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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 2738 | . . . 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 19825 | . . 3 ⊢ (𝑈 ∈ 𝑆 ↔ (𝑈 ⊆ (Base‘𝑊) ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈)) |
8 | 7 | simp3bi 1148 | . 2 ⊢ (𝑈 ∈ 𝑆 → ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈) |
9 | oveq1 7177 | . . . . 5 ⊢ (𝑥 = 𝑍 → (𝑥 · 𝑎) = (𝑍 · 𝑎)) | |
10 | 9 | oveq1d 7185 | . . . 4 ⊢ (𝑥 = 𝑍 → ((𝑥 · 𝑎) + 𝑏) = ((𝑍 · 𝑎) + 𝑏)) |
11 | 10 | eleq1d 2817 | . . 3 ⊢ (𝑥 = 𝑍 → (((𝑥 · 𝑎) + 𝑏) ∈ 𝑈 ↔ ((𝑍 · 𝑎) + 𝑏) ∈ 𝑈)) |
12 | oveq2 7178 | . . . . 5 ⊢ (𝑎 = 𝑋 → (𝑍 · 𝑎) = (𝑍 · 𝑋)) | |
13 | 12 | oveq1d 7185 | . . . 4 ⊢ (𝑎 = 𝑋 → ((𝑍 · 𝑎) + 𝑏) = ((𝑍 · 𝑋) + 𝑏)) |
14 | 13 | eleq1d 2817 | . . 3 ⊢ (𝑎 = 𝑋 → (((𝑍 · 𝑎) + 𝑏) ∈ 𝑈 ↔ ((𝑍 · 𝑋) + 𝑏) ∈ 𝑈)) |
15 | oveq2 7178 | . . . 4 ⊢ (𝑏 = 𝑌 → ((𝑍 · 𝑋) + 𝑏) = ((𝑍 · 𝑋) + 𝑌)) | |
16 | 15 | eleq1d 2817 | . . 3 ⊢ (𝑏 = 𝑌 → (((𝑍 · 𝑋) + 𝑏) ∈ 𝑈 ↔ ((𝑍 · 𝑋) + 𝑌) ∈ 𝑈)) |
17 | 11, 14, 16 | rspc3v 3539 | . 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 510 | 1 ⊢ ((𝑈 ∈ 𝑆 ∧ (𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈)) → ((𝑍 · 𝑋) + 𝑌) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∀wral 3053 ⊆ wss 3843 ∅c0 4211 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 +gcplusg 16668 Scalarcsca 16671 ·𝑠 cvsca 16672 LSubSpclss 19822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-iota 6297 df-fun 6341 df-fv 6347 df-ov 7173 df-lss 19823 |
This theorem is referenced by: lssvsubcl 19834 lssvacl 19845 lssvscl 19846 islss3 19850 lssintcl 19855 lspsolvlem 20033 lbsextlem2 20050 isphld 20470 |
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