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Theorem islindeps 47633
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 47632 . . 3 ((𝑆 ∈ 𝒫 𝐡 ∧ 𝑀 ∈ π‘Š) β†’ (𝑆 linDepS 𝑀 ↔ Β¬ 𝑆 linIndS 𝑀))
21ancoms 457 . 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 47626 . . . . 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 457 . . . 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 527 . . . . . 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 480 . . . 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 278 . . 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 317 . 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 3090 . . . 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 2931 . . . . . . . . 9 ((π‘“β€˜π‘₯) β‰  0 ↔ Β¬ (π‘“β€˜π‘₯) = 0 )
1716rexbii 3084 . . . . . . . 8 (βˆƒπ‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) β‰  0 ↔ βˆƒπ‘₯ ∈ 𝑆 Β¬ (π‘“β€˜π‘₯) = 0 )
18 rexnal 3090 . . . . . . . 8 (βˆƒπ‘₯ ∈ 𝑆 Β¬ (π‘“β€˜π‘₯) = 0 ↔ Β¬ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 )
1917, 18bitr2i 275 . . . . . . 7 (Β¬ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ↔ βˆƒπ‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) β‰  0 )
2019anbi2i 621 . . . . . 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 pm4.61 403 . . . . . 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 1086 . . . . . 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 302 . . . . 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 3084 . . . 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 276 . . 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 304 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆ€wral 3051  βˆƒwrex 3060  π’« cpw 4598   class class class wbr 5143  β€˜cfv 6543  (class class class)co 7416   ↑m cmap 8843   finSupp cfsupp 9385  Basecbs 17179  Scalarcsca 17235  0gc0g 17420   linC clinc 47584   linIndS clininds 47620   linDepS clindeps 47621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-iota 6495  df-fv 6551  df-ov 7419  df-lininds 47622  df-lindeps 47624
This theorem is referenced by:  el0ldep  47646  ldepspr  47653  islindeps2  47663  isldepslvec2  47665  zlmodzxzldep  47684
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