Theorem lindff 21867

Description: Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)

Hypothesis

Ref	Expression
lindff.b	⊢ 𝐵 = (Base‘𝑊)

Assertion

Ref	Expression
lindff	⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → 𝐹:dom 𝐹⟶𝐵)

Proof of Theorem lindff

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

Step	Hyp	Ref	Expression
1		simpl 486	. . 3 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → 𝐹 LIndF 𝑊)
2		rellindf 21860	. . . . . 6 ⊢ Rel LIndF
3	2	brrelex1i 5703	. . . . 5 ⊢ (𝐹 LIndF 𝑊 → 𝐹 ∈ V)
4		lindff.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
5		eqid 2762	. . . . . 6 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
6		eqid 2762	. . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
7		eqid 2762	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
8		eqid 2762	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
9		eqid 2762	. . . . . 6 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
10	4, 5, 6, 7, 8, 9	islindf 21864	. . . . 5 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
11	3, 10	sylan2 602	. . . 4 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 LIndF 𝑊) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
12	11	ancoms 462	. . 3 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
13	1, 12	mpbid 234	. 2 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
14	13	simpld 498	1 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → 𝐹:dom 𝐹⟶𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  Vcvv 3454   ∖ cdif 3901  {csn 4582   class class class wbr 5100  dom cdm 5647   “ cima 5650  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  Scalarcsca 17289   ·_𝑠 cvsca 17290  0_gc0g 17468  LSpanclspn 21038   LIndF clindf 21856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-ov 7399  df-lindf 21858

This theorem is referenced by:  lindfind2  21870  lindff1  21872  lindfrn  21873  f1lindf  21874  indlcim  21892