Theorem linindsi 49247

Description: The implications 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
linindsi	⊢ (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)   0 (𝑥,𝑓)   𝑍(𝑥,𝑓)

Proof of Theorem linindsi

Step	Hyp	Ref	Expression
1		linindsv 49245	. . 3 ⊢ (𝑆 linIndS 𝑀 → (𝑆 ∈ V ∧ 𝑀 ∈ V))
2		islininds.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
3		islininds.z	. . . 4 ⊢ 𝑍 = (0g‘𝑀)
4		islininds.r	. . . 4 ⊢ 𝑅 = (Scalar‘𝑀)
5		islininds.e	. . . 4 ⊢ 𝐸 = (Base‘𝑅)
6		islininds.0	. . . 4 ⊢ 0 = (0g‘𝑅)
7	2, 3, 4, 5, 6	islininds 49246	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑀 ∈ V) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))))
8	1, 7	syl 18	. 2 ⊢ (𝑆 linIndS 𝑀 → (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	ibi 270	1 ⊢ (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 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455  𝒫 cpw 4562   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8820   finSupp cfsupp 9317  Basecbs 17264  Scalarcsca 17308  0_gc0g 17487   linC clinc 49204   linIndS clininds 49240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-iota 6492  df-fv 6544  df-ov 7413  df-lininds 49242

This theorem is referenced by:  linindslinci  49248  linindscl  49251