Theorem islinindfis 45790

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 45787	. 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 1001	. . . . . . 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 8637	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐸 ↑m 𝑆) → 𝑓:𝑆⟶𝐸)
9	8	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → 𝑓:𝑆⟶𝐸)
10		simpll 764	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → 𝑆 ∈ Fin)
11	5	fvexi 6788	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
12	11	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → 0 ∈ V)
13	9, 10, 12	fdmfifsupp 9138	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → 𝑓 finSupp 0 )
14	13	adantr 481	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → 𝑓 finSupp 0 )
15	14	imim1i 63	. . . . . . . . 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 416	. . . . . . . 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 854	. . . . . . 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 234	. . . . . 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 32	. . . . 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		pm3.42 494	. . . . 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 224	. . . 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 3111	. . 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 629	. 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 278	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 ))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432  𝒫 cpw 4533   class class class wbr 5074  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  Fincfn 8733   finSupp cfsupp 9128  Basecbs 16912  Scalarcsca 16965  0_gc0g 17150   linC clinc 45745   linIndS clininds 45781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-1o 8297  df-map 8617  df-en 8734  df-fin 8737  df-fsupp 9129  df-lininds 45783

This theorem is referenced by:  islinindfiss  45791