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Theorem islinindfis 43246
 Description: The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.)
Hypotheses
Ref Expression
islininds.b 𝐵 = (Base‘𝑀)
islininds.z 𝑍 = (0g𝑀)
islininds.r 𝑅 = (Scalar‘𝑀)
islininds.e 𝐸 = (Base‘𝑅)
islininds.0 0 = (0g𝑅)
Assertion
Ref Expression
islinindfis ((𝑆 ∈ Fin ∧ 𝑀𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
Distinct variable groups:   𝑓,𝐸   𝑓,𝑀,𝑥   𝑆,𝑓,𝑥   0 ,𝑓   𝑓,𝑍   𝑓,𝑊
Allowed substitution hints:   𝐵(𝑥,𝑓)   𝑅(𝑥,𝑓)   𝐸(𝑥)   𝑊(𝑥)   0 (𝑥)   𝑍(𝑥)

Proof of Theorem islinindfis
StepHypRef Expression
1 islininds.b . . 3 𝐵 = (Base‘𝑀)
2 islininds.z . . 3 𝑍 = (0g𝑀)
3 islininds.r . . 3 𝑅 = (Scalar‘𝑀)
4 islininds.e . . 3 𝐸 = (Base‘𝑅)
5 islininds.0 . . 3 0 = (0g𝑅)
61, 2, 3, 4, 5islininds 43243 . 2 ((𝑆 ∈ Fin ∧ 𝑀𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
7 pm4.79 989 . . . . . . 7 (((𝑓 finSupp 0 → ∀𝑥𝑆 (𝑓𝑥) = 0 ) ∨ ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 )) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))
8 elmapi 8162 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐸𝑚 𝑆) → 𝑓:𝑆𝐸)
98adantl 475 . . . . . . . . . . . 12 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → 𝑓:𝑆𝐸)
10 simpll 757 . . . . . . . . . . . 12 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → 𝑆 ∈ Fin)
115fvexi 6460 . . . . . . . . . . . . 13 0 ∈ V
1211a1i 11 . . . . . . . . . . . 12 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → 0 ∈ V)
139, 10, 12fdmfifsupp 8573 . . . . . . . . . . 11 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → 𝑓 finSupp 0 )
1413adantr 474 . . . . . . . . . 10 ((((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → 𝑓 finSupp 0 )
1514imim1i 63 . . . . . . . . 9 ((𝑓 finSupp 0 → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → ((((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))
1615expd 406 . . . . . . . 8 ((𝑓 finSupp 0 → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
17 ax-1 6 . . . . . . . 8 (((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
1816, 17jaoi 846 . . . . . . 7 (((𝑓 finSupp 0 → ∀𝑥𝑆 (𝑓𝑥) = 0 ) ∨ ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 )) → (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
197, 18sylbir 227 . . . . . 6 (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
2019com12 32 . . . . 5 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
21 pm3.42 489 . . . . 5 (((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))
2220, 21impbid1 217 . . . 4 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) ↔ ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
2322ralbidva 3166 . . 3 ((𝑆 ∈ Fin ∧ 𝑀𝑊) → (∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) ↔ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
2423anbi2d 622 . 2 ((𝑆 ∈ Fin ∧ 𝑀𝑊) → ((𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )) ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
256, 24bitrd 271 1 ((𝑆 ∈ Fin ∧ 𝑀𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 836   = wceq 1601   ∈ wcel 2106  ∀wral 3089  Vcvv 3397  𝒫 cpw 4378   class class class wbr 4886  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922   ↑𝑚 cmap 8140  Fincfn 8241   finSupp cfsupp 8563  Basecbs 16255  Scalarcsca 16341  0gc0g 16486   linC clinc 43201   linIndS clininds 43237 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-supp 7577  df-er 8026  df-map 8142  df-en 8242  df-fin 8245  df-fsupp 8564  df-lininds 43239 This theorem is referenced by:  islinindfiss  43247
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