expression |
|---|---|
| lss0cl.z | ⊢ 0 = (0g‘𝑊) |
| lss0cl.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
| Ref | Expression |
|---|---|
| lssne0 | ⊢ (𝑋 ∈ 𝑆 → (𝑋 ≠ { 0 } ↔ ∃𝑦 ∈ 𝑋 𝑦 ≠ 0 )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lss0cl.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 2 | 1 | lssn0 20938 | . . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ≠ ∅) |
| 3 | eqsn 4829 | . . . 4 ⊢ (𝑋 ≠ ∅ → (𝑋 = { 0 } ↔ ∀𝑦 ∈ 𝑋 𝑦 = 0 )) | |
| 4 | 2, 3 | syl 17 | . . 3 ⊢ (𝑋 ∈ 𝑆 → (𝑋 = { 0 } ↔ ∀𝑦 ∈ 𝑋 𝑦 = 0 )) |
| 5 | nne 2944 | . . . . 5 ⊢ (¬ 𝑦 ≠ 0 ↔ 𝑦 = 0 ) | |
| 6 | 5 | ralbii 3093 | . . . 4 ⊢ (∀𝑦 ∈ 𝑋 ¬ 𝑦 ≠ 0 ↔ ∀𝑦 ∈ 𝑋 𝑦 = 0 ) |
| 7 | ralnex 3072 | . . . 4 ⊢ (∀𝑦 ∈ 𝑋 ¬ 𝑦 ≠ 0 ↔ ¬ ∃𝑦 ∈ 𝑋 𝑦 ≠ 0 ) | |
| 8 | 6, 7 | bitr3i 277 | . . 3 ⊢ (∀𝑦 ∈ 𝑋 𝑦 = 0 ↔ ¬ ∃𝑦 ∈ 𝑋 𝑦 ≠ 0 ) |
| 9 | 4, 8 | bitr2di 288 | . 2 ⊢ (𝑋 ∈ 𝑆 → (¬ ∃𝑦 ∈ 𝑋 𝑦 ≠ 0 ↔ 𝑋 = { 0 })) |
| 10 | 9 | necon1abid 2979 | 1 ⊢ (𝑋 ∈ 𝑆 → (𝑋 ≠ { 0 } ↔ ∃𝑦 ∈ 𝑋 𝑦 ≠ 0 )) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∅c0 4333 {csn 4626 ‘cfv 6561 0gc0g 17484 LSubSpclss 20929 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-lss 20930 |
| This theorem is referenced by: lsmsat 39009 lssatomic 39012 dochsatshpb 41454 hgmapvvlem3 41927 |
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