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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 2818 | . . . 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 19635 | . . 3 ⊢ (𝑈 ∈ 𝑆 ↔ (𝑈 ⊆ (Base‘𝑊) ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈)) |
8 | 7 | simp3bi 1139 | . 2 ⊢ (𝑈 ∈ 𝑆 → ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈) |
9 | oveq1 7152 | . . . . 5 ⊢ (𝑥 = 𝑍 → (𝑥 · 𝑎) = (𝑍 · 𝑎)) | |
10 | 9 | oveq1d 7160 | . . . 4 ⊢ (𝑥 = 𝑍 → ((𝑥 · 𝑎) + 𝑏) = ((𝑍 · 𝑎) + 𝑏)) |
11 | 10 | eleq1d 2894 | . . 3 ⊢ (𝑥 = 𝑍 → (((𝑥 · 𝑎) + 𝑏) ∈ 𝑈 ↔ ((𝑍 · 𝑎) + 𝑏) ∈ 𝑈)) |
12 | oveq2 7153 | . . . . 5 ⊢ (𝑎 = 𝑋 → (𝑍 · 𝑎) = (𝑍 · 𝑋)) | |
13 | 12 | oveq1d 7160 | . . . 4 ⊢ (𝑎 = 𝑋 → ((𝑍 · 𝑎) + 𝑏) = ((𝑍 · 𝑋) + 𝑏)) |
14 | 13 | eleq1d 2894 | . . 3 ⊢ (𝑎 = 𝑋 → (((𝑍 · 𝑎) + 𝑏) ∈ 𝑈 ↔ ((𝑍 · 𝑋) + 𝑏) ∈ 𝑈)) |
15 | oveq2 7153 | . . . 4 ⊢ (𝑏 = 𝑌 → ((𝑍 · 𝑋) + 𝑏) = ((𝑍 · 𝑋) + 𝑌)) | |
16 | 15 | eleq1d 2894 | . . 3 ⊢ (𝑏 = 𝑌 → (((𝑍 · 𝑋) + 𝑏) ∈ 𝑈 ↔ ((𝑍 · 𝑋) + 𝑌) ∈ 𝑈)) |
17 | 11, 14, 16 | rspc3v 3633 | . 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 507 | 1 ⊢ ((𝑈 ∈ 𝑆 ∧ (𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈)) → ((𝑍 · 𝑋) + 𝑌) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ⊆ wss 3933 ∅c0 4288 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Scalarcsca 16556 ·𝑠 cvsca 16557 LSubSpclss 19632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-lss 19633 |
This theorem is referenced by: lssvsubcl 19644 lssvacl 19655 lssvscl 19656 islss3 19660 lssintcl 19665 lspsolvlem 19843 lbsextlem2 19860 isphld 20726 |
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