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Mirrors > Home > MPE Home > Th. List > lssn0 | Structured version Visualization version GIF version |
Description: A subspace is not empty. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.) |
Ref | Expression |
---|---|
lssn0.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
Ref | Expression |
---|---|
lssn0 | ⊢ (𝑈 ∈ 𝑆 → 𝑈 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
2 | eqid 2821 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
3 | eqid 2821 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2821 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
5 | eqid 2821 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
6 | lssn0.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | islss 19705 | . 2 ⊢ (𝑈 ∈ 𝑆 ↔ (𝑈 ⊆ (Base‘𝑊) ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ 𝑈)) |
8 | 7 | simp2bi 1142 | 1 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ⊆ wss 3935 ∅c0 4290 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 +gcplusg 16564 Scalarcsca 16567 ·𝑠 cvsca 16568 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: 00lss 19712 lss0cl 19717 lssne0 19721 lsssubg 19728 lbsextlem2 19930 minveclem1 24026 |
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