Theorem islinindfis 49505

Description: The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-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
islinindfis	⊢ ((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))))

Distinct variable groups:   𝑓,𝐸   𝑓,𝑀,𝑥   𝑆,𝑓,𝑥   0 ,𝑓   𝑓,𝑍   𝑓,𝑊

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

Proof of Theorem islinindfis

Step	Hyp	Ref	Expression
1		islininds.b	. . 3 ⊢ 𝐵 = (Base‘𝑀)
2		islininds.z	. . 3 ⊢ 𝑍 = (0g‘𝑀)
3		islininds.r	. . 3 ⊢ 𝑅 = (Scalar‘𝑀)
4		islininds.e	. . 3 ⊢ 𝐸 = (Base‘𝑅)
5		islininds.0	. . 3 ⊢ 0 = (0g‘𝑅)
6	1, 2, 3, 4, 5	islininds 49502	. 2 ⊢ ((𝑆 ∈ Fin ∧ 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 ))))
7		pm4.79 1021	. . . . . . 7 ⊢ (((𝑓 finSupp 0 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) ∨ ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))
8		elmapi 8853	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐸 ↑m 𝑆) → 𝑓:𝑆⟶𝐸)
9	8	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → 𝑓:𝑆⟶𝐸)
10		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → 𝑆 ∈ Fin)
11	5	fvexi 6891	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
12	11	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → 0 ∈ V)
13	9, 10, 12	fdmfifsupp 9351	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → 𝑓 finSupp 0 )
14	13	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → 𝑓 finSupp 0 )
15	14	imim1i 64	. . . . . . . . 9 ⊢ ((𝑓 finSupp 0 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) → ((((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))
16	15	expd 421	. . . . . . . 8 ⊢ ((𝑓 finSupp 0 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) → (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
17		ax-1 6	. . . . . . . 8 ⊢ (((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) → (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
18	16, 17	jaoi 871	. . . . . . 7 ⊢ (((𝑓 finSupp 0 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) ∨ ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )) → (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
19	7, 18	sylbir 238	. . . . . 6 ⊢ (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) → (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
20	19	com12 33	. . . . 5 ⊢ (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
21		pm3.42 499	. . . . 5 ⊢ (((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))
22	20, 21	impbid1 228	. . . 4 ⊢ (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) ↔ ((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
23	22	ralbidva 3184	. . 3 ⊢ ((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) → (∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ) ↔ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )))
24	23	anbi2d 642	. 2 ⊢ ((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) → ((𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 )) ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))))
25	6, 24	bitrd 282	1 ⊢ ((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥 ∈ 𝑆 (𝑓‘𝑥) = 0 ))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451  𝒫 cpw 4557   class class class wbr 5103  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412   ↑_m cmap 8831  Fincfn 8957   finSupp cfsupp 9337  Basecbs 17367  Scalarcsca 17411  0_gc0g 17590   linC clinc 49460   linIndS clininds 49496

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-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8162  df-1o 8460  df-map 8833  df-en 8958  df-fin 8961  df-fsupp 9338  df-lininds 49498

This theorem is used by:  islinindfiss  49506