Theorem lindfind 22122

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 781	. 2 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐸 ∈ dom 𝐹)
2		eldifsn 4747	. . 3 ⊢ (𝐴 ∈ (𝐾 ∖ { 0 }) ↔ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 ))
3	2	bilanri 512	. 2 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐴 ∈ (𝐾 ∖ { 0 }))
4		simpll 779	. . . 4 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐹 LIndF 𝑊)
5		lindfind.l	. . . . . . 7 ⊢ 𝐿 = (Scalar‘𝑊)
6		lindfind.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐿)
7	5, 6	elbasfv 17393	. . . . . 6 ⊢ (𝐴 ∈ 𝐾 → 𝑊 ∈ V)
8	7	ad2antrl 741	. . . . 5 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝑊 ∈ V)
9		rellindf 22114	. . . . . . 7 ⊢ Rel LIndF
10	9	brrelex1i 5706	. . . . . 6 ⊢ (𝐹 LIndF 𝑊 → 𝐹 ∈ V)
11	10	ad2antrr 739	. . . . 5 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → 𝐹 ∈ V)
12		eqid 2760	. . . . . 6 ⊢ (Base‘𝑊) = (Base‘𝑊)
13		lindfind.s	. . . . . 6 ⊢ · = ( ·𝑠 ‘𝑊)
14		lindfind.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑊)
15		lindfind.z	. . . . . 6 ⊢ 0 = (0g‘𝐿)
16	12, 13, 14, 5, 6, 15	islindf 22118	. . . . 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}))))))
17	8, 11, 16	syl2anc 596	. . . 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}))))))
18	4, 17	mpbid 235	. . 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})))))
19	18	simprd 501	. 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}))))
20		fveq2 6885	. . . . . 6 ⊢ (𝑒 = 𝐸 → (𝐹‘𝑒) = (𝐹‘𝐸))
21	20	oveq2d 7436	. . . . 5 ⊢ (𝑒 = 𝐸 → (𝑎 · (𝐹‘𝑒)) = (𝑎 · (𝐹‘𝐸)))
22		sneq 4593	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → {𝑒} = {𝐸})
23	22	difeq2d 4063	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (dom 𝐹 ∖ {𝑒}) = (dom 𝐹 ∖ {𝐸}))
24	23	imaeq2d 6051	. . . . . 6 ⊢ (𝑒 = 𝐸 → (𝐹 “ (dom 𝐹 ∖ {𝑒})) = (𝐹 “ (dom 𝐹 ∖ {𝐸})))
25	24	fveq2d 6889	. . . . 5 ⊢ (𝑒 = 𝐸 → (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) = (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
26	21, 25	eleq12d 2854	. . . 4 ⊢ (𝑒 = 𝐸 → ((𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) ↔ (𝑎 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
27	26	notbid 321	. . 3 ⊢ (𝑒 = 𝐸 → (¬ (𝑎 · (𝐹‘𝑒)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝑒}))) ↔ ¬ (𝑎 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
28		oveq1 7427	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 · (𝐹‘𝐸)) = (𝐴 · (𝐹‘𝐸)))
29	28	eleq1d 2845	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑎 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))) ↔ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
30	29	notbid 321	. . 3 ⊢ (𝑎 = 𝐴 → (¬ (𝑎 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))) ↔ ¬ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))))
31	27, 30	rspc2va 3587	. 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}))))
32	1, 3, 19, 31	syl21anc 851	1 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  Vcvv 3450   ∖ cdif 4045  {csn 4583   class class class wbr 5102  dom cdm 5650   “ cima 5653  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  Scalarcsca 17431   ·_𝑠 cvsca 17432  0_gc0g 17610  LSpanclspn 21246   LIndF clindf 22110

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-1cn 11258  ax-addcl 11260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-nn 12336  df-slot 17360  df-ndx 17372  df-base 17388  df-lindf 22112

This theorem is used by:  lindfind2  22124  lindfrn  22127  f1lindf  22128