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Theorem suppmptcfin 44355
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 2822 . . . . . 6 (𝑥 = 𝑣 → (𝑥 = 𝑋𝑣 = 𝑋))
32ifbid 4485 . . . . 5 (𝑥 = 𝑣 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑣 = 𝑋, 1 , 0 ))
43cbvmptv 5160 . . . 4 (𝑥𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 )) = (𝑣𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
51, 4eqtri 2841 . . 3 𝐹 = (𝑣𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
6 simp2 1129 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → 𝑉 ∈ 𝒫 𝐵)
7 suppmptcfin.0 . . . . 5 0 = (0g𝑅)
87fvexi 6677 . . . 4 0 ∈ V
98a1i 11 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → 0 ∈ V)
10 suppmptcfin.1 . . . . . 6 1 = (1r𝑅)
1110fvexi 6677 . . . . 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 4508 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉) → if(𝑣 = 𝑋, 1 , 0 ) ∈ V)
155, 6, 9, 14mptsuppd 7842 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → (𝐹 supp 0 ) = {𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 })
16 snfi 8582 . . 3 {𝑋} ∈ Fin
17 2a1 28 . . . . . 6 (𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋)))
18 iffalse 4472 . . . . . . . . . 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 3072 . . . . . . . 8 ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉)) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 000 ))
21 eqid 2818 . . . . . . . . 9 0 = 0
22 eqneqall 3024 . . . . . . . . 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 3179 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → ∀𝑣𝑉 (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋))
28 rabsssn 4597 . . . 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 8726 . . 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 2910 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 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  {crab 3139  Vcvv 3492  wss 3933  ifcif 4463  𝒫 cpw 4535  {csn 4557  cmpt 5137  cfv 6348  (class class class)co 7145   supp csupp 7819  Fincfn 8497  Basecbs 16471  Scalarcsca 16556  0gc0g 16701  1rcur 19180  LModclmod 19563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-supp 7820  df-1o 8091  df-er 8278  df-en 8498  df-fin 8501
This theorem is referenced by:  mptcfsupp  44356
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