Theorem islininds 45460

Description: The property of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)

Hypotheses

Ref	Expression
islininds.b	⊢ 𝐵 = (Base‘𝑀)
islininds.z	⊢ 𝑍 = (0g‘𝑀)
islininds.r	⊢ 𝑅 = (Scalar‘𝑀)
islininds.e	⊢ 𝐸 = (Base‘𝑅)
islininds.0	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
islininds	⊢ ((𝑆 ∈ 𝑉 ∧ 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 ))))

Distinct variable groups:   𝑓,𝐸   𝑓,𝑀,𝑥   𝑆,𝑓,𝑥

Allowed substitution hints:   𝐵(𝑥,𝑓)   𝑅(𝑥,𝑓)   𝐸(𝑥)   𝑉(𝑥,𝑓)   𝑊(𝑥,𝑓)   0 (𝑥,𝑓)   𝑍(𝑥,𝑓)

Proof of Theorem islininds

Dummy variables 𝑚 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 486	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝑠 = 𝑆)
2		fveq2 6717	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3		islininds.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑀)
4	2, 3	eqtr4di 2796	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
5	4	adantl 485	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (Base‘𝑚) = 𝐵)
6	5	pweqd 4532	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝒫 (Base‘𝑚) = 𝒫 𝐵)
7	1, 6	eleq12d 2832	. . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (𝑠 ∈ 𝒫 (Base‘𝑚) ↔ 𝑆 ∈ 𝒫 𝐵))
8		fveq2 6717	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (Scalar‘𝑚) = (Scalar‘𝑀))
9		islininds.r	. . . . . . . . 9 ⊢ 𝑅 = (Scalar‘𝑀)
10	8, 9	eqtr4di 2796	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (Scalar‘𝑚) = 𝑅)
11	10	fveq2d 6721	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘𝑅))
12		islininds.e	. . . . . . 7 ⊢ 𝐸 = (Base‘𝑅)
13	11, 12	eqtr4di 2796	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = 𝐸)
14	13	adantl 485	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (Base‘(Scalar‘𝑚)) = 𝐸)
15	14, 1	oveq12d 7231	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ((Base‘(Scalar‘𝑚)) ↑m 𝑠) = (𝐸 ↑m 𝑆))
16	8	adantl 485	. . . . . . . . . 10 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (Scalar‘𝑚) = (Scalar‘𝑀))
17	16, 9	eqtr4di 2796	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (Scalar‘𝑚) = 𝑅)
18	17	fveq2d 6721	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (0g‘(Scalar‘𝑚)) = (0g‘𝑅))
19		islininds.0	. . . . . . . 8 ⊢ 0 = (0g‘𝑅)
20	18, 19	eqtr4di 2796	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (0g‘(Scalar‘𝑚)) = 0 )
21	20	breq2d 5065	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (𝑓 finSupp (0g‘(Scalar‘𝑚)) ↔ 𝑓 finSupp 0 ))
22		fveq2 6717	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → ( linC ‘𝑚) = ( linC ‘𝑀))
23	22	adantl 485	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ( linC ‘𝑚) = ( linC ‘𝑀))
24		eqidd 2738	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → 𝑓 = 𝑓)
25	23, 24, 1	oveq123d 7234	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (𝑓( linC ‘𝑚)𝑠) = (𝑓( linC ‘𝑀)𝑆))
26		fveq2 6717	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (0g‘𝑚) = (0g‘𝑀))
27	26	adantl 485	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (0g‘𝑚) = (0g‘𝑀))
28		islininds.z	. . . . . . . 8 ⊢ 𝑍 = (0g‘𝑀)
29	27, 28	eqtr4di 2796	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (0g‘𝑚) = 𝑍)
30	25, 29	eqeq12d 2753	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ((𝑓( linC ‘𝑚)𝑠) = (0g‘𝑚) ↔ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
31	21, 30	anbi12d 634	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g‘𝑚)) ↔ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)))
32	10	fveq2d 6721	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g‘𝑅))
33	32, 19	eqtr4di 2796	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = 0 )
34	33	adantl 485	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (0g‘(Scalar‘𝑚)) = 0 )
35	34	eqeq2d 2748	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ((𝑓‘𝑥) = (0g‘(Scalar‘𝑚)) ↔ (𝑓‘𝑥) = 0 ))
36	1, 35	raleqbidv 3313	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (∀𝑥 ∈ 𝑠 (𝑓‘𝑥) = (0g‘(Scalar‘𝑚)) ↔ ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))
37	31, 36	imbi12d 348	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g‘𝑚)) → ∀𝑥 ∈ 𝑠 (𝑓‘𝑥) = (0g‘(Scalar‘𝑚))) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
38	15, 37	raleqbidv 3313	. . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → (∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑠)((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g‘𝑚)) → ∀𝑥 ∈ 𝑠 (𝑓‘𝑥) = (0g‘(Scalar‘𝑚))) ↔ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
39	7, 38	anbi12d 634	. 2 ⊢ ((𝑠 = 𝑆 ∧ 𝑚 = 𝑀) → ((𝑠 ∈ 𝒫 (Base‘𝑚) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑠)((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g‘𝑚)) → ∀𝑥 ∈ 𝑠 (𝑓‘𝑥) = (0g‘(Scalar‘𝑚)))) ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))))
40		df-lininds 45456	. 2 ⊢ linIndS = {⟨𝑠, 𝑚⟩ ∣ (𝑠 ∈ 𝒫 (Base‘𝑚) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑠)((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g‘𝑚)) → ∀𝑥 ∈ 𝑠 (𝑓‘𝑥) = (0g‘(Scalar‘𝑚))))}
41	39, 40	brabga 5415	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 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 ))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∀wral 3061  𝒫 cpw 4513   class class class wbr 5053  ‘cfv 6380  (class class class)co 7213   ↑_m cmap 8508   finSupp cfsupp 8985  Basecbs 16760  Scalarcsca 16805  0_gc0g 16944   linC clinc 45418   linIndS clininds 45454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-iota 6338  df-fv 6388  df-ov 7216  df-lininds 45456

This theorem is referenced by:  linindsi  45461  islinindfis  45463  islindeps  45467  lindslininds  45478  linds0  45479  lindsrng01  45482  snlindsntor  45485