Theorem suppmptcfin 48406

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 2735	. . . . . 6 ⊢ (𝑥 = 𝑣 → (𝑥 = 𝑋 ↔ 𝑣 = 𝑋))
3	2	ifbid 4499	. . . . 5 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑣 = 𝑋, 1 , 0 ))
4	3	cbvmptv 5195	. . . 4 ⊢ (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 )) = (𝑣 ∈ 𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
5	1, 4	eqtri 2754	. . 3 ⊢ 𝐹 = (𝑣 ∈ 𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
6		simp2 1137	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑉 ∈ 𝒫 𝐵)
7		suppmptcfin.0	. . . . 5 ⊢ 0 = (0g‘𝑅)
8	7	fvexi 6836	. . . 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 6836	. . . . 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 4522	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → if(𝑣 = 𝑋, 1 , 0 ) ∈ V)
15	5, 6, 9, 14	mptsuppd 8117	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 supp 0 ) = {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 })
16		snfi 8965	. . 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 4484	. . . . . . . . . 10 ⊢ (¬ 𝑣 = 𝑋 → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
19	18	adantr 480	. . . . . . . . 9 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
20	19	neeq1d 2987	. . . . . . . 8 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 ↔ 0 ≠ 0 ))
21		eqid 2731	. . . . . . . . 9 ⊢ 0 = 0
22		eqneqall 2939	. . . . . . . . 9 ⊢ ( 0 = 0 → ( 0 ≠ 0 → 𝑣 = 𝑋))
23	21, 22	ax-mp 5	. . . . . . . 8 ⊢ ( 0 ≠ 0 → 𝑣 = 𝑋)
24	20, 23	biimtrdi 253	. . . . . . 7 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋))
25	24	ex 412	. . . . . 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 182	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋))
27	26	ralrimiva 3124	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ∀𝑣 ∈ 𝑉 (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋))
28		rabsssn 4621	. . . 4 ⊢ ({𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋} ↔ ∀𝑣 ∈ 𝑉 (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋))
29	27, 28	sylibr 234	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋})
30		ssfi 9082	. . 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 587	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ∈ Fin)
32	15, 31	eqeltrd 2831	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 supp 0 ) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  {crab 3395  Vcvv 3436   ⊆ wss 3902  ifcif 4475  𝒫 cpw 4550  {csn 4576   ↦ cmpt 5172  ‘cfv 6481  (class class class)co 7346   supp csupp 8090  Fincfn 8869  Basecbs 17117  Scalarcsca 17161  0_gc0g 17340  1_rcur 20097  LModclmod 20791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-supp 8091  df-1o 8385  df-en 8870  df-fin 8873

This theorem is referenced by:  mptcfsupp  48407