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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 2726 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
2 | eqid 2726 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
3 | eqid 2726 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2726 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
5 | eqid 2726 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
6 | lssn0.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | islss 20911 | . 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 1143 | 1 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ⊆ wss 3947 ∅c0 4325 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 +gcplusg 17266 Scalarcsca 17269 ·𝑠 cvsca 17270 LSubSpclss 20908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6506 df-fun 6556 df-fv 6562 df-ov 7427 df-lss 20909 |
This theorem is referenced by: 00lss 20918 lss0cl 20924 lssne0 20928 lsssubg 20934 lbsextlem2 21140 minveclem1 25443 |
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