Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lssne0 | Structured version Visualization version GIF version |
Description: A nonzero subspace has a nonzero vector. (shne0i 29224 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 19711 | . . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ≠ ∅) |
3 | eqsn 4761 | . . . 4 ⊢ (𝑋 ≠ ∅ → (𝑋 = { 0 } ↔ ∀𝑦 ∈ 𝑋 𝑦 = 0 )) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝑋 ∈ 𝑆 → (𝑋 = { 0 } ↔ ∀𝑦 ∈ 𝑋 𝑦 = 0 )) |
5 | nne 3020 | . . . . 5 ⊢ (¬ 𝑦 ≠ 0 ↔ 𝑦 = 0 ) | |
6 | 5 | ralbii 3165 | . . . 4 ⊢ (∀𝑦 ∈ 𝑋 ¬ 𝑦 ≠ 0 ↔ ∀𝑦 ∈ 𝑋 𝑦 = 0 ) |
7 | ralnex 3236 | . . . 4 ⊢ (∀𝑦 ∈ 𝑋 ¬ 𝑦 ≠ 0 ↔ ¬ ∃𝑦 ∈ 𝑋 𝑦 ≠ 0 ) | |
8 | 6, 7 | bitr3i 279 | . . 3 ⊢ (∀𝑦 ∈ 𝑋 𝑦 = 0 ↔ ¬ ∃𝑦 ∈ 𝑋 𝑦 ≠ 0 ) |
9 | 4, 8 | syl6rbb 290 | . 2 ⊢ (𝑋 ∈ 𝑆 → (¬ ∃𝑦 ∈ 𝑋 𝑦 ≠ 0 ↔ 𝑋 = { 0 })) |
10 | 9 | necon1abid 3054 | 1 ⊢ (𝑋 ∈ 𝑆 → (𝑋 ≠ { 0 } ↔ ∃𝑦 ∈ 𝑋 𝑦 ≠ 0 )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ∅c0 4290 {csn 4566 ‘cfv 6354 0gc0g 16712 LSubSpclss 19702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-ov 7158 df-lss 19703 |
This theorem is referenced by: lsmsat 36143 lssatomic 36146 dochsatshpb 38587 hgmapvvlem3 39060 |
Copyright terms: Public domain | W3C validator |