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Theorem islindeps 45028
 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 45027 . . 3 ((𝑆 ∈ 𝒫 𝐵𝑀𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))
21ancoms 462 . 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 45021 . . . . 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 462 . . . 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 532 . . . . . 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 226 . . . . 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 485 . . . 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 282 . . 3 ((𝑀𝑊𝑆 ∈ 𝒫 𝐵) → (𝑆 linIndS 𝑀 ↔ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
1413notbid 321 . 2 ((𝑀𝑊𝑆 ∈ 𝒫 𝐵) → (¬ 𝑆 linIndS 𝑀 ↔ ¬ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
15 rexnal 3201 . . . 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 2988 . . . . . . . . 9 ((𝑓𝑥) ≠ 0 ↔ ¬ (𝑓𝑥) = 0 )
1716rexbii 3211 . . . . . . . 8 (∃𝑥𝑆 (𝑓𝑥) ≠ 0 ↔ ∃𝑥𝑆 ¬ (𝑓𝑥) = 0 )
18 rexnal 3201 . . . . . . . 8 (∃𝑥𝑆 ¬ (𝑓𝑥) = 0 ↔ ¬ ∀𝑥𝑆 (𝑓𝑥) = 0 )
1917, 18bitr2i 279 . . . . . . 7 (¬ ∀𝑥𝑆 (𝑓𝑥) = 0 ↔ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 )
2019anbi2i 625 . . . . . 6 (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ∧ ¬ ∀𝑥𝑆 (𝑓𝑥) = 0 ) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 ))
21 pm4.61 408 . . . . . 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 1086 . . . . . 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 306 . . . . 5 (¬ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) ↔ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 ))
2423rexbii 3211 . . . 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 280 . . 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 308 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  𝒫 cpw 4500   class class class wbr 5034  ‘cfv 6332  (class class class)co 7145   ↑m cmap 8407   finSupp cfsupp 8835  Basecbs 16495  Scalarcsca 16580  0gc0g 16725   linC clinc 44979   linIndS clininds 45015   linDepS clindeps 45016 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-iota 6291  df-fv 6340  df-ov 7148  df-lininds 45017  df-lindeps 45019 This theorem is referenced by:  el0ldep  45041  ldepspr  45048  islindeps2  45058  isldepslvec2  45060  zlmodzxzldep  45079
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