Theorem lsssetm 13912

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
lsssetm	⊢ (𝑊 ∈ 𝑋 → 𝑆 = {𝑠 ∈ 𝒫 𝑉 ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀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   𝑗,𝑎,𝑏,𝑠,𝑥

Allowed substitution hints:   𝐵(𝑗,𝑎,𝑏)   + (𝑥,𝑗,𝑎,𝑏)   𝑆(𝑥,𝑗,𝑠,𝑎,𝑏)   · (𝑥,𝑗,𝑎,𝑏)   𝐹(𝑥,𝑗,𝑠,𝑎,𝑏)   𝑉(𝑥,𝑗,𝑎,𝑏)   𝑊(𝑗)   𝑋(𝑥,𝑗,𝑠,𝑎,𝑏)

Proof of Theorem lsssetm

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lssset.s	. 2 ⊢ 𝑆 = (LSubSp‘𝑊)
2		df-lssm 13909	. . 3 ⊢ LSubSp = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠)})
3		fveq2 5558	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4		lssset.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2247	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6	5	pweqd 3610	. . . 4 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7		fveq2 5558	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
8		lssset.f	. . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊)
9	7, 8	eqtr4di 2247	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
10	9	fveq2d 5562	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
11		lssset.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐹)
12	10, 11	eqtr4di 2247	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐵)
13		fveq2 5558	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
14		lssset.t	. . . . . . . . . . . 12 ⊢ · = ( ·𝑠 ‘𝑊)
15	13, 14	eqtr4di 2247	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
16	15	oveqd 5939	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)𝑎) = (𝑥 · 𝑎))
17	16	oveq1d 5937	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) = ((𝑥 · 𝑎)(+g‘𝑤)𝑏))
18		fveq2 5558	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (+g‘𝑤) = (+g‘𝑊))
19		lssset.p	. . . . . . . . . . 11 ⊢ + = (+g‘𝑊)
20	18, 19	eqtr4di 2247	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (+g‘𝑤) = + )
21	20	oveqd 5939	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((𝑥 · 𝑎)(+g‘𝑤)𝑏) = ((𝑥 · 𝑎) + 𝑏))
22	17, 21	eqtrd 2229	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) = ((𝑥 · 𝑎) + 𝑏))
23	22	eleq1d 2265	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠 ↔ ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠))
24	23	2ralbidv 2521	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠 ↔ ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠))
25	12, 24	raleqbidv 2709	. . . . 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))
26	25	anbi2d 464	. . . 4 ⊢ (𝑤 = 𝑊 → ((∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠) ↔ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠)))
27	6, 26	rabeqbidv 2758	. . 3 ⊢ (𝑤 = 𝑊 → {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠)} = {𝑠 ∈ 𝒫 𝑉 ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠)})
28		elex 2774	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
29		basfn 12736	. . . . . . 7 ⊢ Base Fn V
30		funfvex 5575	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑊 ∈ dom Base) → (Base‘𝑊) ∈ V)
31	30	funfni 5358	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑊 ∈ V) → (Base‘𝑊) ∈ V)
32	29, 28, 31	sylancr 414	. . . . . 6 ⊢ (𝑊 ∈ 𝑋 → (Base‘𝑊) ∈ V)
33	4, 32	eqeltrid 2283	. . . . 5 ⊢ (𝑊 ∈ 𝑋 → 𝑉 ∈ V)
34	33	pwexd 4214	. . . 4 ⊢ (𝑊 ∈ 𝑋 → 𝒫 𝑉 ∈ V)
35		rabexg 4176	. . . 4 ⊢ (𝒫 𝑉 ∈ V → {𝑠 ∈ 𝒫 𝑉 ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠)} ∈ V)
36	34, 35	syl 14	. . 3 ⊢ (𝑊 ∈ 𝑋 → {𝑠 ∈ 𝒫 𝑉 ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠)} ∈ V)
37	2, 27, 28, 36	fvmptd3 5655	. 2 ⊢ (𝑊 ∈ 𝑋 → (LSubSp‘𝑊) = {𝑠 ∈ 𝒫 𝑉 ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠)})
38	1, 37	eqtrid 2241	1 ⊢ (𝑊 ∈ 𝑋 → 𝑆 = {𝑠 ∈ 𝒫 𝑉 ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠)})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  {crab 2479  Vcvv 2763  𝒫 cpw 3605   Fn wfn 5253  ‘cfv 5258  (class class class)co 5922  Basecbs 12678  +_gcplusg 12755  Scalarcsca 12758   ·_𝑠 cvsca 12759  LSubSpclss 13908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-inn 8991  df-ndx 12681  df-slot 12682  df-base 12684  df-lssm 13909

This theorem is referenced by:  islssm  13913  islssmg  13914