Theorem islbs4 21869

Description: A basis is an independent spanning set. This could have been used as alternative definition of a basis: LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ (((LSpan‘𝑤) ‘𝑏) = (Base‘𝑤) ∧ 𝑏 ∈ (LIndS‘𝑤))}). (Contributed by Stefan O'Rear, 24-Feb-2015.)

Hypotheses

Ref	Expression
islbs4.b	⊢ 𝐵 = (Base‘𝑊)
islbs4.j	⊢ 𝐽 = (LBasis‘𝑊)
islbs4.k	⊢ 𝐾 = (LSpan‘𝑊)

Assertion

Ref	Expression
islbs4	⊢ (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵))

Proof of Theorem islbs4

Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6944	. . 3 ⊢ (𝑋 ∈ (LBasis‘𝑊) → 𝑊 ∈ V)
2		islbs4.j	. . 3 ⊢ 𝐽 = (LBasis‘𝑊)
3	1, 2	eleq2s 2856	. 2 ⊢ (𝑋 ∈ 𝐽 → 𝑊 ∈ V)
4		elfvex 6944	. . 3 ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ V)
5	4	adantr 480	. 2 ⊢ ((𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵) → 𝑊 ∈ V)
6		islbs4.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
7		eqid 2734	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
8		eqid 2734	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
9		eqid 2734	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
10		islbs4.k	. . . 4 ⊢ 𝐾 = (LSpan‘𝑊)
11		eqid 2734	. . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
12	6, 7, 8, 9, 2, 10, 11	islbs 21092	. . 3 ⊢ (𝑊 ∈ V → (𝑋 ∈ 𝐽 ↔ (𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵 ∧ ∀𝑥 ∈ 𝑋 ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥})))))
13		3anan32 1096	. . . 4 ⊢ ((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵 ∧ ∀𝑥 ∈ 𝑋 ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥}))) ↔ ((𝑋 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑋 ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥}))) ∧ (𝐾‘𝑋) = 𝐵))
14	6, 8, 10, 7, 9, 11	islinds2 21850	. . . . 5 ⊢ (𝑊 ∈ V → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑋 ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥})))))
15	14	anbi1d 631	. . . 4 ⊢ (𝑊 ∈ V → ((𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵) ↔ ((𝑋 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑋 ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥}))) ∧ (𝐾‘𝑋) = 𝐵)))
16	13, 15	bitr4id 290	. . 3 ⊢ (𝑊 ∈ V → ((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵 ∧ ∀𝑥 ∈ 𝑋 ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)𝑥) ∈ (𝐾‘(𝑋 ∖ {𝑥}))) ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵)))
17	12, 16	bitrd 279	. 2 ⊢ (𝑊 ∈ V → (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵)))
18	3, 5, 17	pm5.21nii 378	1 ⊢ (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105  ∀wral 3058  Vcvv 3477   ∖ cdif 3959   ⊆ wss 3962  {csn 4630  ‘cfv 6562  (class class class)co 7430  Basecbs 17244  Scalarcsca 17300   ·_𝑠 cvsca 17301  0_gc0g 17485  LSpanclspn 20986  LBasisclbs 21090  LIndSclinds 21842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-lbs 21091  df-lindf 21843  df-linds 21844

This theorem is referenced by:  lbslinds  21870  islinds3  21871  lmimlbs  21873  lindflbs  33386  rlmdim  33636  rgmoddimOLD  33637  dimkerim  33654  fedgmullem1  33656  fedgmul  33658  ccfldextdgrr  33696  lindsenlbs  37601