Theorem suppmptcfin 49133

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 2767	. . . . . 6 ⊢ (𝑥 = 𝑣 → (𝑥 = 𝑋 ↔ 𝑣 = 𝑋))
3	2	ifbid 4512	. . . . 5 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑣 = 𝑋, 1 , 0 ))
4	3	cbvmptv 5216	. . . 4 ⊢ (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 )) = (𝑣 ∈ 𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
5	1, 4	eqtri 2786	. . 3 ⊢ 𝐹 = (𝑣 ∈ 𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
6		simp2 1155	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝑉 ∈ 𝒫 𝐵)
7		suppmptcfin.0	. . . . 5 ⊢ 0 = (0g‘𝑅)
8	7	fvexi 6897	. . . 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 6897	. . . . 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 4535	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → if(𝑣 = 𝑋, 1 , 0 ) ∈ V)
15	5, 6, 9, 14	mptsuppd 8184	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 supp 0 ) = {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 })
16		snfi 9041	. . 3 ⊢ {𝑋} ∈ Fin
17		2a1 29	. . . . . 6 ⊢ (𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋)))
18		iffalse 4497	. . . . . . . . . 10 ⊢ (¬ 𝑣 = 𝑋 → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
19	18	adantr 485	. . . . . . . . 9 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
20	19	neeq1d 3017	. . . . . . . 8 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 ↔ 0 ≠ 0 ))
21		eqid 2763	. . . . . . . . 9 ⊢ 0 = 0
22		eqneqall 2969	. . . . . . . . 9 ⊢ ( 0 = 0 → ( 0 ≠ 0 → 𝑣 = 𝑋))
23	21, 22	ax-mp 5	. . . . . . . 8 ⊢ ( 0 ≠ 0 → 𝑣 = 𝑋)
24	20, 23	biimtrdi 256	. . . . . . 7 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋))
25	24	ex 417	. . . . . 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 184	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋))
27	26	ralrimiva 3157	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ∀𝑣 ∈ 𝑉 (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋))
28		rabsssn 4635	. . . 4 ⊢ ({𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋} ↔ ∀𝑣 ∈ 𝑉 (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋))
29	27, 28	sylibr 237	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋})
30		ssfi 9158	. . 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 598	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ∈ Fin)
32	15, 31	eqeltrd 2863	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 supp 0 ) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  {crab 3416  Vcvv 3455   ⊆ wss 3906  ifcif 4488  𝒫 cpw 4563  {csn 4590   ↦ cmpt 5193  ‘cfv 6538  (class class class)co 7412   supp csupp 8157  Fincfn 8944  Basecbs 17270  Scalarcsca 17314  0_gc0g 17493  1_rcur 20264  LModclmod 20962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-supp 8158  df-1o 8454  df-en 8945  df-fin 8948

This theorem is referenced by:  mptcfsupp  49134