Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hbtlem1 Structured version   Visualization version   GIF version

Theorem hbtlem1 43135
Description: Value of the leading coefficient sequence function. (Contributed by Stefan O'Rear, 31-Mar-2015.)
Hypotheses
Ref Expression
hbtlem.p 𝑃 = (Poly1𝑅)
hbtlem.u 𝑈 = (LIdeal‘𝑃)
hbtlem.s 𝑆 = (ldgIdlSeq‘𝑅)
hbtlem.d 𝐷 = (deg1𝑅)
Assertion
Ref Expression
hbtlem1 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})
Distinct variable groups:   𝑗,𝐼,𝑘   𝑅,𝑗,𝑘   𝑗,𝑋,𝑘
Allowed substitution hints:   𝐷(𝑗,𝑘)   𝑃(𝑗,𝑘)   𝑆(𝑗,𝑘)   𝑈(𝑗,𝑘)   𝑉(𝑗,𝑘)

Proof of Theorem hbtlem1
Dummy variables 𝑖 𝑟 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbtlem.s . . . . . 6 𝑆 = (ldgIdlSeq‘𝑅)
2 elex 3501 . . . . . . 7 (𝑅𝑉𝑅 ∈ V)
3 fveq2 6906 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
4 hbtlem.p . . . . . . . . . . . 12 𝑃 = (Poly1𝑅)
53, 4eqtr4di 2795 . . . . . . . . . . 11 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
65fveq2d 6910 . . . . . . . . . 10 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = (LIdeal‘𝑃))
7 hbtlem.u . . . . . . . . . 10 𝑈 = (LIdeal‘𝑃)
86, 7eqtr4di 2795 . . . . . . . . 9 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = 𝑈)
9 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅 → (deg1𝑟) = (deg1𝑅))
10 hbtlem.d . . . . . . . . . . . . . . . 16 𝐷 = (deg1𝑅)
119, 10eqtr4di 2795 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅 → (deg1𝑟) = 𝐷)
1211fveq1d 6908 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → ((deg1𝑟)‘𝑘) = (𝐷𝑘))
1312breq1d 5153 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (((deg1𝑟)‘𝑘) ≤ 𝑥 ↔ (𝐷𝑘) ≤ 𝑥))
1413anbi1d 631 . . . . . . . . . . . 12 (𝑟 = 𝑅 → ((((deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))))
1514rexbidv 3179 . . . . . . . . . . 11 (𝑟 = 𝑅 → (∃𝑘𝑖 (((deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))))
1615abbidv 2808 . . . . . . . . . 10 (𝑟 = 𝑅 → {𝑗 ∣ ∃𝑘𝑖 (((deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})
1716mpteq2dv 5244 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 (((deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
188, 17mpteq12dv 5233 . . . . . . . 8 (𝑟 = 𝑅 → (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 (((deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))
19 df-ldgis 43134 . . . . . . . 8 ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 (((deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))
2018, 19, 7mptfvmpt 7248 . . . . . . 7 (𝑅 ∈ V → (ldgIdlSeq‘𝑅) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))
212, 20syl 17 . . . . . 6 (𝑅𝑉 → (ldgIdlSeq‘𝑅) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))
221, 21eqtrid 2789 . . . . 5 (𝑅𝑉𝑆 = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))
2322fveq1d 6908 . . . 4 (𝑅𝑉 → (𝑆𝐼) = ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼))
2423fveq1d 6908 . . 3 (𝑅𝑉 → ((𝑆𝐼)‘𝑋) = (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋))
25243ad2ant1 1134 . 2 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋))
26 rexeq 3322 . . . . . . 7 (𝑖 = 𝐼 → (∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))))
2726abbidv 2808 . . . . . 6 (𝑖 = 𝐼 → {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})
2827mpteq2dv 5244 . . . . 5 (𝑖 = 𝐼 → (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
29 eqid 2737 . . . . 5 (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
30 nn0ex 12532 . . . . . 6 0 ∈ V
3130mptex 7243 . . . . 5 (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) ∈ V
3228, 29, 31fvmpt 7016 . . . 4 (𝐼𝑈 → ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
3332fveq1d 6908 . . 3 (𝐼𝑈 → (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋) = ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})‘𝑋))
34333ad2ant2 1135 . 2 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋) = ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})‘𝑋))
35 eqid 2737 . . 3 (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})
36 breq2 5147 . . . . . 6 (𝑥 = 𝑋 → ((𝐷𝑘) ≤ 𝑥 ↔ (𝐷𝑘) ≤ 𝑋))
37 fveq2 6906 . . . . . . 7 (𝑥 = 𝑋 → ((coe1𝑘)‘𝑥) = ((coe1𝑘)‘𝑋))
3837eqeq2d 2748 . . . . . 6 (𝑥 = 𝑋 → (𝑗 = ((coe1𝑘)‘𝑥) ↔ 𝑗 = ((coe1𝑘)‘𝑋)))
3936, 38anbi12d 632 . . . . 5 (𝑥 = 𝑋 → (((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))))
4039rexbidv 3179 . . . 4 (𝑥 = 𝑋 → (∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))))
4140abbidv 2808 . . 3 (𝑥 = 𝑋 → {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})
42 simp3 1139 . . 3 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
43 simpr 484 . . . . . 6 (((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋)) → 𝑗 = ((coe1𝑘)‘𝑋))
4443reximi 3084 . . . . 5 (∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋)) → ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋))
4544ss2abi 4067 . . . 4 {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))} ⊆ {𝑗 ∣ ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋)}
46 abrexexg 7985 . . . . 5 (𝐼𝑈 → {𝑗 ∣ ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋)} ∈ V)
47463ad2ant2 1135 . . . 4 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → {𝑗 ∣ ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋)} ∈ V)
48 ssexg 5323 . . . 4 (({𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))} ⊆ {𝑗 ∣ ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋)} ∧ {𝑗 ∣ ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋)} ∈ V) → {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))} ∈ V)
4945, 47, 48sylancr 587 . . 3 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))} ∈ V)
5035, 41, 42, 49fvmptd3 7039 . 2 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})‘𝑋) = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})
5125, 34, 503eqtrd 2781 1 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  Vcvv 3480  wss 3951   class class class wbr 5143  cmpt 5225  cfv 6561  cle 11296  0cn0 12526  LIdealclidl 21216  Poly1cpl1 22178  coe1cco1 22179  deg1cdg1 26093  ldgIdlSeqcldgis 43133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267  df-n0 12527  df-ldgis 43134
This theorem is referenced by:  hbtlem2  43136  hbtlem4  43138  hbtlem3  43139  hbtlem5  43140  hbtlem6  43141
  Copyright terms: Public domain W3C validator