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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 2622 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
2 | eqid 2622 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
3 | eqid 2622 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2622 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
5 | eqid 2622 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
6 | lssn0.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | islss 18935 | . 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 1077 | 1 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ⊆ wss 3574 ∅c0 3915 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 Scalarcsca 15944 ·𝑠 cvsca 15945 LSubSpclss 18932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-lss 18933 |
This theorem is referenced by: 00lss 18942 lss0cl 18947 lssne0 18951 lsssubg 18957 lbsextlem2 19159 minveclem1 23195 |
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