Theorem suppmptcfin 46445

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 2740	. . . . . 6 ⊢ (𝑥 = 𝑣 → (𝑥 = 𝑋 ↔ 𝑣 = 𝑋))
3	2	ifbid 4509	. . . . 5 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑣 = 𝑋, 1 , 0 ))
4	3	cbvmptv 5218	. . . 4 ⊢ (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 )) = (𝑣 ∈ 𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
5	1, 4	eqtri 2764	. . 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 6856	. . . 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 6856	. . . . 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 4532	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) → if(𝑣 = 𝑋, 1 , 0 ) ∈ V)
15	5, 6, 9, 14	mptsuppd 8118	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 supp 0 ) = {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 })
16		snfi 8988	. . 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 4495	. . . . . . . . . 10 ⊢ (¬ 𝑣 = 𝑋 → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
19	18	adantr 481	. . . . . . . . 9 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
20	19	neeq1d 3003	. . . . . . . 8 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 ↔ 0 ≠ 0 ))
21		eqid 2736	. . . . . . . . 9 ⊢ 0 = 0
22		eqneqall 2954	. . . . . . . . 9 ⊢ ( 0 = 0 → ( 0 ≠ 0 → 𝑣 = 𝑋))
23	21, 22	ax-mp 5	. . . . . . . 8 ⊢ ( 0 ≠ 0 → 𝑣 = 𝑋)
24	20, 23	syl6bi 252	. . . . . . 7 ⊢ ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋))
25	24	ex 413	. . . . . 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 3143	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → ∀𝑣 ∈ 𝑉 (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋))
28		rabsssn 4628	. . . 4 ⊢ ({𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋} ↔ ∀𝑣 ∈ 𝑉 (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 → 𝑣 = 𝑋))
29	27, 28	sylibr 233	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋})
30		ssfi 9117	. . 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 2838	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 supp 0 ) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  {crab 3407  Vcvv 3445   ⊆ wss 3910  ifcif 4486  𝒫 cpw 4560  {csn 4586   ↦ cmpt 5188  ‘cfv 6496  (class class class)co 7357   supp csupp 8092  Fincfn 8883  Basecbs 17083  Scalarcsca 17136  0_gc0g 17321  1_rcur 19913  LModclmod 20322

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-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-supp 8093  df-1o 8412  df-en 8884  df-fin 8887

This theorem is referenced by:  mptcfsupp  46446