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Theorem suppmptcfin 47055
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 2737 . . . . . 6 (π‘₯ = 𝑣 β†’ (π‘₯ = 𝑋 ↔ 𝑣 = 𝑋))
32ifbid 4552 . . . . 5 (π‘₯ = 𝑣 β†’ if(π‘₯ = 𝑋, 1 , 0 ) = if(𝑣 = 𝑋, 1 , 0 ))
43cbvmptv 5262 . . . 4 (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑋, 1 , 0 )) = (𝑣 ∈ 𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
51, 4eqtri 2761 . . 3 𝐹 = (𝑣 ∈ 𝑉 ↦ if(𝑣 = 𝑋, 1 , 0 ))
6 simp2 1138 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) β†’ 𝑉 ∈ 𝒫 𝐡)
7 suppmptcfin.0 . . . . 5 0 = (0gβ€˜π‘…)
87fvexi 6906 . . . 4 0 ∈ V
98a1i 11 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) β†’ 0 ∈ V)
10 suppmptcfin.1 . . . . . 6 1 = (1rβ€˜π‘…)
1110fvexi 6906 . . . . 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 4575 . . 3 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉) β†’ if(𝑣 = 𝑋, 1 , 0 ) ∈ V)
155, 6, 9, 14mptsuppd 8172 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) β†’ (𝐹 supp 0 ) = {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) β‰  0 })
16 snfi 9044 . . 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 4538 . . . . . . . . . 10 (Β¬ 𝑣 = 𝑋 β†’ if(𝑣 = 𝑋, 1 , 0 ) = 0 )
1918adantr 482 . . . . . . . . 9 ((Β¬ 𝑣 = 𝑋 ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) ∧ 𝑣 ∈ 𝑉)) β†’ if(𝑣 = 𝑋, 1 , 0 ) = 0 )
2019neeq1d 3001 . . . . . . . 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 2733 . . . . . . . . 9 0 = 0
22 eqneqall 2952 . . . . . . . . 9 ( 0 = 0 β†’ ( 0 β‰  0 β†’ 𝑣 = 𝑋))
2321, 22ax-mp 5 . . . . . . . 8 ( 0 β‰  0 β†’ 𝑣 = 𝑋)
2420, 23syl6bi 253 . . . . . . 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 414 . . . . . 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 3147 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) β†’ βˆ€π‘£ ∈ 𝑉 (if(𝑣 = 𝑋, 1 , 0 ) β‰  0 β†’ 𝑣 = 𝑋))
28 rabsssn 4671 . . . 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 9173 . . 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 588 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) β†’ {𝑣 ∈ 𝑉 ∣ if(𝑣 = 𝑋, 1 , 0 ) β‰  0 } ∈ Fin)
3215, 31eqeltrd 2834 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  {crab 3433  Vcvv 3475   βŠ† wss 3949  ifcif 4529  π’« cpw 4603  {csn 4629   ↦ cmpt 5232  β€˜cfv 6544  (class class class)co 7409   supp csupp 8146  Fincfn 8939  Basecbs 17144  Scalarcsca 17200  0gc0g 17385  1rcur 20004  LModclmod 20471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-supp 8147  df-1o 8466  df-en 8940  df-fin 8943
This theorem is referenced by:  mptcfsupp  47056
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