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Theorem islinindfis 44497
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 44494 . 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 1000 . . . . . . 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 8422 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐸m 𝑆) → 𝑓:𝑆𝐸)
98adantl 484 . . . . . . . . . . . 12 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸m 𝑆)) → 𝑓:𝑆𝐸)
10 simpll 765 . . . . . . . . . . . 12 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸m 𝑆)) → 𝑆 ∈ Fin)
115fvexi 6679 . . . . . . . . . . . . 13 0 ∈ V
1211a1i 11 . . . . . . . . . . . 12 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸m 𝑆)) → 0 ∈ V)
139, 10, 12fdmfifsupp 8837 . . . . . . . . . . 11 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸m 𝑆)) → 𝑓 finSupp 0 )
1413adantr 483 . . . . . . . . . 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 418 . . . . . . . 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 853 . . . . . . 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 237 . . . . . 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 pm3.42 496 . . . . 5 (((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))
2220, 21impbid1 227 . . . 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 3196 . . 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 630 . 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 281 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 208  wa 398  wo 843   = wceq 1533  wcel 2110  wral 3138  Vcvv 3495  𝒫 cpw 4539   class class class wbr 5059  wf 6346  cfv 6350  (class class class)co 7150  m cmap 8400  Fincfn 8503   finSupp cfsupp 8827  Basecbs 16477  Scalarcsca 16562  0gc0g 16707   linC clinc 44452   linIndS clininds 44488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-supp 7825  df-er 8283  df-map 8402  df-en 8504  df-fin 8507  df-fsupp 8828  df-lininds 44490
This theorem is referenced by:  islinindfiss  44498
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