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Theorem lssset 20409
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 = (Scalarβ€˜π‘Š)
lssset.b 𝐡 = (Baseβ€˜πΉ)
lssset.v 𝑉 = (Baseβ€˜π‘Š)
lssset.p + = (+gβ€˜π‘Š)
lssset.t Β· = ( ·𝑠 β€˜π‘Š)
lssset.s 𝑆 = (LSubSpβ€˜π‘Š)
Assertion
Ref Expression
lssset (π‘Š ∈ 𝑋 β†’ 𝑆 = {𝑠 ∈ (𝒫 𝑉 βˆ– {βˆ…}) ∣ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘Ž ∈ 𝑠 βˆ€π‘ ∈ 𝑠 ((π‘₯ Β· π‘Ž) + 𝑏) ∈ 𝑠})
Distinct variable groups:   + ,𝑠   π‘₯,𝑠,𝐡   𝑉,𝑠   π‘Ž,𝑏,𝑠,π‘₯,π‘Š   Β· ,𝑠
Allowed substitution hints:   𝐡(π‘Ž,𝑏)   + (π‘₯,π‘Ž,𝑏)   𝑆(π‘₯,𝑠,π‘Ž,𝑏)   Β· (π‘₯,π‘Ž,𝑏)   𝐹(π‘₯,𝑠,π‘Ž,𝑏)   𝑉(π‘₯,π‘Ž,𝑏)   𝑋(π‘₯,𝑠,π‘Ž,𝑏)

Proof of Theorem lssset
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 lssset.s . 2 𝑆 = (LSubSpβ€˜π‘Š)
2 elex 3462 . . 3 (π‘Š ∈ 𝑋 β†’ π‘Š ∈ V)
3 fveq2 6843 . . . . . . . 8 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
4 lssset.v . . . . . . . 8 𝑉 = (Baseβ€˜π‘Š)
53, 4eqtr4di 2791 . . . . . . 7 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝑉)
65pweqd 4578 . . . . . 6 (𝑀 = π‘Š β†’ 𝒫 (Baseβ€˜π‘€) = 𝒫 𝑉)
76difeq1d 4082 . . . . 5 (𝑀 = π‘Š β†’ (𝒫 (Baseβ€˜π‘€) βˆ– {βˆ…}) = (𝒫 𝑉 βˆ– {βˆ…}))
8 fveq2 6843 . . . . . . . . 9 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = (Scalarβ€˜π‘Š))
9 lssset.f . . . . . . . . 9 𝐹 = (Scalarβ€˜π‘Š)
108, 9eqtr4di 2791 . . . . . . . 8 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) = 𝐹)
1110fveq2d 6847 . . . . . . 7 (𝑀 = π‘Š β†’ (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜πΉ))
12 lssset.b . . . . . . 7 𝐡 = (Baseβ€˜πΉ)
1311, 12eqtr4di 2791 . . . . . 6 (𝑀 = π‘Š β†’ (Baseβ€˜(Scalarβ€˜π‘€)) = 𝐡)
14 fveq2 6843 . . . . . . . . . . . 12 (𝑀 = π‘Š β†’ ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘Š))
15 lssset.t . . . . . . . . . . . 12 Β· = ( ·𝑠 β€˜π‘Š)
1614, 15eqtr4di 2791 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ ( ·𝑠 β€˜π‘€) = Β· )
1716oveqd 7375 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (π‘₯( ·𝑠 β€˜π‘€)π‘Ž) = (π‘₯ Β· π‘Ž))
1817oveq1d 7373 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((π‘₯( ·𝑠 β€˜π‘€)π‘Ž)(+gβ€˜π‘€)𝑏) = ((π‘₯ Β· π‘Ž)(+gβ€˜π‘€)𝑏))
19 fveq2 6843 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (+gβ€˜π‘€) = (+gβ€˜π‘Š))
20 lssset.p . . . . . . . . . . 11 + = (+gβ€˜π‘Š)
2119, 20eqtr4di 2791 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (+gβ€˜π‘€) = + )
