Theorem islbs4 22046

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 6914	. . 3 ⊢ (𝑋 ∈ (LBasis‘𝑊) → 𝑊 ∈ V)
2		islbs4.j	. . 3 ⊢ 𝐽 = (LBasis‘𝑊)
3	1, 2	eleq2s 2878	. 2 ⊢ (𝑋 ∈ 𝐽 → 𝑊 ∈ V)
4		elfvex 6914	. . 3 ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑊 ∈ V)
5	4	adantr 486	. 2 ⊢ ((𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵) → 𝑊 ∈ V)
6		islbs4.b	. . . 4 ⊢ 𝐵 = (Base‘𝑊)
7		eqid 2760	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
8		eqid 2760	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
9		eqid 2760	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
10		islbs4.k	. . . 4 ⊢ 𝐾 = (LSpan‘𝑊)
11		eqid 2760	. . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
12	6, 7, 8, 9, 2, 10, 11	islbs 21261	. . 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 1113	. . . 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 22027	. . . . 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 643	. . . 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 293	. . 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 282	. 2 ⊢ (𝑊 ∈ V → (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵)))
18	3, 5, 17	pm5.21nii 381	1 ⊢ (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ∖ cdif 3896   ⊆ wss 3899  {csn 4584  ‘cfv 6533  (class class class)co 7414  Basecbs 17302  Scalarcsca 17346   ·_𝑠 cvsca 17347  0_gc0g 17525  LSpanclspn 21156  LBasisclbs 21259  LIndSclinds 22019

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-lbs 21260  df-lindf 22020  df-linds 22021

This theorem is used by:  lbslinds  22047  islinds3  22048  lmimlbs  22050  lindsenlbs  22065  lindflbs  33813  rlmdim  34121  dimkerim  34138  fedgmullem1  34140  fedgmul  34142  ccfldextdgrr  34183