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Theorem islindeps 42036
Description: The property of being a linearly dependent subset. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Hypotheses
Ref Expression
islindeps.b 𝐵 = (Base‘𝑀)
islindeps.z 𝑍 = (0g𝑀)
islindeps.r 𝑅 = (Scalar‘𝑀)
islindeps.e 𝐸 = (Base‘𝑅)
islindeps.0 0 = (0g𝑅)
Assertion
Ref Expression
islindeps ((𝑀𝑊𝑆 ∈ 𝒫 𝐵) → (𝑆 linDepS 𝑀 ↔ ∃𝑓 ∈ (𝐸𝑚 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 )))
Distinct variable groups:   𝑓,𝐸   𝑓,𝑀,𝑥   𝑆,𝑓,𝑥
Allowed substitution hints:   𝐵(𝑥,𝑓)   𝑅(𝑥,𝑓)   𝐸(𝑥)   𝑊(𝑥,𝑓)   0 (𝑥,𝑓)   𝑍(𝑥,𝑓)

Proof of Theorem islindeps
StepHypRef Expression
1 lindepsnlininds 42035 . . 3 ((𝑆 ∈ 𝒫 𝐵𝑀𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))
21ancoms 468 . 2 ((𝑀𝑊𝑆 ∈ 𝒫 𝐵) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))
3 islindeps.b . . . . . 6 𝐵 = (Base‘𝑀)
4 islindeps.z . . . . . 6 𝑍 = (0g𝑀)
5 islindeps.r . . . . . 6 𝑅 = (Scalar‘𝑀)
6 islindeps.e . . . . . 6 𝐸 = (Base‘𝑅)
7 islindeps.0 . . . . . 6 0 = (0g𝑅)
83, 4, 5, 6, 7islininds 42029 . . . . 5 ((𝑆 ∈ 𝒫 𝐵𝑀𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
98ancoms 468 . . . 4 ((𝑀𝑊𝑆 ∈ 𝒫 𝐵) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
10 ibar 524 . . . . . 6 (𝑆 ∈ 𝒫 𝐵 → (∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
1110bicomd 212 . . . . 5 (𝑆 ∈ 𝒫 𝐵 → ((𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )) ↔ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
1211adantl 481 . . . 4 ((𝑀𝑊𝑆 ∈ 𝒫 𝐵) → ((𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )) ↔ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
139, 12bitrd 267 . . 3 ((𝑀𝑊𝑆 ∈ 𝒫 𝐵) → (𝑆 linIndS 𝑀 ↔ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
1413notbid 307 . 2 ((𝑀𝑊𝑆 ∈ 𝒫 𝐵) → (¬ 𝑆 linIndS 𝑀 ↔ ¬ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
15 rexnal 2978 . . . 4 (∃𝑓 ∈ (𝐸𝑚 𝑆) ¬ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) ↔ ¬ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))
16 df-ne 2782 . . . . . . . . 9 ((𝑓𝑥) ≠ 0 ↔ ¬ (𝑓𝑥) = 0 )
1716rexbii 3023 . . . . . . . 8 (∃𝑥𝑆 (𝑓𝑥) ≠ 0 ↔ ∃𝑥𝑆 ¬ (𝑓𝑥) = 0 )
18 rexnal 2978 . . . . . . . 8 (∃𝑥𝑆 ¬ (𝑓𝑥) = 0 ↔ ¬ ∀𝑥𝑆 (𝑓𝑥) = 0 )
1917, 18bitr2i 264 . . . . . . 7 (¬ ∀𝑥𝑆 (𝑓𝑥) = 0 ↔ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 )
2019anbi2i 726 . . . . . 6 (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ∧ ¬ ∀𝑥𝑆 (𝑓𝑥) = 0 ) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 ))
21 pm4.61 441 . . . . . 6 (¬ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ∧ ¬ ∀𝑥𝑆 (𝑓𝑥) = 0 ))
22 df-3an 1033 . . . . . 6 ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 ) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 ))
2320, 21, 223bitr4i 291 . . . . 5 (¬ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) ↔ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 ))
2423rexbii 3023 . . . 4 (∃𝑓 ∈ (𝐸𝑚 𝑆) ¬ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) ↔ ∃𝑓 ∈ (𝐸𝑚 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 ))
2515, 24bitr3i 265 . . 3 (¬ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) ↔ ∃𝑓 ∈ (𝐸𝑚 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 ))
2625a1i 11 . 2 ((𝑀𝑊𝑆 ∈ 𝒫 𝐵) → (¬ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) ↔ ∃𝑓 ∈ (𝐸𝑚 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 )))
272, 14, 263bitrd 293 1 ((𝑀𝑊𝑆 ∈ 𝒫 𝐵) → (𝑆 linDepS 𝑀 ↔ ∃𝑓 ∈ (𝐸𝑚 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wral 2896  wrex 2897  𝒫 cpw 4108   class class class wbr 4583  cfv 5804  (class class class)co 6549  𝑚 cmap 7744   finSupp cfsupp 8158  Basecbs 15695  Scalarcsca 15771  0gc0g 15923   linC clinc 41987   linIndS clininds 42023   linDepS clindeps 42024
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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-iota 5768  df-fv 5812  df-ov 6552  df-lininds 42025  df-lindeps 42027
This theorem is referenced by:  el0ldep  42049  ldepspr  42056  islindeps2  42066  isldepslvec2  42068  zlmodzxzldep  42087
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