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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 2610 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
2 | eqid 2610 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
3 | eqid 2610 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2610 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
5 | eqid 2610 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
6 | lssn0.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | islss 18756 | . 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 1070 | 1 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ⊆ wss 3540 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Scalarcsca 15771 ·𝑠 cvsca 15772 LSubSpclss 18753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-lss 18754 |
This theorem is referenced by: 00lss 18763 lss0cl 18768 lssne0 18772 lsssubg 18778 lbsextlem2 18980 minveclem1 23003 |
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