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Theorem islindf2 20355
 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 1131 . . 3 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → 𝑊𝑌)
2 simp3 1133 . . . 4 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → 𝐹:𝐼𝐵)
3 simp2 1132 . . . 4 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → 𝐼𝑋)
4 fex 6653 . . . 4 ((𝐹:𝐼𝐵𝐼𝑋) → 𝐹 ∈ V)
52, 3, 4syl2anc 696 . . 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 20353 . . 3 ((𝑊𝑌𝐹 ∈ V) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
131, 5, 12syl2anc 696 . 2 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
14 ffdm 6223 . . . . 5 (𝐹:𝐼𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐼))
1514simpld 477 . . . 4 (𝐹:𝐼𝐵𝐹:dom 𝐹𝐵)
16153ad2ant3 1130 . . 3 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → 𝐹:dom 𝐹𝐵)
1716biantrurd 530 . 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 6212 . . . 4 (𝐹:𝐼𝐵 → dom 𝐹 = 𝐼)
19183ad2ant3 1130 . . 3 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → dom 𝐹 = 𝐼)
2019difeq1d 3870 . . . . . . . 8 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (dom 𝐹 ∖ {𝑥}) = (𝐼 ∖ {𝑥}))
2120imaeq2d 5624 . . . . . . 7 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (𝐹 “ (dom 𝐹 ∖ {𝑥})) = (𝐹 “ (𝐼 ∖ {𝑥})))
2221fveq2d 6356 . . . . . 6 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) = (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥}))))
2322eleq2d 2825 . . . . 5 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → ((𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥})))))
2423notbid 307 . . . 4 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥})))))
2524ralbidv 3124 . . 3 ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥})))))
2619, 25raleqbidv 3291 . 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 296 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 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∀wral 3050  Vcvv 3340   ∖ cdif 3712   ⊆ wss 3715  {csn 4321   class class class wbr 4804  dom cdm 5266   “ cima 5269  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813  Basecbs 16059  Scalarcsca 16146   ·𝑠 cvsca 16147  0gc0g 16302  LSpanclspn 19173   LIndF clindf 20345 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-lindf 20347 This theorem is referenced by:  lindfmm  20368  islindf4  20379
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