MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lssset Structured version   Visualization version   GIF version

Theorem lssset 18701
Description: The set of all (not necessarily closed) linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 15-Jul-2014.)
Hypotheses
Ref Expression
lssset.f 𝐹 = (Scalar‘𝑊)
lssset.b 𝐵 = (Base‘𝐹)
lssset.v 𝑉 = (Base‘𝑊)
lssset.p + = (+g𝑊)
lssset.t · = ( ·𝑠𝑊)
lssset.s 𝑆 = (LSubSp‘𝑊)
Assertion
Ref Expression
lssset (𝑊𝑋𝑆 = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥𝐵𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})
Distinct variable groups:   + ,𝑠   𝑥,𝑠,𝐵   𝑉,𝑠   𝑎,𝑏,𝑠,𝑥,𝑊   · ,𝑠
Allowed substitution hints:   𝐵(𝑎,𝑏)   + (𝑥,𝑎,𝑏)   𝑆(𝑥,𝑠,𝑎,𝑏)   · (𝑥,𝑎,𝑏)   𝐹(𝑥,𝑠,𝑎,𝑏)   𝑉(𝑥,𝑎,𝑏)   𝑋(𝑥,𝑠,𝑎,𝑏)

Proof of Theorem lssset
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 lssset.s . 2 𝑆 = (LSubSp‘𝑊)
2 elex 3184 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6088 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 lssset.v . . . . . . . 8 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2661 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
65pweqd 4112 . . . . . 6 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
76difeq1d 3688 . . . . 5 (𝑤 = 𝑊 → (𝒫 (Base‘𝑤) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
8 fveq2 6088 . . . . . . . . 9 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
9 lssset.f . . . . . . . . 9 𝐹 = (Scalar‘𝑊)
108, 9syl6eqr 2661 . . . . . . . 8 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
1110fveq2d 6092 . . . . . . 7 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
12 lssset.b . . . . . . 7 𝐵 = (Base‘𝐹)
1311, 12syl6eqr 2661 . . . . . 6 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐵)
14 fveq2 6088 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
15 lssset.t . . . . . . . . . . . 12 · = ( ·𝑠𝑊)
1614, 15syl6eqr 2661 . . . . . . . . . . 11 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
1716oveqd 6544 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑥( ·𝑠𝑤)𝑎) = (𝑥 · 𝑎))
1817oveq1d 6542 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) = ((𝑥 · 𝑎)(+g𝑤)𝑏))
19 fveq2 6088 . . . . . . . . . . 11 (𝑤 = 𝑊 → (+g𝑤) = (+g𝑊))
20 lssset.p . . . . . . . . . . 11 + = (+g𝑊)
2119, 20syl6eqr 2661 . . . . . . . . . 10 (𝑤 = 𝑊 → (+g𝑤) = + )
2221oveqd 6544 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑥 · 𝑎)(+g𝑤)𝑏) = ((𝑥 · 𝑎) + 𝑏))
2318, 22eqtrd 2643 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) = ((𝑥 · 𝑎) + 𝑏))
2423eleq1d 2671 . . . . . . 7 (𝑤 = 𝑊 → (((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) ∈ 𝑠 ↔ ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠))
25242ralbidv 2971 . . . . . 6 (𝑤 = 𝑊 → (∀𝑎𝑠𝑏𝑠 ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) ∈ 𝑠 ↔ ∀𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠))
2613, 25raleqbidv 3128 . . . . 5 (𝑤 = 𝑊 → (∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎𝑠𝑏𝑠 ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) ∈ 𝑠 ↔ ∀𝑥𝐵𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠))
277, 26rabeqbidv 3167 . . . 4 (𝑤 = 𝑊 → {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎𝑠𝑏𝑠 ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) ∈ 𝑠} = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥𝐵𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})
28 df-lss 18700 . . . 4 LSubSp = (𝑤 ∈ V ↦ {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎𝑠𝑏𝑠 ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) ∈ 𝑠})
29 fvex 6098 . . . . . . . 8 (Base‘𝑊) ∈ V
304, 29eqeltri 2683 . . . . . . 7 𝑉 ∈ V
3130pwex 4769 . . . . . 6 𝒫 𝑉 ∈ V
32 difexg 4730 . . . . . 6 (𝒫 𝑉 ∈ V → (𝒫 𝑉 ∖ {∅}) ∈ V)
3331, 32ax-mp 5 . . . . 5 (𝒫 𝑉 ∖ {∅}) ∈ V
3433rabex 4735 . . . 4 {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥𝐵𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠} ∈ V
3527, 28, 34fvmpt 6176 . . 3 (𝑊 ∈ V → (LSubSp‘𝑊) = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥𝐵𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})
362, 35syl 17 . 2 (𝑊𝑋 → (LSubSp‘𝑊) = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥𝐵𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})
371, 36syl5eq 2655 1 (𝑊𝑋𝑆 = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥𝐵𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  wral 2895  {crab 2899  Vcvv 3172  cdif 3536  c0 3873  𝒫 cpw 4107  {csn 4124  cfv 5790  (class class class)co 6527  Basecbs 15641  +gcplusg 15714  Scalarcsca 15717   ·𝑠 cvsca 15718  LSubSpclss 18699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-lss 18700
This theorem is referenced by:  islss  18702
  Copyright terms: Public domain W3C validator