Theorem linindslinci 42237

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 42236	. . 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 4656	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓 finSupp 0 ↔ 𝐹 finSupp 0 ))
8		oveq1 6657	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑓( linC ‘𝑀)𝑆) = (𝐹( linC ‘𝑀)𝑆))
9	8	eqeq1d 2624	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 ↔ (𝐹( linC ‘𝑀)𝑆) = 𝑍))
10	7, 9	anbi12d 747	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ↔ (𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍)))
11		fveq1 6190	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
12	11	eqeq1d 2624	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) = 0 ↔ (𝐹‘𝑥) = 0 ))
13	12	ralbidv 2986	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = 0 ))
14	10, 13	imbi12d 334	. . . . . . . 8 ⊢ (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 3305	. . . . . . 7 ⊢ (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	. . . . . 6 ⊢ (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 1262	. . . . 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 ))
18	17	com12 32	. . . 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 ))
19	18	adantl 482	. . 3 ⊢ ((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 ))
20	6, 19	syl 17	. 2 ⊢ (𝑆 linIndS 𝑀 → ((𝐹 ∈ (𝐸 ↑𝑚 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) = 0 ))
21	20	imp 445	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 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  𝒫 cpw 4158   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857   finSupp cfsupp 8275  Basecbs 15857  Scalarcsca 15944  0_gc0g 16100   linC clinc 42193   linIndS clininds 42229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-iota 5851  df-fv 5896  df-ov 6653  df-lininds 42231

This theorem is referenced by: (None)