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Theorem islindf2 20886
Description: Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.)
Hypotheses
Ref Expression
islindf.b 𝐵 = (Base‘𝑊)
islindf.v · = ( ·𝑠𝑊)
islindf.k 𝐾 = (LSpan‘𝑊)
islindf.s 𝑆 = (Scalar‘𝑊)
islindf.n 𝑁 = (Base‘𝑆)
islindf.z 0 = (0g𝑆)
Assertion
Ref Expression
islindf2 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥𝐼𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥})))))
Distinct variable groups:   𝑘,𝐹,𝑥   𝑘,𝑁   𝑘,𝑊,𝑥   0 ,𝑘   𝐵,𝑘,𝑥   𝑘,𝐼,𝑥   𝑘,𝑋,𝑥   𝑘,𝑌,𝑥
Allowed substitution hints:   𝑆(𝑥,𝑘)   · (𝑥,𝑘)   𝐾(𝑥,𝑘)   𝑁(𝑥)   0 (𝑥)

Proof of Theorem islindf2
StepHypRef Expression
1 simp1 1128 . . 3 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → 𝑊𝑌)
2 simp3 1130 . . . 4 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → 𝐹:𝐼𝐵)
3 simp2 1129 . . . 4 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → 𝐼𝑋)
4 fex 6980 . . . 4 ((𝐹:𝐼𝐵𝐼𝑋) → 𝐹 ∈ V)
52, 3, 4syl2anc 584 . . 3 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → 𝐹 ∈ V)
6 islindf.b . . . 4 𝐵 = (Base‘𝑊)
7 islindf.v . . . 4 · = ( ·𝑠𝑊)
8 islindf.k . . . 4 𝐾 = (LSpan‘𝑊)
9 islindf.s . . . 4 𝑆 = (Scalar‘𝑊)
10 islindf.n . . . 4 𝑁 = (Base‘𝑆)
11 islindf.z . . . 4 0 = (0g𝑆)
126, 7, 8, 9, 10, 11islindf 20884 . . 3 ((𝑊𝑌𝐹 ∈ V) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
131, 5, 12syl2anc 584 . 2 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
14 ffdm 6529 . . . . 5 (𝐹:𝐼𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐼))
1514simpld 495 . . . 4 (𝐹:𝐼𝐵𝐹:dom 𝐹𝐵)
16153ad2ant3 1127 . . 3 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → 𝐹:dom 𝐹𝐵)
1716biantrurd 533 . 2 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (∀𝑥 ∈ dom 𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
18 fdm 6515 . . . 4 (𝐹:𝐼𝐵 → dom 𝐹 = 𝐼)
19183ad2ant3 1127 . . 3 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → dom 𝐹 = 𝐼)
2019difeq1d 4095 . . . . . . . 8 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (dom 𝐹 ∖ {𝑥}) = (𝐼 ∖ {𝑥}))
2120imaeq2d 5922 . . . . . . 7 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (𝐹 “ (dom 𝐹 ∖ {𝑥})) = (𝐹 “ (𝐼 ∖ {𝑥})))
2221fveq2d 6667 . . . . . 6 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) = (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥}))))
2322eleq2d 2895 . . . . 5 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → ((𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥})))))
2423notbid 319 . . . 4 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥})))))
2524ralbidv 3194 . . 3 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥})))))
2619, 25raleqbidv 3399 . 2 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (∀𝑥 ∈ dom 𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ ∀𝑥𝐼𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥})))))
2713, 17, 263bitr2d 308 1 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥𝐼𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥})))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cdif 3930  wss 3933  {csn 4557   class class class wbr 5057  dom cdm 5548  cima 5551  wf 6344  cfv 6348  (class class class)co 7145  Basecbs 16471  Scalarcsca 16556   ·𝑠 cvsca 16557  0gc0g 16701  LSpanclspn 19672   LIndF clindf 20876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-lindf 20878
This theorem is referenced by:  lindfmm  20899  islindf4  20910
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