Theorem lindfind 21753

Description: A linearly independent family is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)

Hypotheses

Ref	Expression
lindfind.s	⊢ · = ( ·𝑠 ‘𝑊)
lindfind.n	⊢ 𝑁 = (LSpan‘𝑊)
lindfind.l	⊢ 𝐿 = (Scalar‘𝑊)
lindfind.z	⊢ 0 = (0g‘𝐿)
lindfind.k	⊢ 𝐾 = (Base‘𝐿)

Assertion

Ref	Expression
lindfind	⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))

Proof of Theorem lindfind

Dummy variables 𝑎 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 768	. 2 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐸 ∈ dom 𝐹)
2		eldifsn 4735	. . . 4 ⊢ (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ))
3	2	biimpri 228	. . 3 ⊢ ((𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ) → 𝐴 ∈ (𝐾 ∖ { 0 }))
4	3	adantl 481	. 2 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐴 ∈ (𝐾 ∖ { 0 }))
5		simpll 766	. . . 4 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐹 LIndF 𝑊)
6		lindfind.l	. . . . . . 7 ⊢ 𝐿 = (Scalar‘𝑊)
7		lindfind.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐿)
8	6, 7	elbasfv 17126	. . . . . 6 ⊢ (𝐴 ∈ 𝐾 → 𝑊 ∈ V)
9	8	ad2antrl 728	. . . . 5 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝑊 ∈ V)
10		rellindf 21745	. . . . . . 7 ⊢ Rel LIndF
11	10	brrelex1i 5670	. . . . . 6 ⊢ (𝐹 LIndF 𝑊 → 𝐹 ∈ V)
12	11	ad2antrr 726	. . . . 5 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐹 ∈ V)
13		eqid 2731	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
14		lindfind.s	. . . . . 6 ⊢ · = ( ·𝑠 ‘𝑊)
15		lindfind.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑊)
16		lindfind.z	. . . . . 6 ⊢ 0 = (0g‘𝐿)
17	13, 14, 15, 6, 7, 16	islindf 21749	. . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶(Base‘𝑊) ∧ ∀𝑒 ∈ dom 𝐹∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))))))
18	9, 12, 17	syl2anc 584	. . . 4 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶(Base‘𝑊) ∧ ∀𝑒 ∈ dom 𝐹∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))))))
19	5, 18	mpbid 232	. . 3 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → (𝐹:dom 𝐹⟶(Base‘𝑊) ∧ ∀𝑒 ∈ dom 𝐹∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒})))))
20	19	simprd 495	. 2 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ∀𝑒 ∈ dom 𝐹∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))))
21		fveq2 6822	. . . . . 6 ⊢ (𝑒 = 𝐸 → (𝐹‘𝑒) = (𝐹‘𝐸))
22	21	oveq2d 7362	. . . . 5 ⊢ (𝑒 = 𝐸 → (𝑎 · (𝐹‘𝑒)) = (𝑎 · (𝐹‘𝐸)))
23		sneq 4583	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → {𝑒} = {𝐸})
24	23	difeq2d 4073	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (dom 𝐹 ∖ {𝑒}) = (dom 𝐹 ∖ {𝐸}))
25	24	imaeq2d 6008	. . . . . 6 ⊢ (𝑒 = 𝐸 → (𝐹 “ (dom 𝐹 ∖ {𝑒})) = (𝐹 “ (dom 𝐹 ∖ {𝐸})))
26	25	fveq2d 6826	. . . . 5 ⊢ (𝑒 = 𝐸 → (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) = (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
27	22, 26	eleq12d 2825	. . . 4 ⊢ (𝑒 = 𝐸 → ((𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) ↔ (𝑎 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
28	27	notbid 318	. . 3 ⊢ (𝑒 = 𝐸 → (¬ (𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) ↔ ¬ (𝑎 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
29		oveq1 7353	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 · (𝐹‘𝐸)) = (𝐴 · (𝐹‘𝐸)))
30	29	eleq1d 2816	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑎 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))) ↔ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
31	30	notbid 318	. . 3 ⊢ (𝑎 = 𝐴 → (¬ (𝑎 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))) ↔ ¬ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
32	28, 31	rspc2va 3584	. 2 ⊢ (((𝐸 ∈ dom 𝐹 ∧ 𝐴 ∈ (𝐾 ∖ { 0 })) ∧ ∀𝑒 ∈ dom 𝐹∀𝑎 ∈ (𝐾 ∖ { 0 }) ¬ (𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒})))) → ¬ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
33	1, 4, 20, 32	syl21anc 837	1 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  Vcvv 3436   ∖ cdif 3894  {csn 4573   class class class wbr 5089  dom cdm 5614   “ cima 5617  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  Scalarcsca 17164   ·_𝑠 cvsca 17165  0_gc0g 17343  LSpanclspn 20904   LIndF clindf 21741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-1cn 11064  ax-addcl 11066

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-nn 12126  df-slot 17093  df-ndx 17105  df-base 17121  df-lindf 21743

This theorem is referenced by:  lindfind2  21755  lindfrn  21758  f1lindf  21759