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Theorem hbtlem1 36712
 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 3185 . . . . . . 7 (𝑅𝑉𝑅 ∈ V)
3 fveq2 6103 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
4 hbtlem.p . . . . . . . . . . . 12 𝑃 = (Poly1𝑅)
53, 4syl6eqr 2662 . . . . . . . . . . 11 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
65fveq2d 6107 . . . . . . . . . 10 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = (LIdeal‘𝑃))
7 hbtlem.u . . . . . . . . . 10 𝑈 = (LIdeal‘𝑃)
86, 7syl6eqr 2662 . . . . . . . . 9 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = 𝑈)
9 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
10 hbtlem.d . . . . . . . . . . . . . . . 16 𝐷 = ( deg1𝑅)
119, 10syl6eqr 2662 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅 → ( deg1𝑟) = 𝐷)
1211fveq1d 6105 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → (( deg1𝑟)‘𝑘) = (𝐷𝑘))
1312breq1d 4593 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → ((( deg1𝑟)‘𝑘) ≤ 𝑥 ↔ (𝐷𝑘) ≤ 𝑥))
1413anbi1d 737 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))))
1514rexbidv 3034 . . . . . . . . . . 11 (𝑟 = 𝑅 → (∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))))
1615abbidv 2728 . . . . . . . . . 10 (𝑟 = 𝑅 → {𝑗 ∣ ∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})
1716mpteq2dv 4673 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
188, 17mpteq12dv 4663 . . . . . . . 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 36711 . . . . . . . 8 ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))
20 fvex 6113 . . . . . . . . . 10 (LIdeal‘𝑃) ∈ V
217, 20eqeltri 2684 . . . . . . . . 9 𝑈 ∈ V
2221mptex 6390 . . . . . . . 8 (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})) ∈ V
2318, 19, 22fvmpt 6191 . . . . . . 7 (𝑅 ∈ V → (ldgIdlSeq‘𝑅) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))
242, 23syl 17 . . . . . 6 (𝑅𝑉 → (ldgIdlSeq‘𝑅) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))
251, 24syl5eq 2656 . . . . 5 (𝑅𝑉𝑆 = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))
2625fveq1d 6105 . . . 4 (𝑅𝑉 → (𝑆𝐼) = ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼))
2726fveq1d 6105 . . 3 (𝑅𝑉 → ((𝑆𝐼)‘𝑋) = (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋))
28273ad2ant1 1075 . 2 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋))
29 rexeq 3116 . . . . . . 7 (𝑖 = 𝐼 → (∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))))
3029abbidv 2728 . . . . . 6 (𝑖 = 𝐼 → {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})
3130mpteq2dv 4673 . . . . 5 (𝑖 = 𝐼 → (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
32 eqid 2610 . . . . 5 (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
33 nn0ex 11175 . . . . . 6 0 ∈ V
3433mptex 6390 . . . . 5 (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) ∈ V
3531, 32, 34fvmpt 6191 . . . 4 (𝐼𝑈 → ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
3635fveq1d 6105 . . 3 (𝐼𝑈 → (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋) = ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})‘𝑋))
37363ad2ant2 1076 . 2 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋) = ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})‘𝑋))
38 simp3 1056 . . 3 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
39 simpr 476 . . . . . 6 (((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋)) → 𝑗 = ((coe1𝑘)‘𝑋))
4039reximi 2994 . . . . 5 (∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋)) → ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋))
4140ss2abi 3637 . . . 4 {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))} ⊆ {𝑗 ∣ ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋)}
42 abrexexg 7034 . . . . 5 (𝐼𝑈 → {𝑗 ∣ ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋)} ∈ V)
43423ad2ant2 1076 . . . 4 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → {𝑗 ∣ ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋)} ∈ V)
44 ssexg 4732 . . . 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)
4541, 43, 44sylancr 694 . . 3 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))} ∈ V)
46 breq2 4587 . . . . . . 7 (𝑥 = 𝑋 → ((𝐷𝑘) ≤ 𝑥 ↔ (𝐷𝑘) ≤ 𝑋))
47 fveq2 6103 . . . . . . . 8 (𝑥 = 𝑋 → ((coe1𝑘)‘𝑥) = ((coe1𝑘)‘𝑋))
4847eqeq2d 2620 . . . . . . 7 (𝑥 = 𝑋 → (𝑗 = ((coe1𝑘)‘𝑥) ↔ 𝑗 = ((coe1𝑘)‘𝑋)))
4946, 48anbi12d 743 . . . . . 6 (𝑥 = 𝑋 → (((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))))
5049rexbidv 3034 . . . . 5 (𝑥 = 𝑋 → (∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))))
5150abbidv 2728 . . . 4 (𝑥 = 𝑋 → {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})
52 eqid 2610 . . . 4 (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})
5351, 52fvmptg 6189 . . 3 ((𝑋 ∈ ℕ0 ∧ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))} ∈ V) → ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})‘𝑋) = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})
5438, 45, 53syl2anc 691 . 2 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})‘𝑋) = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})
5528, 37, 543eqtrd 2648 1 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804   ≤ cle 9954  ℕ0cn0 11169  LIdealclidl 18991  Poly1cpl1 19368  coe1cco1 19369   deg1 cdg1 23618  ldgIdlSeqcldgis 36710 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rrecex 9887  ax-cnre 9888 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-nn 10898  df-n0 11170  df-ldgis 36711 This theorem is referenced by:  hbtlem2  36713  hbtlem4  36715  hbtlem3  36716  hbtlem5  36717  hbtlem6  36718
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