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Theorem suppmptcfin 44256
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 2828 . . . . . 6 (𝑥 = 𝑣 → (𝑥 = 𝑋𝑣 = 𝑋))
32ifbid 4491 . . . . 5 (𝑥 = 𝑣 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑣 = 𝑋, 1 , 0 ))
43cbvmptv 5165 . . . 4 (𝑥𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 )) = (𝑣𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
51, 4eqtri 2848 . . 3 𝐹 = (𝑣𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
6 simp2 1131 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → 𝑉 ∈ 𝒫 𝐵)
7 suppmptcfin.0 . . . . 5 0 = (0g𝑅)
87fvexi 6680 . . . 4 0 ∈ V
98a1i 11 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → 0 ∈ V)
10 suppmptcfin.1 . . . . . 6 1 = (1r𝑅)
1110fvexi 6680 . . . . 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 4514 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉) → if(𝑣 = 𝑋, 1 , 0 ) ∈ V)
155, 6, 9, 14mptsuppd 7847 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → (𝐹 supp 0 ) = {𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 })
16 snfi 8586 . . 3 {𝑋} ∈ Fin
17 2a1 28 . . . . . 6 (𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋)))
18 iffalse 4478 . . . . . . . . . 10 𝑣 = 𝑋 → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
1918adantr 481 . . . . . . . . 9 ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉)) → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
2019neeq1d 3079 . . . . . . . 8 ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉)) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 000 ))
21 eqid 2824 . . . . . . . . 9 0 = 0
22 eqneqall 3031 . . . . . . . . 9 ( 0 = 0 → ( 00𝑣 = 𝑋))
2321, 22ax-mp 5 . . . . . . . 8 ( 00𝑣 = 𝑋)
2420, 23syl6bi 254 . . . . . . 7 ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉)) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋))
2524ex 413 . . . . . 6 𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋)))
2617, 25pm2.61i 183 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋))
2726ralrimiva 3186 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → ∀𝑣𝑉 (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋))
28 rabsssn 4603 . . . 4 ({𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋} ↔ ∀𝑣𝑉 (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋))
2927, 28sylibr 235 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → {𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋})
30 ssfi 8730 . . 3 (({𝑋} ∈ Fin ∧ {𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋}) → {𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ∈ Fin)
3116, 29, 30sylancr 587 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → {𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ∈ Fin)
3215, 31eqeltrd 2917 1 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → (𝐹 supp 0 ) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2106  wne 3020  wral 3142  {crab 3146  Vcvv 3499  wss 3939  ifcif 4469  𝒫 cpw 4541  {csn 4563  cmpt 5142  cfv 6351  (class class class)co 7151   supp csupp 7824  Fincfn 8501  Basecbs 16475  Scalarcsca 16560  0gc0g 16705  1rcur 19173  LModclmod 19556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-supp 7825  df-1o 8096  df-er 8282  df-en 8502  df-fin 8505
This theorem is referenced by:  mptcfsupp  44257
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