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Theorem islindeps 41527
 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 41526 . . 3 ((𝑆 ∈ 𝒫 𝐵𝑀𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))
21ancoms 469 . 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 41520 . . . . 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 469 . . . 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 525 . . . . . 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 213 . . . . 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 482 . . . 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 268 . . 3 ((𝑀𝑊𝑆 ∈ 𝒫 𝐵) → (𝑆 linIndS 𝑀 ↔ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
1413notbid 308 . 2 ((𝑀𝑊𝑆 ∈ 𝒫 𝐵) → (¬ 𝑆 linIndS 𝑀 ↔ ¬ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
15 rexnal 2989 . . . 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 2791 . . . . . . . . 9 ((𝑓𝑥) ≠ 0 ↔ ¬ (𝑓𝑥) = 0 )
1716rexbii 3034 . . . . . . . 8 (∃𝑥𝑆 (𝑓𝑥) ≠ 0 ↔ ∃𝑥𝑆 ¬ (𝑓𝑥) = 0 )
18 rexnal 2989 . . . . . . . 8 (∃𝑥𝑆 ¬ (𝑓𝑥) = 0 ↔ ¬ ∀𝑥𝑆 (𝑓𝑥) = 0 )
1917, 18bitr2i 265 . . . . . . 7 (¬ ∀𝑥𝑆 (𝑓𝑥) = 0 ↔ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 )
2019anbi2i 729 . . . . . 6 (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ∧ ¬ ∀𝑥𝑆 (𝑓𝑥) = 0 ) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 ))
21 pm4.61 442 . . . . . 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 1038 . . . . . 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 292 . . . . 5 (¬ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) ↔ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 ))
2423rexbii 3034 . . . 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 266 . . 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 294 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 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  ∃wrex 2908  𝒫 cpw 4130   class class class wbr 4613  ‘cfv 5847  (class class class)co 6604   ↑𝑚 cmap 7802   finSupp cfsupp 8219  Basecbs 15781  Scalarcsca 15865  0gc0g 16021   linC clinc 41478   linIndS clininds 41514   linDepS clindeps 41515 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-iota 5810  df-fv 5855  df-ov 6607  df-lininds 41516  df-lindeps 41518 This theorem is referenced by:  el0ldep  41540  ldepspr  41547  islindeps2  41557  isldepslvec2  41559  zlmodzxzldep  41578
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