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Theorem lsscl 19643
Description: Closure property of a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
Hypotheses
Ref Expression
lsscl.f 𝐹 = (Scalar‘𝑊)
lsscl.b 𝐵 = (Base‘𝐹)
lsscl.p + = (+g𝑊)
lsscl.t · = ( ·𝑠𝑊)
lsscl.s 𝑆 = (LSubSp‘𝑊)
Assertion
Ref Expression
lsscl ((𝑈𝑆 ∧ (𝑍𝐵𝑋𝑈𝑌𝑈)) → ((𝑍 · 𝑋) + 𝑌) ∈ 𝑈)

Proof of Theorem lsscl
Dummy variables 𝑥 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef 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 + = (+g𝑊)
5 lsscl.t . . . 4 · = ( ·𝑠𝑊)
6 lsscl.s . . . 4 𝑆 = (LSubSp‘𝑊)
71, 2, 3, 4, 5, 6islss 19635 . . 3 (𝑈𝑆 ↔ (𝑈 ⊆ (Base‘𝑊) ∧ 𝑈 ≠ ∅ ∧ ∀𝑥𝐵𝑎𝑈𝑏𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈))
87simp3bi 1139 . 2 (𝑈𝑆 → ∀𝑥𝐵𝑎𝑈𝑏𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈)
9 oveq1 7152 . . . . 5 (𝑥 = 𝑍 → (𝑥 · 𝑎) = (𝑍 · 𝑎))
109oveq1d 7160 . . . 4 (𝑥 = 𝑍 → ((𝑥 · 𝑎) + 𝑏) = ((𝑍 · 𝑎) + 𝑏))
1110eleq1d 2894 . . 3 (𝑥 = 𝑍 → (((𝑥 · 𝑎) + 𝑏) ∈ 𝑈 ↔ ((𝑍 · 𝑎) + 𝑏) ∈ 𝑈))
12 oveq2 7153 . . . . 5 (𝑎 = 𝑋 → (𝑍 · 𝑎) = (𝑍 · 𝑋))
1312oveq1d 7160 . . . 4 (𝑎 = 𝑋 → ((𝑍 · 𝑎) + 𝑏) = ((𝑍 · 𝑋) + 𝑏))
1413eleq1d 2894 . . 3 (𝑎 = 𝑋 → (((𝑍 · 𝑎) + 𝑏) ∈ 𝑈 ↔ ((𝑍 · 𝑋) + 𝑏) ∈ 𝑈))
15 oveq2 7153 . . . 4 (𝑏 = 𝑌 → ((𝑍 · 𝑋) + 𝑏) = ((𝑍 · 𝑋) + 𝑌))
1615eleq1d 2894 . . 3 (𝑏 = 𝑌 → (((𝑍 · 𝑋) + 𝑏) ∈ 𝑈 ↔ ((𝑍 · 𝑋) + 𝑌) ∈ 𝑈))
1711, 14, 16rspc3v 3633 . 2 ((𝑍𝐵𝑋𝑈𝑌𝑈) → (∀𝑥𝐵𝑎𝑈𝑏𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈 → ((𝑍 · 𝑋) + 𝑌) ∈ 𝑈))
188, 17mpan9 507 1 ((𝑈𝑆 ∧ (𝑍𝐵𝑋𝑈𝑌𝑈)) → ((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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