Theorem lssex 14392

Description: Existence of a linear subspace. (Contributed by Jim Kingdon, 27-Apr-2025.)

Assertion

Ref	Expression
lssex	⊢ (𝑊 ∈ 𝑉 → (LSubSp‘𝑊) ∈ V)

Proof of Theorem lssex

Dummy variables 𝑤 𝑎 𝑏 𝑗 𝑠 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		basfn 13164	. . . . . 6 ⊢ Base Fn V
2		vex 2804	. . . . . . 7 ⊢ 𝑤 ∈ V
3	2	a1i 9	. . . . . 6 ⊢ (𝑊 ∈ 𝑉 → 𝑤 ∈ V)
4		funfvex 5659	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑤 ∈ dom Base) → (Base‘𝑤) ∈ V)
5	4	funfni 5434	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑤 ∈ V) → (Base‘𝑤) ∈ V)
6	1, 3, 5	sylancr 414	. . . . 5 ⊢ (𝑊 ∈ 𝑉 → (Base‘𝑤) ∈ V)
7	6	pwexd 4273	. . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝒫 (Base‘𝑤) ∈ V)
8		rabexg 4234	. . . 4 ⊢ (𝒫 (Base‘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)} ∈ V)
9	7, 8	syl 14	. . 3 ⊢ (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)} ∈ V)
10	9	alrimiv 1921	. 2 ⊢ (𝑊 ∈ 𝑉 → ∀𝑤{𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠)} ∈ V)
11		df-lssm 14391	. . 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)})
12	11	mptfvex 5735	. 2 ⊢ ((∀𝑤{𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (∃𝑗 𝑗 ∈ 𝑠 ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠)} ∈ V ∧ 𝑊 ∈ 𝑉) → (LSubSp‘𝑊) ∈ V)
13	10, 12	mpancom 422	1 ⊢ (𝑊 ∈ 𝑉 → (LSubSp‘𝑊) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1395  ∃wex 1540   ∈ wcel 2201  ∀wral 2509  {crab 2513  Vcvv 2801  𝒫 cpw 3653   Fn wfn 5323  ‘cfv 5328  (class class class)co 6023  Basecbs 13105  +_gcplusg 13183  Scalarcsca 13186   ·_𝑠 cvsca 13187  LSubSpclss 14390

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-cnex 8128  ax-resscn 8129  ax-1re 8131  ax-addrcl 8134

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-iota 5288  df-fun 5330  df-fn 5331  df-fv 5336  df-inn 9149  df-ndx 13108  df-slot 13109  df-base 13111  df-lssm 14391

This theorem is referenced by:  lidlex  14511