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Theorem suppmptcfin 45279
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 = (0g𝑅)
suppmptcfin.1 1 = (1r𝑅)
suppmptcfin.f 𝐹 = (𝑥𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))
Assertion
Ref Expression
suppmptcfin ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → (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.
StepHypRef Expression
1 suppmptcfin.f . . . 4 𝐹 = (𝑥𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))
2 eqeq1 2743 . . . . . 6 (𝑥 = 𝑣 → (𝑥 = 𝑋𝑣 = 𝑋))
32ifbid 4438 . . . . 5 (𝑥 = 𝑣 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑣 = 𝑋, 1 , 0 ))
43cbvmptv 5134 . . . 4 (𝑥𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 )) = (𝑣𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
51, 4eqtri 2762 . . 3 𝐹 = (𝑣𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
6 simp2 1138 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → 𝑉 ∈ 𝒫 𝐵)
7 suppmptcfin.0 . . . . 5 0 = (0g𝑅)
87fvexi 6691 . . . 4 0 ∈ V
98a1i 11 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → 0 ∈ V)
10 suppmptcfin.1 . . . . . 6 1 = (1r𝑅)
1110fvexi 6691 . . . . 5 1 ∈ V
1211a1i 11 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉) → 1 ∈ V)
138a1i 11 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉) → 0 ∈ V)
1412, 13ifcld 4461 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉) → if(𝑣 = 𝑋, 1 , 0 ) ∈ V)
155, 6, 9, 14mptsuppd 7885 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → (𝐹 supp 0 ) = {𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 })
16 snfi 8645 . . 3 {𝑋} ∈ Fin
17 2a1 28 . . . . . 6 (𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋)))
18 iffalse 4424 . . . . . . . . . 10 𝑣 = 𝑋 → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
1918adantr 484 . . . . . . . . 9 ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉)) → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
2019neeq1d 2994 . . . . . . . 8 ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉)) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 000 ))
21 eqid 2739 . . . . . . . . 9 0 = 0
22 eqneqall 2946 . . . . . . . . 9 ( 0 = 0 → ( 00𝑣 = 𝑋))
2321, 22ax-mp 5 . . . . . . . 8 ( 00𝑣 = 𝑋)
2420, 23syl6bi 256 . . . . . . 7 ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉)) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋))
2524ex 416 . . . . . 6 𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋)))
2617, 25pm2.61i 185 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋))
2726ralrimiva 3097 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → ∀𝑣𝑉 (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋))
28 rabsssn 4559 . . . 4 ({𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋} ↔ ∀𝑣𝑉 (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋))
2927, 28sylibr 237 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → {𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋})
30 ssfi 8775 . . 3 (({𝑋} ∈ Fin ∧ {𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋}) → {𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ∈ Fin)
3116, 29, 30sylancr 590 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → {𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ∈ Fin)
3215, 31eqeltrd 2834 1 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → (𝐹 supp 0 ) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  wne 2935  wral 3054  {crab 3058  Vcvv 3399  wss 3844  ifcif 4415  𝒫 cpw 4489  {csn 4517  cmpt 5111  cfv 6340  (class class class)co 7173   supp csupp 7859  Fincfn 8558  Basecbs 16589  Scalarcsca 16674  0gc0g 16819  1rcur 19373  LModclmod 19756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-om 7603  df-supp 7860  df-1o 8134  df-en 8559  df-fin 8562
This theorem is referenced by:  mptcfsupp  45280
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