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Theorem suppmptcfin 43021
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 2829 . . . . . 6 (𝑥 = 𝑣 → (𝑥 = 𝑋𝑣 = 𝑋))
32ifbid 4330 . . . . 5 (𝑥 = 𝑣 → if(𝑥 = 𝑋, 1 , 0 ) = if(𝑣 = 𝑋, 1 , 0 ))
43cbvmptv 4975 . . . 4 (𝑥𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 )) = (𝑣𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
51, 4eqtri 2849 . . 3 𝐹 = (𝑣𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
6 simp2 1171 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → 𝑉 ∈ 𝒫 𝐵)
7 suppmptcfin.0 . . . . 5 0 = (0g𝑅)
87fvexi 6451 . . . 4 0 ∈ V
98a1i 11 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → 0 ∈ V)
10 suppmptcfin.1 . . . . . 6 1 = (1r𝑅)
1110fvexi 6451 . . . . 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 4353 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉) → if(𝑣 = 𝑋, 1 , 0 ) ∈ V)
155, 6, 9, 14mptsuppd 7587 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → (𝐹 supp 0 ) = {𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 })
16 snfi 8313 . . 3 {𝑋} ∈ Fin
17 2a1 28 . . . . . 6 (𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋)))
18 iffalse 4317 . . . . . . . . . 10 𝑣 = 𝑋 → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
1918adantr 474 . . . . . . . . 9 ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉)) → if(𝑣 = 𝑋, 1 , 0 ) = 0 )
2019neeq1d 3058 . . . . . . . 8 ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉)) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 000 ))
21 eqid 2825 . . . . . . . . 9 0 = 0
22 eqneqall 3010 . . . . . . . . 9 ( 0 = 0 → ( 00𝑣 = 𝑋))
2321, 22ax-mp 5 . . . . . . . 8 ( 00𝑣 = 𝑋)
2420, 23syl6bi 245 . . . . . . 7 ((¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉)) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋))
2524ex 403 . . . . . 6 𝑣 = 𝑋 → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋)))
2617, 25pm2.61i 177 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ 𝑣𝑉) → (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋))
2726ralrimiva 3175 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → ∀𝑣𝑉 (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋))
28 rabsssn 4437 . . . 4 ({𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋} ↔ ∀𝑣𝑉 (if(𝑣 = 𝑋, 1 , 0 ) ≠ 0𝑣 = 𝑋))
2927, 28sylibr 226 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → {𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋})
30 ssfi 8455 . . 3 (({𝑋} ∈ Fin ∧ {𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ⊆ {𝑋}) → {𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ∈ Fin)
3116, 29, 30sylancr 581 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → {𝑣𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) ≠ 0 } ∈ Fin)
3215, 31eqeltrd 2906 1 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → (𝐹 supp 0 ) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386  w3a 1111   = wceq 1656  wcel 2164  wne 2999  wral 3117  {crab 3121  Vcvv 3414  wss 3798  ifcif 4308  𝒫 cpw 4380  {csn 4399  cmpt 4954  cfv 6127  (class class class)co 6910   supp csupp 7564  Fincfn 8228  Basecbs 16229  Scalarcsca 16315  0gc0g 16460  1rcur 18862  LModclmod 19226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-supp 7565  df-1o 7831  df-er 8014  df-en 8229  df-fin 8232
This theorem is referenced by:  mptcfsupp  43022
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