Theorem linindslinci 48479

Description: The implications of being a linearly independent subset and a linear combination of this subset being 0. (Contributed by AV, 24-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
linindslinci	⊢ ((𝑆 linIndS 𝑀 ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍)) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = 0 )

Distinct variable groups:   𝑥,𝑀   𝑥,𝑆   𝑥,𝐹

Allowed substitution hints:   𝐵(𝑥)   𝑅(𝑥)   𝐸(𝑥)   0 (𝑥)   𝑍(𝑥)

Proof of Theorem linindslinci

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		islininds.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
2		islininds.z	. . . 4 ⊢ 𝑍 = (0g‘𝑀)
3		islininds.r	. . . 4 ⊢ 𝑅 = (Scalar‘𝑀)
4		islininds.e	. . . 4 ⊢ 𝐸 = (Base‘𝑅)
5		islininds.0	. . . 4 ⊢ 0 = (0g‘𝑅)
6	1, 2, 3, 4, 5	linindsi 48478	. . 3 ⊢ (𝑆 linIndS 𝑀 → (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
7		breq1 5094	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓 finSupp 0 ↔ 𝐹 finSupp 0 ))
8		oveq1 7353	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓( linC ‘𝑀)𝑆) = (𝐹( linC ‘𝑀)𝑆))
9	8	eqeq1d 2733	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 ↔ (𝐹( linC ‘𝑀)𝑆) = 𝑍))
10	7, 9	anbi12d 632	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ↔ (𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍)))
11		fveq1 6821	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
12	11	eqeq1d 2733	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) = 0 ↔ (𝐹‘𝑥) = 0 ))
13	12	ralbidv 3155	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = 0 ))
14	10, 13	imbi12d 344	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (((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 )))
15	14	rspcv 3573	. . . . . 6 ⊢ (𝐹 ∈ (𝐸 ↑m 𝑆) → (∀𝑓 ∈ (𝐸 ↑m 𝑆)((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 )))
16	15	com23 86	. . . . 5 ⊢ (𝐹 ∈ (𝐸 ↑m 𝑆) → ((𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = 0 )))
17	16	3impib 1116	. . . 4 ⊢ ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = 0 ))
18	17	com12 32	. . 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 ))
19	6, 18	simpl2im 503	. 2 ⊢ (𝑆 linIndS 𝑀 → ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = 0 ))
20	19	imp 406	1 ⊢ ((𝑆 linIndS 𝑀 ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍)) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = 0 )

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  𝒫 cpw 4550   class class class wbr 5091  ‘cfv 6481  (class class class)co 7346   ↑_m cmap 8750   finSupp cfsupp 9245  Basecbs 17117  Scalarcsca 17161  0_gc0g 17340   linC clinc 48435   linIndS clininds 48471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-iota 6437  df-fv 6489  df-ov 7349  df-lininds 48473

This theorem is referenced by: (None)