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Theorem lssset 19634
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 3510 . . 3 (𝑊𝑋𝑊 ∈ V)
3 fveq2 6663 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 lssset.v . . . . . . . 8 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2871 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
65pweqd 4540 . . . . . 6 (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
76difeq1d 4095 . . . . 5 (𝑤 = 𝑊 → (𝒫 (Base‘𝑤) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
8 fveq2 6663 . . . . . . . . 9 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
9 lssset.f . . . . . . . . 9 𝐹 = (Scalar‘𝑊)
108, 9syl6eqr 2871 . . . . . . . 8 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
1110fveq2d 6667 . . . . . . 7 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
12 lssset.b . . . . . . 7 𝐵 = (Base‘𝐹)
1311, 12syl6eqr 2871 . . . . . 6 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐵)
14 fveq2 6663 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
15 lssset.t . . . . . . . . . . . 12 · = ( ·𝑠𝑊)
1614, 15syl6eqr 2871 . . . . . . . . . . 11 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
1716oveqd 7162 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑥( ·𝑠𝑤)𝑎) = (𝑥 · 𝑎))
1817oveq1d 7160 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) = ((𝑥 · 𝑎)(+g𝑤)𝑏))
19 fveq2 6663 . . . . . . . . . . 11 (𝑤 = 𝑊 → (+g𝑤) = (+g𝑊))
20 lssset.p . . . . . . . . . . 11 + = (+g𝑊)
2119, 20syl6eqr 2871 . . . . . . . . . 10 (𝑤 = 𝑊 → (+g𝑤) = + )
2221oveqd 7162 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑥 · 𝑎)(+g𝑤)𝑏) = ((𝑥 · 𝑎) + 𝑏))
2318, 22eqtrd 2853 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) = ((𝑥 · 𝑎) + 𝑏))
2423eleq1d 2894 . . . . . . 7 (𝑤 = 𝑊 → (((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) ∈ 𝑠 ↔ ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠))
25242ralbidv 3196 . . . . . 6 (𝑤 = 𝑊 → (∀𝑎𝑠𝑏𝑠 ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) ∈ 𝑠 ↔ ∀𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠))
2613, 25raleqbidv 3399 . . . . 5 (𝑤 = 𝑊 → (∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎𝑠𝑏𝑠 ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) ∈ 𝑠 ↔ ∀𝑥𝐵𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠))
277, 26rabeqbidv 3483 . . . 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 19633 . . . 4 LSubSp = (𝑤 ∈ V ↦ {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎𝑠𝑏𝑠 ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) ∈ 𝑠})
294fvexi 6677 . . . . . . 7 𝑉 ∈ V
3029pwex 5272 . . . . . 6 𝒫 𝑉 ∈ V
3130difexi 5223 . . . . 5 (𝒫 𝑉 ∖ {∅}) ∈ V
3231rabex 5226 . . . 4 {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥𝐵𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠} ∈ V
3327, 28, 32fvmpt 6761 . . 3 (𝑊 ∈ V → (LSubSp‘𝑊) = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥𝐵𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})
342, 33syl 17 . 2 (𝑊𝑋 → (LSubSp‘𝑊) = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥𝐵𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})
351, 34syl5eq 2865 1 (𝑊𝑋𝑆 = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥𝐵𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  cdif 3930  c0 4288  𝒫 cpw 4535  {csn 4557  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-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:  islss  19635
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