Theorem islinindfis 48442

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 48439	. 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 1005	. . . . . . 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 8825	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐸 ↑m 𝑆) → 𝑓:𝑆⟶𝐸)
9	8	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → 𝑓:𝑆⟶𝐸)
10		simpll 766	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → 𝑆 ∈ Fin)
11	5	fvexi 6875	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
12	11	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → 0 ∈ V)
13	9, 10, 12	fdmfifsupp 9333	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → 𝑓 finSupp 0 )
14	13	adantr 480	. . . . . . . . . 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 415	. . . . . . . 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 857	. . . . . . 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 235	. . . . . 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		pm3.42 493	. . . . 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 225	. . . 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 3155	. . 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 630	. 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 279	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 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450  𝒫 cpw 4566   class class class wbr 5110  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ↑_m cmap 8802  Fincfn 8921   finSupp cfsupp 9319  Basecbs 17186  Scalarcsca 17230  0_gc0g 17409   linC clinc 48397   linIndS clininds 48433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-1o 8437  df-map 8804  df-en 8922  df-fin 8925  df-fsupp 9320  df-lininds 48435

This theorem is referenced by:  islinindfiss  48443