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Theorem islinindfis 45790
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 45787 . 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 1001 . . . . . . 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 8637 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐸m 𝑆) → 𝑓:𝑆𝐸)
98adantl 482 . . . . . . . . . . . 12 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸m 𝑆)) → 𝑓:𝑆𝐸)
10 simpll 764 . . . . . . . . . . . 12 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸m 𝑆)) → 𝑆 ∈ Fin)
115fvexi 6788 . . . . . . . . . . . . 13 0 ∈ V
1211a1i 11 . . . . . . . . . . . 12 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸m 𝑆)) → 0 ∈ V)
139, 10, 12fdmfifsupp 9138 . . . . . . . . . . 11 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸m 𝑆)) → 𝑓 finSupp 0 )
1413adantr 481 . . . . . . . . . 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 416 . . . . . . . 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 854 . . . . . . 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 234 . . . . . 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 pm3.42 494 . . . . 5 (((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))
2220, 21impbid1 224 . . . 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 3111 . . 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 629 . 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 278 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 205  wa 396  wo 844   = wceq 1539  wcel 2106  wral 3064  Vcvv 3432  𝒫 cpw 4533   class class class wbr 5074  wf 6429  cfv 6433  (class class class)co 7275  m cmap 8615  Fincfn 8733   finSupp cfsupp 9128  Basecbs 16912  Scalarcsca 16965  0gc0g 17150   linC clinc 45745   linIndS clininds 45781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-1o 8297  df-map 8617  df-en 8734  df-fin 8737  df-fsupp 9129  df-lininds 45783
This theorem is referenced by:  islinindfiss  45791
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