Theorem islindeps 47409

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 = (0g‘𝑀)
islindeps.r	⊢ 𝑅 = (Scalar‘𝑀)
islindeps.e	⊢ 𝐸 = (Base‘𝑅)
islindeps.0	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
islindeps	⊢ ((𝑀 ∈ 𝑊 ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑆 linDepS 𝑀 ↔ ∃𝑓 ∈ (𝐸 ↑m 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ 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

Step	Hyp	Ref	Expression
1		lindepsnlininds 47408	. . 3 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ 𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))
2	1	ancoms 458	. 2 ⊢ ((𝑀 ∈ 𝑊 ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))
3		islindeps.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
4		islindeps.z	. . . . . 6 ⊢ 𝑍 = (0g‘𝑀)
5		islindeps.r	. . . . . 6 ⊢ 𝑅 = (Scalar‘𝑀)
6		islindeps.e	. . . . . 6 ⊢ 𝐸 = (Base‘𝑅)
7		islindeps.0	. . . . . 6 ⊢ 0 = (0g‘𝑅)
8	3, 4, 5, 6, 7	islininds 47402	. . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ 𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))))
9	8	ancoms 458	. . . 4 ⊢ ((𝑀 ∈ 𝑊 ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))))
10		ibar 528	. . . . . 6 ⊢ (𝑆 ∈ 𝒫 𝐵 → (∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))))
11	10	bicomd 222	. . . . 5 ⊢ (𝑆 ∈ 𝒫 𝐵 → ((𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )) ↔ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
12	11	adantl 481	. . . 4 ⊢ ((𝑀 ∈ 𝑊 ∧ 𝑆 ∈ 𝒫 𝐵) → ((𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )) ↔ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
13	9, 12	bitrd 279	. . 3 ⊢ ((𝑀 ∈ 𝑊 ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑆 linIndS 𝑀 ↔ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
14	13	notbid 318	. 2 ⊢ ((𝑀 ∈ 𝑊 ∧ 𝑆 ∈ 𝒫 𝐵) → (¬ 𝑆 linIndS 𝑀 ↔ ¬ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
15		rexnal 3094	. . . 4 ⊢ (∃𝑓 ∈ (𝐸 ↑m 𝑆) ¬ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) ↔ ¬ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))
16		df-ne 2935	. . . . . . . . 9 ⊢ ((𝑓‘𝑥) ≠ 0 ↔ ¬ (𝑓‘𝑥) = 0 )
17	16	rexbii 3088	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ 0 ↔ ∃𝑥 ∈ 𝑆 ¬ (𝑓‘𝑥) = 0 )
18		rexnal 3094	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝑆 ¬ (𝑓‘𝑥) = 0 ↔ ¬ ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )
19	17, 18	bitr2i 276	. . . . . . 7 ⊢ (¬ ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ↔ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ 0 )
20	19	anbi2i 622	. . . . . 6 ⊢ (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ∧ ¬ ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ∧ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ 0 ))
21		pm4.61 404	. . . . . 6 ⊢ (¬ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ∧ ¬ ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))
22		df-3an 1086	. . . . . 6 ⊢ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ 0 ) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ∧ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ 0 ))
23	20, 21, 22	3bitr4i 303	. . . . 5 ⊢ (¬ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) ↔ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ 0 ))
24	23	rexbii 3088	. . . 4 ⊢ (∃𝑓 ∈ (𝐸 ↑m 𝑆) ¬ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) ↔ ∃𝑓 ∈ (𝐸 ↑m 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ 0 ))
25	15, 24	bitr3i 277	. . 3 ⊢ (¬ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) ↔ ∃𝑓 ∈ (𝐸 ↑m 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ 0 ))
26	25	a1i 11	. 2 ⊢ ((𝑀 ∈ 𝑊 ∧ 𝑆 ∈ 𝒫 𝐵) → (¬ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) ↔ ∃𝑓 ∈ (𝐸 ↑m 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ 0 )))
27	2, 14, 26	3bitrd 305	1 ⊢ ((𝑀 ∈ 𝑊 ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑆 linDepS 𝑀 ↔ ∃𝑓 ∈ (𝐸 ↑m 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ 0 )))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2934  ∀wral 3055  ∃wrex 3064  𝒫 cpw 4597   class class class wbr 5141  ‘cfv 6537  (class class class)co 7405   ↑_m cmap 8822   finSupp cfsupp 9363  Basecbs 17153  Scalarcsca 17209  0_gc0g 17394   linC clinc 47360   linIndS clininds 47396   linDepS clindeps 47397

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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-iota 6489  df-fv 6545  df-ov 7408  df-lininds 47398  df-lindeps 47400

This theorem is referenced by:  el0ldep  47422  ldepspr  47429  islindeps2  47439  isldepslvec2  47441  zlmodzxzldep  47460