Theorem lssset 20954

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	⊢ + = (+g‘𝑊)
lssset.t	⊢ · = ( ·𝑠 ‘𝑊)
lssset.s	⊢ 𝑆 = (LSubSp‘𝑊)

Assertion

Ref	Expression
lssset	⊢ (𝑊 ∈ 𝑋 → 𝑆 = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((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.

Step	Hyp	Ref	Expression
1		lssset.s	. 2 ⊢ 𝑆 = (LSubSp‘𝑊)
2		elex 3509	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		fveq2 6920	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4		lssset.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2798	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6	5	pweqd 4639	. . . . . 6 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7	6	difeq1d 4148	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝒫 (Base‘𝑤) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
8		fveq2 6920	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
9		lssset.f	. . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊)
10	8, 9	eqtr4di 2798	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
11	10	fveq2d 6924	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
12		lssset.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐹)
13	11, 12	eqtr4di 2798	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐵)
14		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
15		lssset.t	. . . . . . . . . . . 12 ⊢ · = ( ·𝑠 ‘𝑊)
16	14, 15	eqtr4di 2798	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
17	16	oveqd 7465	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)𝑎) = (𝑥 · 𝑎))
18	17	oveq1d 7463	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) = ((𝑥 · 𝑎)(+g‘𝑤)𝑏))
19		fveq2 6920	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (+g‘𝑤) = (+g‘𝑊))
20		lssset.p	. . . . . . . . . . 11 ⊢ + = (+g‘𝑊)
21	19, 20	eqtr4di 2798	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (+g‘𝑤) = + )
22	21	oveqd 7465	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑥 · 𝑎)(+g‘𝑤)𝑏) = ((𝑥 · 𝑎) + 𝑏))
23	18, 22	eqtrd 2780	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) = ((𝑥 · 𝑎) + 𝑏))
24	23	eleq1d 2829	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠 ↔ ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠))
25	24	2ralbidv 3227	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠 ↔ ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠))
26	13, 25	raleqbidv 3354	. . . . 5 ⊢ (𝑤 = 𝑊 → (∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠))
27	7, 26	rabeqbidv 3462	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠} = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})
28		df-lss 20953	. . . 4 ⊢ LSubSp = (𝑤 ∈ V ↦ {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠})
29	4	fvexi 6934	. . . . . . 7 ⊢ 𝑉 ∈ V
30	29	pwex 5398	. . . . . 6 ⊢ 𝒫 𝑉 ∈ V
31	30	difexi 5348	. . . . 5 ⊢ (𝒫 𝑉 ∖ {∅}) ∈ V
32	31	rabex 5357	. . . 4 ⊢ {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠} ∈ V
33	27, 28, 32	fvmpt 7029	. . 3 ⊢ (𝑊 ∈ V → (LSubSp‘𝑊) = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})
34	2, 33	syl 17	. 2 ⊢ (𝑊 ∈ 𝑋 → (LSubSp‘𝑊) = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})
35	1, 34	eqtrid 2792	1 ⊢ (𝑊 ∈ 𝑋 → 𝑆 = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443  Vcvv 3488   ∖ cdif 3973  ∅c0 4352  𝒫 cpw 4622  {csn 4648  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  Scalarcsca 17314   ·_𝑠 cvsca 17315  LSubSpclss 20952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-lss 20953

This theorem is referenced by:  islss  20955