Theorem islinindfis 49655

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 49652	. 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 8873	. . . . . . . . . . . . 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 6900	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
12	11	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ Fin ∧ 𝑀 ∈ 𝑊) ∧ 𝑓 ∈ (𝐸 ↑m 𝑆)) → 0 ∈ V)
13	9, 10, 12	fdmfifsupp 9371	. . . . . . . . . . 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 3183	. . 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 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450  𝒫 cpw 4556   class class class wbr 5102  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421   ↑_m cmap 8851  Fincfn 8977   finSupp cfsupp 9357  Basecbs 17393  Scalarcsca 17437  0_gc0g 17616   linC clinc 49610   linIndS clininds 49646

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-supp 8181  df-1o 8480  df-map 8853  df-en 8978  df-fin 8981  df-fsupp 9358  df-lininds 49648

This theorem is used by:  islinindfiss  49656