Theorem islinindfis 46520

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 46517	. 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 1002	. . . . . . 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 8787	. . . . . . . . . . . . 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 765	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → 𝑆 ∈ Fin)
11	5	fvexi 6856	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
12	11	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → 0 ∈ V)
13	9, 10, 12	fdmfifsupp 9315	. . . . . . . . . . 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 855	. . . . . . 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 3172	. . 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 845   = wceq 1541   ∈ wcel 2106  ∀wral 3064  Vcvv 3445  𝒫 cpw 4560   class class class wbr 5105  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ↑_m cmap 8765  Fincfn 8883   finSupp cfsupp 9305  Basecbs 17083  Scalarcsca 17136  0_gc0g 17321   linC clinc 46475   linIndS clininds 46511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-1o 8412  df-map 8767  df-en 8884  df-fin 8887  df-fsupp 9306  df-lininds 46513

This theorem is referenced by:  islinindfiss  46521