2221oveqd 7375 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((π‘₯ Β· π‘Ž)(+gβ€˜π‘€)𝑏) = ((π‘₯ Β· π‘Ž) + 𝑏))
2318, 22eqtrd 2773 . . . . . . . 8 (𝑀 = π‘Š β†’ ((π‘₯( ·𝑠 β€˜π‘€)π‘Ž)(+gβ€˜π‘€)𝑏) = ((π‘₯ Β· π‘Ž) + 𝑏))
2423eleq1d 2819 . . . . . . 7 (𝑀 = π‘Š β†’ (((π‘₯( ·𝑠 β€˜π‘€)π‘Ž)(+gβ€˜π‘€)𝑏) ∈ 𝑠 ↔ ((π‘₯ Β· π‘Ž) + 𝑏) ∈ 𝑠))
25242ralbidv 3209 . . . . . 6 (𝑀 = π‘Š β†’ (βˆ€π‘Ž ∈ 𝑠 βˆ€π‘ ∈ 𝑠 ((π‘₯( ·𝑠 β€˜π‘€)π‘Ž)(+gβ€˜π‘€)𝑏) ∈ 𝑠 ↔ βˆ€π‘Ž ∈ 𝑠 βˆ€π‘ ∈ 𝑠 ((π‘₯ Β· π‘Ž) + 𝑏) ∈ 𝑠))
2613, 25raleqbidv 3318 . . . . 5 (𝑀 = π‘Š β†’ (βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€))βˆ€π‘Ž ∈ 𝑠 βˆ€π‘ ∈ 𝑠 ((π‘₯( ·𝑠 β€˜π‘€)π‘Ž)(+gβ€˜π‘€)𝑏) ∈ 𝑠 ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘Ž ∈ 𝑠 βˆ€π‘ ∈ 𝑠 ((π‘₯ Β· π‘Ž) + 𝑏) ∈ 𝑠))
277, 26rabeqbidv 3423 . . . 4 (𝑀 = π‘Š β†’ {𝑠 ∈ (𝒫 (Baseβ€˜π‘€) βˆ– {βˆ…}) ∣ βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€))βˆ€π‘Ž ∈ 𝑠 βˆ€π‘ ∈ 𝑠 ((π‘₯( ·𝑠 β€˜π‘€)π‘Ž)(+gβ€˜π‘€)𝑏) ∈ 𝑠} = {𝑠 ∈ (𝒫 𝑉 βˆ– {βˆ…}) ∣ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘Ž ∈ 𝑠 βˆ€π‘ ∈ 𝑠 ((π‘₯ Β· π‘Ž) + 𝑏) ∈ 𝑠})
28 df-lss 20408 . . . 4 LSubSp = (𝑀 ∈ V ↦ {𝑠 ∈ (𝒫 (Baseβ€˜π‘€) βˆ– {βˆ…}) ∣ βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€))βˆ€π‘Ž ∈ 𝑠 βˆ€π‘ ∈ 𝑠 ((π‘₯( ·𝑠 β€˜π‘€)π‘Ž)(+gβ€˜π‘€)𝑏) ∈ 𝑠})
294fvexi 6857 . . . . . . 7 𝑉 ∈ V
3029pwex 5336 . . . . . 6 𝒫 𝑉 ∈ V
3130difexi 5286 . . . . 5 (𝒫 𝑉 βˆ– {βˆ…}) ∈ V
3231rabex 5290 . . . 4 {𝑠 ∈ (𝒫 𝑉 βˆ– {βˆ…}) ∣ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘Ž ∈ 𝑠 βˆ€π‘ ∈ 𝑠 ((π‘₯ Β· π‘Ž) + 𝑏) ∈ 𝑠} ∈ V
3327, 28, 32fvmpt 6949 . . 3 (π‘Š ∈ V β†’ (LSubSpβ€˜π‘Š) = {𝑠 ∈ (𝒫 𝑉 βˆ– {βˆ…}) ∣ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘Ž ∈ 𝑠 βˆ€π‘ ∈ 𝑠 ((π‘₯ Β· π‘Ž) + 𝑏) ∈ 𝑠})
342, 33syl 17 . 2 (π‘Š ∈ 𝑋 β†’ (LSubSpβ€˜π‘Š) = {𝑠 ∈ (𝒫 𝑉 βˆ– {βˆ…}) ∣ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘Ž ∈ 𝑠 βˆ€π‘ ∈ 𝑠 ((π‘₯ Β· π‘Ž) + 𝑏) ∈ 𝑠})
351, 34eqtrid 2785 1 (π‘Š ∈ 𝑋 β†’ 𝑆 = {𝑠 ∈ (𝒫 𝑉 βˆ– {βˆ…}) ∣ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘Ž ∈ 𝑠 βˆ€π‘ ∈ 𝑠 ((π‘₯ Β· π‘Ž) + 𝑏) ∈ 𝑠})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406  Vcvv 3444   βˆ– cdif 3908  βˆ…c0 4283  π’« cpw 4561  {csn 4587  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  Scalarcsca 17141   ·𝑠 cvsca 17142  LSubSpclss 20407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-lss 20408
This theorem is referenced by:  islss  20410
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