Theorem lindff 21782

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 482	. . 3 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → 𝐹 LIndF 𝑊)
2		rellindf 21775	. . . . . 6 ⊢ Rel LIndF
3	2	brrelex1i 5688	. . . . 5 ⊢ (𝐹 LIndF 𝑊 → 𝐹 ∈ V)
4		lindff.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
5		eqid 2737	. . . . . 6 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
6		eqid 2737	. . . . . 6 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
7		eqid 2737	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
8		eqid 2737	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
9		eqid 2737	. . . . . 6 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
10	4, 5, 6, 7, 8, 9	islindf 21779	. . . . 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 594	. . . 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 458	. . 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 232	. 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 494	1 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → 𝐹:dom 𝐹⟶𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ∖ cdif 3900  {csn 4582   class class class wbr 5100  dom cdm 5632   “ cima 5635  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  Scalarcsca 17192   ·_𝑠 cvsca 17193  0_gc0g 17371  LSpanclspn 20934   LIndF clindf 21771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-lindf 21773

This theorem is referenced by:  lindfind2  21785  lindff1  21787  lindfrn  21788  f1lindf  21789  indlcim  21807