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Theorem hbtlem1 37160
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 3203 . . . . . . 7 (𝑅𝑉𝑅 ∈ V)
3 fveq2 6150 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
4 hbtlem.p . . . . . . . . . . . 12 𝑃 = (Poly1𝑅)
53, 4syl6eqr 2678 . . . . . . . . . . 11 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
65fveq2d 6154 . . . . . . . . . 10 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = (LIdeal‘𝑃))
7 hbtlem.u . . . . . . . . . 10 𝑈 = (LIdeal‘𝑃)
86, 7syl6eqr 2678 . . . . . . . . 9 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = 𝑈)
9 fveq2 6150 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
10 hbtlem.d . . . . . . . . . . . . . . . 16 𝐷 = ( deg1𝑅)
119, 10syl6eqr 2678 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅 → ( deg1𝑟) = 𝐷)
1211fveq1d 6152 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → (( deg1𝑟)‘𝑘) = (𝐷𝑘))
1312breq1d 4628 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → ((( deg1𝑟)‘𝑘) ≤ 𝑥 ↔ (𝐷𝑘) ≤ 𝑥))
1413anbi1d 740 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))))
1514rexbidv 3050 . . . . . . . . . . 11 (𝑟 = 𝑅 → (∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))))
1615abbidv 2744 . . . . . . . . . 10 (𝑟 = 𝑅 → {𝑗 ∣ ∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})
1716mpteq2dv 4710 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
188, 17mpteq12dv 4698 . . . . . . . 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 37159 . . . . . . . 8 ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))
20 fvex 6160 . . . . . . . . . 10 (LIdeal‘𝑃) ∈ V
217, 20eqeltri 2700 . . . . . . . . 9 𝑈 ∈ V
2221mptex 6441 . . . . . . . 8 (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})) ∈ V
2318, 19, 22fvmpt 6240 . . . . . . 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 2672 . . . . 5 (𝑅𝑉𝑆 = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))
2625fveq1d 6152 . . . 4 (𝑅𝑉 → (𝑆𝐼) = ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼))
2726fveq1d 6152 . . 3 (𝑅𝑉 → ((𝑆𝐼)‘𝑋) = (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋))
28273ad2ant1 1080 . 2 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋))
29 rexeq 3133 . . . . . . 7 (𝑖 = 𝐼 → (∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))))
3029abbidv 2744 . . . . . 6 (𝑖 = 𝐼 → {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})
3130mpteq2dv 4710 . . . . 5 (𝑖 = 𝐼 → (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
32 eqid 2626 . . . . 5 (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
33 nn0ex 11243 . . . . . 6 0 ∈ V
3433mptex 6441 . . . . 5 (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) ∈ V
3531, 32, 34fvmpt 6240 . . . 4 (𝐼𝑈 → ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
3635fveq1d 6152 . . 3 (𝐼𝑈 → (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋) = ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})‘𝑋))
37363ad2ant2 1081 . 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 1061 . . 3 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
39 simpr 477 . . . . . 6 (((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋)) → 𝑗 = ((coe1𝑘)‘𝑋))
4039reximi 3010 . . . . 5 (∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋)) → ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋))
4140ss2abi 3658 . . . 4 {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))} ⊆ {𝑗 ∣ ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋)}
42 abrexexg 7091 . . . . 5 (𝐼𝑈 → {𝑗 ∣ ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋)} ∈ V)
43423ad2ant2 1081 . . . 4 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → {𝑗 ∣ ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋)} ∈ V)
44 ssexg 4769 . . . 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 4622 . . . . . . 7 (𝑥 = 𝑋 → ((𝐷𝑘) ≤ 𝑥 ↔ (𝐷𝑘) ≤ 𝑋))
47 fveq2 6150 . . . . . . . 8 (𝑥 = 𝑋 → ((coe1𝑘)‘𝑥) = ((coe1𝑘)‘𝑋))
4847eqeq2d 2636 . . . . . . 7 (𝑥 = 𝑋 → (𝑗 = ((coe1𝑘)‘𝑥) ↔ 𝑗 = ((coe1𝑘)‘𝑋)))
4946, 48anbi12d 746 . . . . . 6 (𝑥 = 𝑋 → (((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))))
5049rexbidv 3050 . . . . 5 (𝑥 = 𝑋 → (∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))))
5150abbidv 2744 . . . 4 (𝑥 = 𝑋 → {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})
52 eqid 2626 . . . 4 (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})
5351, 52fvmptg 6238 . . 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 692 . 2 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})‘𝑋) = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})
5528, 37, 543eqtrd 2664 1 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  {cab 2612  wrex 2913  Vcvv 3191  wss 3560   class class class wbr 4618  cmpt 4678  cfv 5850  cle 10020  0cn0 11237  LIdealclidl 19084  Poly1cpl1 19461  coe1cco1 19462   deg1 cdg1 23713  ldgIdlSeqcldgis 37158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-i2m1 9949  ax-1ne0 9950  ax-rrecex 9953  ax-cnre 9954
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-nn 10966  df-n0 11238  df-ldgis 37159
This theorem is referenced by:  hbtlem2  37161  hbtlem4  37163  hbtlem3  37164  hbtlem5  37165  hbtlem6  37166
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