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Theorem lssset 18982
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 3243 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6229 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 lssset.v . . . . . . . 8 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2703 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
65pweqd 4196 . . . . . 6 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
76difeq1d 3760 . . . . 5 (𝑤 = 𝑊 → (𝒫 (Base‘𝑤) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
8 fveq2 6229 . . . . . . . . 9 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
9 lssset.f . . . . . . . . 9 𝐹 = (Scalar‘𝑊)
108, 9syl6eqr 2703 . . . . . . . 8 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
1110fveq2d 6233 . . . . . . 7 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
12 lssset.b . . . . . . 7 𝐵 = (Base‘𝐹)
1311, 12syl6eqr 2703 . . . . . 6 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐵)
14 fveq2 6229 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
15 lssset.t . . . . . . . . . . . 12 · = ( ·𝑠𝑊)
1614, 15syl6eqr 2703 . . . . . . . . . . 11 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
1716oveqd 6707 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑥( ·𝑠𝑤)𝑎) = (𝑥 · 𝑎))
1817oveq1d 6705 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) = ((𝑥 · 𝑎)(+g𝑤)𝑏))
19 fveq2 6229 . . . . . . . . . . 11 (𝑤 = 𝑊 → (+g𝑤) = (+g𝑊))
20 lssset.p . . . . . . . . . . 11 + = (+g𝑊)
2119, 20syl6eqr 2703 . . . . . . . . . 10 (𝑤 = 𝑊 → (+g𝑤) = + )
2221oveqd 6707 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑥 · 𝑎)(+g𝑤)𝑏) = ((𝑥 · 𝑎) + 𝑏))
2318, 22eqtrd 2685 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) = ((𝑥 · 𝑎) + 𝑏))
2423eleq1d 2715 . . . . . . 7 (𝑤 = 𝑊 → (((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) ∈ 𝑠 ↔ ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠))
25242ralbidv 3018 . . . . . 6 (𝑤 = 𝑊 → (∀𝑎𝑠𝑏𝑠 ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) ∈ 𝑠 ↔ ∀𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠))
2613, 25raleqbidv 3182 . . . . 5 (𝑤 = 𝑊 → (∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎𝑠𝑏𝑠 ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) ∈ 𝑠 ↔ ∀𝑥𝐵𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠))
277, 26rabeqbidv 3226 . . . 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 18981 . . . 4 LSubSp = (𝑤 ∈ V ↦ {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎𝑠𝑏𝑠 ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) ∈ 𝑠})
29 fvex 6239 . . . . . . . 8 (Base‘𝑊) ∈ V
304, 29eqeltri 2726 . . . . . . 7 𝑉 ∈ V
3130pwex 4878 . . . . . 6 𝒫 𝑉 ∈ V
32 difexg 4841 . . . . . 6 (𝒫 𝑉 ∈ V → (𝒫 𝑉 ∖ {∅}) ∈ V)
3331, 32ax-mp 5 . . . . 5 (𝒫 𝑉 ∖ {∅}) ∈ V
3433rabex 4845 . . . 4 {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥𝐵𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠} ∈ V
3527, 28, 34fvmpt 6321 . . 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 2697 1 (𝑊𝑋𝑆 = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥𝐵𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  cdif 3604  c0 3948  𝒫 cpw 4191  {csn 4210  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Scalarcsca 15991   ·𝑠 cvsca 15992  LSubSpclss 18980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-lss 18981
This theorem is referenced by:  islss  18983
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