Theorem linindsi 49475

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 49473	. . 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 49474	. . 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 )))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450  𝒫 cpw 4557   class class class wbr 5103  ‘cfv 6528  (class class class)co 7409   ↑_m cmap 8826   finSupp cfsupp 9331  Basecbs 17334  Scalarcsca 17378  0_gc0g 17557   linC clinc 49432   linIndS clininds 49468

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 5249  ax-pr 5391

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-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5654  df-rel 5655  df-iota 6484  df-fv 6536  df-ov 7412  df-lininds 49470

This theorem is used by:  linindslinci  49476  linindscl  49479