Theorem suppmptcfin 43021

Description: The support of a mapping with value 0 except of one is finite. (Contributed by AV, 27-Apr-2019.)

Hypotheses

Ref	Expression
suppmptcfin.b	⊢ 𝐵 = (Base‘𝑀)
suppmptcfin.r	⊢ 𝑅 = (Scalar‘𝑀)
suppmptcfin.0	⊢ 0 = (0g‘𝑅)
suppmptcfin.1	⊢ 1 = (1r‘𝑅)
suppmptcfin.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))

Assertion

Ref	Expression
suppmptcfin	⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 supp 0 ) ∈ Fin)

Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝑀   𝑥,𝑉   𝑥,𝑋   𝑥, 1   𝑥, 0

Allowed substitution hint:   𝑅(𝑥)

Proof of Theorem suppmptcfin

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppmptcfin.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))
2		eqeq1 2829	. . . . . 6 ⊢ (𝑥 = 𝑣 → (𝑥 = 𝑋 ↔ 𝑣 = 𝑋))
3	2	ifbid 4330	. . . . 5 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑣 = 𝑋, 1 , 0 ))
4	3	cbvmptv 4975	. . . 4 ⊢ (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 )) = (𝑣 ∈ 𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
5	1, 4	eqtri 2849	. . 3 ⊢ 𝐹 = (𝑣 ∈ 𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
6		simp2 1171	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑉 ∈ 𝒫 𝐵)
7		suppmptcfin.0	. . . . 5 ⊢ 0 = (0g‘𝑅)
8	7	fvexi 6451	. . . 4 ⊢ 0 ∈ V
9	8	a1i 11	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 0 ∈ V)
10		suppmptcfin.1	. . . . . 6 ⊢ 1 = (1r‘𝑅)
11	10	fvexi 6451	. . . . 5 ⊢ 1 ∈ V
12	11	a1i 11	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → 1 ∈ V)
13	8	a1i 11	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → 0 ∈ V)
14	12, 13	ifcld 4353	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → if(𝑣 = 𝑋, 1 , 0 ) ∈ V)
15	5, 6, 9, 14	mptsuppd 7587	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 supp 0 ) = {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 })
16		snfi 8313	. . 3 ⊢ {𝑋} ∈ Fin
17		2a1 28	. . . . . 6 ⊢ (𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋)))
18		iffalse 4317	. . . . . . . . . 10 ⊢ (¬ 𝑣 = 𝑋 → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
19	18	adantr 474	. . . . . . . . 9 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
20	19	neeq1d 3058	. . . . . . . 8 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 ↔ 0 ≠ 0 ))
21		eqid 2825	. . . . . . . . 9 ⊢ 0 = 0
22		eqneqall 3010	. . . . . . . . 9 ⊢ ( 0 = 0 → ( 0 ≠ 0 → 𝑣 = 𝑋))
23	21, 22	ax-mp 5	. . . . . . . 8 ⊢ ( 0 ≠ 0 → 𝑣 = 𝑋)
24	20, 23	syl6bi 245	. . . . . . 7 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋))
25	24	ex 403	. . . . . 6 ⊢ (¬ 𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋)))
26	17, 25	pm2.61i 177	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋))
27	26	ralrimiva 3175	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ∀𝑣 ∈ 𝑉 (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋))
28		rabsssn 4437	. . . 4 ⊢ ({𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋} ↔ ∀𝑣 ∈ 𝑉 (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋))
29	27, 28	sylibr 226	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋})
30		ssfi 8455	. . 3 ⊢ (({𝑋} ∈ Fin ∧ {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋}) → {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ∈ Fin)
31	16, 29, 30	sylancr 581	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ∈ Fin)
32	15, 31	eqeltrd 2906	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 supp 0 ) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  ∀wral 3117  {crab 3121  Vcvv 3414   ⊆ wss 3798  ifcif 4308  𝒫 cpw 4380  {csn 4399   ↦ cmpt 4954  ‘cfv 6127  (class class class)co 6910   supp csupp 7564  Fincfn 8228  Basecbs 16229  Scalarcsca 16315  0_gc0g 16460  1_rcur 18862  LModclmod 19226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-supp 7565  df-1o 7831  df-er 8014  df-en 8229  df-fin 8232

This theorem is referenced by:  mptcfsupp  43022