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Theorem lsscl 19634
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 2826 . . . 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 19626 . . 3 (𝑈𝑆 ↔ (𝑈 ⊆ (Base‘𝑊) ∧ 𝑈 ≠ ∅ ∧ ∀𝑥𝐵𝑎𝑈𝑏𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈))
87simp3bi 1141 . 2 (𝑈𝑆 → ∀𝑥𝐵𝑎𝑈𝑏𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈)
9 oveq1 7155 . . . . 5 (𝑥 = 𝑍 → (𝑥 · 𝑎) = (𝑍 · 𝑎))
109oveq1d 7163 . . . 4 (𝑥 = 𝑍 → ((𝑥 · 𝑎) + 𝑏) = ((𝑍 · 𝑎) + 𝑏))
1110eleq1d 2902 . . 3 (𝑥 = 𝑍 → (((𝑥 · 𝑎) + 𝑏) ∈ 𝑈 ↔ ((𝑍 · 𝑎) + 𝑏) ∈ 𝑈))
12 oveq2 7156 . . . . 5 (𝑎 = 𝑋 → (𝑍 · 𝑎) = (𝑍 · 𝑋))
1312oveq1d 7163 . . . 4 (𝑎 = 𝑋 → ((𝑍 · 𝑎) + 𝑏) = ((𝑍 · 𝑋) + 𝑏))
1413eleq1d 2902 . . 3 (𝑎 = 𝑋 → (((𝑍 · 𝑎) + 𝑏) ∈ 𝑈 ↔ ((𝑍 · 𝑋) + 𝑏) ∈ 𝑈))
15 oveq2 7156 . . . 4 (𝑏 = 𝑌 → ((𝑍 · 𝑋) + 𝑏) = ((𝑍 · 𝑋) + 𝑌))
1615eleq1d 2902 . . 3 (𝑏 = 𝑌 → (((𝑍 · 𝑋) + 𝑏) ∈ 𝑈 ↔ ((𝑍 · 𝑋) + 𝑌) ∈ 𝑈))
1711, 14, 16rspc3v 3640 . 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 1081   = wceq 1530  wcel 2107  wne 3021  wral 3143  wss 3940  c0 4295  cfv 6352  (class class class)co 7148  Basecbs 16473  +gcplusg 16555  Scalarcsca 16558   ·𝑠 cvsca 16559  LSubSpclss 19623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6312  df-fun 6354  df-fv 6360  df-ov 7151  df-lss 19624
This theorem is referenced by:  lssvsubcl  19635  lssvacl  19646  lssvscl  19647  islss3  19651  lssintcl  19656  lspsolvlem  19834  lbsextlem2  19851  isphld  20714
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