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Theorem islinindfis 42032
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 42029 . 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 pm4.79 605 . . . . . . 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 7765 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐸𝑚 𝑆) → 𝑓:𝑆𝐸)
98adantl 481 . . . . . . . . . . . 12 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → 𝑓:𝑆𝐸)
10 simpll 786 . . . . . . . . . . . 12 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → 𝑆 ∈ Fin)
11 fvex 6113 . . . . . . . . . . . . . 14 (0g𝑅) ∈ V
125, 11eqeltri 2684 . . . . . . . . . . . . 13 0 ∈ V
1312a1i 11 . . . . . . . . . . . 12 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → 0 ∈ V)
149, 10, 13fdmfifsupp 8168 . . . . . . . . . . 11 (((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) → 𝑓 finSupp 0 )
1514adantr 480 . . . . . . . . . 10 ((((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → 𝑓 finSupp 0 )
1615imim1i 61 . . . . . . . . 9 ((𝑓 finSupp 0 → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → ((((𝑆 ∈ Fin ∧ 𝑀𝑊) ∧ 𝑓 ∈ (𝐸𝑚 𝑆)) ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))
1716expd 451 . . . . . . . 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 393 . . . . . . 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 224 . . . . . 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 pm3.42 581 . . . . 5 (((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))
2321, 22impbid1 214 . . . 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 2968 . . 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 736 . 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 267 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 195  wo 382  wa 383   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  𝒫 cpw 4108   class class class wbr 4583  wf 5800  cfv 5804  (class class class)co 6549  𝑚 cmap 7744  Fincfn 7841   finSupp cfsupp 8158  Basecbs 15695  Scalarcsca 15771  0gc0g 15923   linC clinc 41987   linIndS clininds 42023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-er 7629  df-map 7746  df-en 7842  df-fin 7845  df-fsupp 8159  df-lininds 42025
This theorem is referenced by:  islinindfiss  42033
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