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Theorem islinindfis 42756
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 42753 . 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 pm4.79 932 . . . . . . 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 8030 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐸𝑚 𝑆) → 𝑓:𝑆𝐸)
98adantl 467 . . . . . . . . . . . 12 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → 𝑓:𝑆𝐸)
10 simpll 742 . . . . . . . . . . . 12 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → 𝑆 ∈ Fin)
11 fvex 6342 . . . . . . . . . . . . . 14 (0g𝑅) ∈ V
125, 11eqeltri 2845 . . . . . . . . . . . . 13 0 ∈ V
1312a1i 11 . . . . . . . . . . . 12 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → 0 ∈ V)
149, 10, 13fdmfifsupp 8440 . . . . . . . . . . 11 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → 𝑓 finSupp 0 )
1514adantr 466 . . . . . . . . . 10 ((((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → 𝑓 finSupp 0 )
1615imim1i 63 . . . . . . . . 9 ((𝑓 finSupp 0 → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → ((((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))
1716expd 400 . . . . . . . 8 ((𝑓 finSupp 0 → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
18 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 )))
1917, 18jaoi 837 . . . . . . 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 )))
207, 19sylbir 225 . . . . . 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 )))
2120com12 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 )))
22 pm3.42 813 . . . . 5 (((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))
2321, 22impbid1 215 . . . 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 )))
2423ralbidva 3133 . . 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 )))
2524anbi2d 606 . 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 ))))
266, 25bitrd 268 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 196  wa 382  wo 826   = wceq 1630  wcel 2144  wral 3060  Vcvv 3349  𝒫 cpw 4295   class class class wbr 4784  wf 6027  cfv 6031  (class class class)co 6792  𝑚 cmap 8008  Fincfn 8108   finSupp cfsupp 8430  Basecbs 16063  Scalarcsca 16151  0gc0g 16307   linC clinc 42711   linIndS clininds 42747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-supp 7446  df-er 7895  df-map 8010  df-en 8109  df-fin 8112  df-fsupp 8431  df-lininds 42749
This theorem is referenced by:  islinindfiss  42757
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