Theorem islindeps 49487

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 49486	. . 3 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ 𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))
2	1	ancoms 464	. 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 49480	. . . . 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 464	. . . 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 538	. . . . . 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 226	. . . . 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 487	. . . 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 282	. . 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 321	. 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 3114	. . . 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 2956	. . . . . . . . 9 ⊢ ((𝑓‘𝑥) ≠ 0 ↔ ¬ (𝑓‘𝑥) = 0 )
17	16	rexbii 3109	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ 0 ↔ ∃𝑥 ∈ 𝑆 ¬ (𝑓‘𝑥) = 0 )
18		rexnal 3114	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝑆 ¬ (𝑓‘𝑥) = 0 ↔ ¬ ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )
19	17, 18	bitr2i 279	. . . . . . 7 ⊢ (¬ ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ↔ ∃𝑥 ∈ 𝑆 (𝑓‘𝑥) ≠ 0 )
20	19	anbi2i 635	. . . . . 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		pm4.61 410	. . . . . 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 1105	. . . . . 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 306	. . . . 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 3109	. . . 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 280	. . 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 308	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 )))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  𝒫 cpw 4556   class class class wbr 5102  ‘cfv 6527  (class class class)co 7408   ↑_m cmap 8825   finSupp cfsupp 9331  Basecbs 17349  Scalarcsca 17393  0_gc0g 17572   linC clinc 49438   linIndS clininds 49474   linDepS clindeps 49475

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-iota 6483  df-fv 6535  df-ov 7411  df-lininds 49476  df-lindeps 49478

This theorem is used by:  el0ldep  49500  ldepspr  49507  islindeps2  49517  isldepslvec2  49519  zlmodzxzldep  49538