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Theorem islindeps 47134
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 𝐡 = (Baseβ€˜π‘€)
islindeps.z 𝑍 = (0gβ€˜π‘€)
islindeps.r 𝑅 = (Scalarβ€˜π‘€)
islindeps.e 𝐸 = (Baseβ€˜π‘…)
islindeps.0 0 = (0gβ€˜π‘…)
Assertion
Ref Expression
islindeps ((𝑀 ∈ π‘Š ∧ 𝑆 ∈ 𝒫 𝐡) β†’ (𝑆 linDepS 𝑀 ↔ βˆƒπ‘“ ∈ (𝐸 ↑m 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍 ∧ βˆƒπ‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) β‰  0 )))
Distinct variable groups:   𝑓,𝐸   𝑓,𝑀,π‘₯   𝑆,𝑓,π‘₯
Allowed substitution hints:   𝐡(π‘₯,𝑓)   𝑅(π‘₯,𝑓)   𝐸(π‘₯)   π‘Š(π‘₯,𝑓)   0 (π‘₯,𝑓)   𝑍(π‘₯,𝑓)

Proof of Theorem islindeps
StepHypRef Expression
1 lindepsnlininds 47133 . . 3 ((𝑆 ∈ 𝒫 𝐡 ∧ 𝑀 ∈ π‘Š) β†’ (𝑆 linDepS 𝑀 ↔ Β¬ 𝑆 linIndS 𝑀))
21ancoms 460 . 2 ((𝑀 ∈ π‘Š ∧ 𝑆 ∈ 𝒫 𝐡) β†’ (𝑆 linDepS 𝑀 ↔ Β¬ 𝑆 linIndS 𝑀))
3 islindeps.b . . . . . 6 𝐡 = (Baseβ€˜π‘€)
4 islindeps.z . . . . . 6 𝑍 = (0gβ€˜π‘€)
5 islindeps.r . . . . . 6 𝑅 = (Scalarβ€˜π‘€)
6 islindeps.e . . . . . 6 𝐸 = (Baseβ€˜π‘…)
7 islindeps.0 . . . . . 6 0 = (0gβ€˜π‘…)
83, 4, 5, 6, 7islininds 47127 . . . . 5 ((𝑆 ∈ 𝒫 𝐡 ∧ 𝑀 ∈ π‘Š) β†’ (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐡 ∧ βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))))
98ancoms 460 . . . 4 ((𝑀 ∈ π‘Š ∧ 𝑆 ∈ 𝒫 𝐡) β†’ (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐡 ∧ βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))))
10 ibar 530 . . . . . 6 (𝑆 ∈ 𝒫 𝐡 β†’ (βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) ↔ (𝑆 ∈ 𝒫 𝐡 ∧ βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))))
1110bicomd 222 . . . . 5 (𝑆 ∈ 𝒫 𝐡 β†’ ((𝑆 ∈ 𝒫 𝐡 ∧ βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 )) ↔ βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 )))
1211adantl 483 . . . 4 ((𝑀 ∈ π‘Š ∧ 𝑆 ∈ 𝒫 𝐡) β†’ ((𝑆 ∈ 𝒫 𝐡 ∧ βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 )) ↔ βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 )))
139, 12bitrd 279 . . 3 ((𝑀 ∈ π‘Š ∧ 𝑆 ∈ 𝒫 𝐡) β†’ (𝑆 linIndS 𝑀 ↔ βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 )))
1413notbid 318 . 2 ((𝑀 ∈ π‘Š ∧ 𝑆 ∈ 𝒫 𝐡) β†’ (Β¬ 𝑆 linIndS 𝑀 ↔ Β¬ βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 )))
15 rexnal 3101 . . . 4 (βˆƒπ‘“ ∈ (𝐸 ↑m 𝑆) Β¬ ((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) ↔ Β¬ βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))
16 df-ne 2942 . . . . . . . . 9 ((π‘“β€˜π‘₯) β‰  0 ↔ Β¬ (π‘“β€˜π‘₯) = 0 )
1716rexbii 3095 . . . . . . . 8 (βˆƒπ‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) β‰  0 ↔ βˆƒπ‘₯ ∈ 𝑆 Β¬ (π‘“β€˜π‘₯) = 0 )
18 rexnal 3101 . . . . . . . 8 (βˆƒπ‘₯ ∈ 𝑆 Β¬ (π‘“β€˜π‘₯) = 0 ↔ Β¬ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 )
1917, 18bitr2i 276 . . . . . . 7 (Β¬ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ↔ βˆƒπ‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) β‰  0 )
2019anbi2i 624 . . . . . 6 (((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) ∧ Β¬ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) ∧ βˆƒπ‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) β‰  0 ))
21 pm4.61 406 . . . . . 6 (Β¬ ((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) ∧ Β¬ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ))
22 df-3an 1090 . . . . . 6 ((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍 ∧ βˆƒπ‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) β‰  0 ) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) ∧ βˆƒπ‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) β‰  0 ))
2320, 21, 223bitr4i 303 . . . . 5 (Β¬ ((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) ↔ (𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍 ∧ βˆƒπ‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) β‰  0 ))
2423rexbii 3095 . . . 4 (βˆƒπ‘“ ∈ (𝐸 ↑m 𝑆) Β¬ ((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) ↔ βˆƒπ‘“ ∈ (𝐸 ↑m 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍 ∧ βˆƒπ‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) β‰  0 ))
2515, 24bitr3i 277 . . 3 (Β¬ βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) ↔ βˆƒπ‘“ ∈ (𝐸 ↑m 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍 ∧ βˆƒπ‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) β‰  0 ))
2625a1i 11 . 2 ((𝑀 ∈ π‘Š ∧ 𝑆 ∈ 𝒫 𝐡) β†’ (Β¬ βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) ↔ βˆƒπ‘“ ∈ (𝐸 ↑m 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍 ∧ βˆƒπ‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) β‰  0 )))
272, 14, 263bitrd 305 1 ((𝑀 ∈ π‘Š ∧ 𝑆 ∈ 𝒫 𝐡) β†’ (𝑆 linDepS 𝑀 ↔ βˆƒπ‘“ ∈ (𝐸 ↑m 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍 ∧ βˆƒπ‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) β‰  0 )))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  π’« cpw 4603   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820   finSupp cfsupp 9361  Basecbs 17144  Scalarcsca 17200  0gc0g 17385   linC clinc 47085   linIndS clininds 47121   linDepS clindeps 47122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-iota 6496  df-fv 6552  df-ov 7412  df-lininds 47123  df-lindeps 47125
This theorem is referenced by:  el0ldep  47147  ldepspr  47154  islindeps2  47164  isldepslvec2  47166  zlmodzxzldep  47185
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