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Theorem suppmptcfin 47045
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 𝐡 = (Baseβ€˜π‘€)
suppmptcfin.r 𝑅 = (Scalarβ€˜π‘€)
suppmptcfin.0 0 = (0gβ€˜π‘…)
suppmptcfin.1 1 = (1rβ€˜π‘…)
suppmptcfin.f 𝐹 = (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑋, 1 , 0 ))
Assertion
Ref Expression
suppmptcfin ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) β†’ (𝐹 supp 0 ) ∈ Fin)
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝐹   π‘₯,𝑀   π‘₯,𝑉   π‘₯,𝑋   π‘₯, 1   π‘₯, 0
Allowed substitution hint:   𝑅(π‘₯)
Proof of Theorem suppmptcfin
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 suppmptcfin.f . . . 4 𝐹 = (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑋, 1 , 0 ))
2 eqeq1 2736 . . . . . 6 (π‘₯ = 𝑣 β†’ (π‘₯ = 𝑋 ↔ 𝑣 = 𝑋))
32ifbid 4551 . . . . 5 (π‘₯ = 𝑣 β†’ if(π‘₯ = 𝑋, 1 , 0 ) = if(𝑣 = 𝑋, 1 , 0 ))
43cbvmptv 5261 . . . 4 (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑋, 1 , 0 )) = (𝑣 ∈ 𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
51, 4eqtri 2760 . . 3 𝐹 = (𝑣 ∈ 𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
6 simp2 1137 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) β†’ 𝑉 ∈ 𝒫 𝐡)
7 suppmptcfin.0 . . . . 5 0 = (0gβ€˜π‘…)
87fvexi 6905 . . . 4 0 ∈ V
98a1i 11 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) β†’ 0 ∈ V)
10 suppmptcfin.1 . . . . . 6 1 = (1rβ€˜π‘…)
1110fvexi 6905 . . . . 5 1 ∈ V
1211a1i 11 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) β†’ 1 ∈ V)
138a1i 11 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) β†’ 0 ∈ V)
1412, 13ifcld 4574 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) β†’ if(𝑣 = 𝑋, 1 , 0 ) ∈ V)
155, 6, 9, 14mptsuppd 8171 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) β†’ (𝐹 supp 0 ) = {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) β‰  0 })
16 snfi 9043 . . 3 {𝑋} ∈ Fin
17 2a1 28 . . . . . 6 (𝑣 = 𝑋 β†’ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) β†’ (if(𝑣 = 𝑋, 1 , 0 ) β‰  0 β†’ 𝑣 = 𝑋)))
18 iffalse 4537 . . . . . . . . . 10 (Β¬ 𝑣 = 𝑋 β†’ if(𝑣 = 𝑋, 1 , 0 ) = 0 )
1918adantr 481 . . . . . . . . 9 ((Β¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) β†’ if(𝑣 = 𝑋, 1 , 0 ) = 0 )
2019neeq1d 3000 . . . . . . . 8 ((Β¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) β†’ (if(𝑣 = 𝑋, 1 , 0 ) β‰  0 ↔ 0 β‰  0 ))
21 eqid 2732 . . . . . . . . 9 0 = 0
22 eqneqall 2951 . . . . . . . . 9 ( 0 = 0 β†’ ( 0 β‰  0 β†’ 𝑣 = 𝑋))
2321, 22ax-mp 5 . . . . . . . 8 ( 0 β‰  0 β†’ 𝑣 = 𝑋)
2420, 23syl6bi 252 . . . . . . 7 ((Β¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) β†’ (if(𝑣 = 𝑋, 1 , 0 ) β‰  0 β†’ 𝑣 = 𝑋))
2524ex 413 . . . . . 6 (Β¬ 𝑣 = 𝑋 β†’ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) β†’ (if(𝑣 = 𝑋, 1 , 0 ) β‰  0 β†’ 𝑣 = 𝑋)))
2617, 25pm2.61i 182 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) β†’ (if(𝑣 = 𝑋, 1 , 0 ) β‰  0 β†’ 𝑣 = 𝑋))
2726ralrimiva 3146 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) β†’ βˆ€π‘£ ∈ 𝑉 (if(𝑣 = 𝑋, 1 , 0 ) β‰  0 β†’ 𝑣 = 𝑋))
28 rabsssn 4670 . . . 4 ({𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) β‰  0 } βŠ† {𝑋} ↔ βˆ€π‘£ ∈ 𝑉 (if(𝑣 = 𝑋, 1 , 0 ) β‰  0 β†’ 𝑣 = 𝑋))
2927, 28sylibr 233 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) β†’ {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) β‰  0 } βŠ† {𝑋})
30 ssfi 9172 . . 3 (({𝑋} ∈ Fin ∧ {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) β‰  0 } βŠ† {𝑋}) β†’ {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) β‰  0 } ∈ Fin)
3116, 29, 30sylancr 587 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) β†’ {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) β‰  0 } ∈ Fin)
3215, 31eqeltrd 2833 1 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) β†’ (𝐹 supp 0 ) ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  {crab 3432  Vcvv 3474   βŠ† wss 3948  ifcif 4528  π’« cpw 4602  {csn 4628   ↦ cmpt 5231  β€˜cfv 6543  (class class class)co 7408   supp csupp 8145  Fincfn 8938  Basecbs 17143  Scalarcsca 17199  0gc0g 17384  1rcur 20003  LModclmod 20470
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-supp 8146  df-1o 8465  df-en 8939  df-fin 8942
This theorem is referenced by:  mptcfsupp  47046
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