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Mirrors > Home > MPE Home > Th. List > lsscl | Structured version Visualization version GIF version |
Description: Closure property of a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.) |
Ref | Expression |
---|---|
lsscl.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lsscl.b | ⊢ 𝐵 = (Base‘𝐹) |
lsscl.p | ⊢ + = (+g‘𝑊) |
lsscl.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lsscl.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
Ref | Expression |
---|---|
lsscl | ⊢ ((𝑈 ∈ 𝑆 ∧ (𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈)) → ((𝑍 · 𝑋) + 𝑌) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lsscl.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | lsscl.b | . . . 4 ⊢ 𝐵 = (Base‘𝐹) | |
3 | eqid 2799 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | lsscl.p | . . . 4 ⊢ + = (+g‘𝑊) | |
5 | lsscl.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
6 | lsscl.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | islss 19253 | . . 3 ⊢ (𝑈 ∈ 𝑆 ↔ (𝑈 ⊆ (Base‘𝑊) ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈)) |
8 | 7 | simp3bi 1178 | . 2 ⊢ (𝑈 ∈ 𝑆 → ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈) |
9 | oveq1 6885 | . . . . 5 ⊢ (𝑥 = 𝑍 → (𝑥 · 𝑎) = (𝑍 · 𝑎)) | |
10 | 9 | oveq1d 6893 | . . . 4 ⊢ (𝑥 = 𝑍 → ((𝑥 · 𝑎) + 𝑏) = ((𝑍 · 𝑎) + 𝑏)) |
11 | 10 | eleq1d 2863 | . . 3 ⊢ (𝑥 = 𝑍 → (((𝑥 · 𝑎) + 𝑏) ∈ 𝑈 ↔ ((𝑍 · 𝑎) + 𝑏) ∈ 𝑈)) |
12 | oveq2 6886 | . . . . 5 ⊢ (𝑎 = 𝑋 → (𝑍 · 𝑎) = (𝑍 · 𝑋)) | |
13 | 12 | oveq1d 6893 | . . . 4 ⊢ (𝑎 = 𝑋 → ((𝑍 · 𝑎) + 𝑏) = ((𝑍 · 𝑋) + 𝑏)) |
14 | 13 | eleq1d 2863 | . . 3 ⊢ (𝑎 = 𝑋 → (((𝑍 · 𝑎) + 𝑏) ∈ 𝑈 ↔ ((𝑍 · 𝑋) + 𝑏) ∈ 𝑈)) |
15 | oveq2 6886 | . . . 4 ⊢ (𝑏 = 𝑌 → ((𝑍 · 𝑋) + 𝑏) = ((𝑍 · 𝑋) + 𝑌)) | |
16 | 15 | eleq1d 2863 | . . 3 ⊢ (𝑏 = 𝑌 → (((𝑍 · 𝑋) + 𝑏) ∈ 𝑈 ↔ ((𝑍 · 𝑋) + 𝑌) ∈ 𝑈)) |
17 | 11, 14, 16 | rspc3v 3513 | . 2 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈 → ((𝑍 · 𝑋) + 𝑌) ∈ 𝑈)) |
18 | 8, 17 | mpan9 503 | 1 ⊢ ((𝑈 ∈ 𝑆 ∧ (𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈)) → ((𝑍 · 𝑋) + 𝑌) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 ∀wral 3089 ⊆ wss 3769 ∅c0 4115 ‘cfv 6101 (class class class)co 6878 Basecbs 16184 +gcplusg 16267 Scalarcsca 16270 ·𝑠 cvsca 16271 LSubSpclss 19250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-iota 6064 df-fun 6103 df-fv 6109 df-ov 6881 df-lss 19251 |
This theorem is referenced by: lssvsubcl 19262 lssvacl 19275 lssvscl 19276 islss3 19280 lssintcl 19285 lspsolvlem 19464 lbsextlem2 19482 isphld 20323 |
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