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Theorem hbtlem1 41436
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 3463 . . . . . . 7 (𝑅𝑉𝑅 ∈ V)
3 fveq2 6842 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
4 hbtlem.p . . . . . . . . . . . 12 𝑃 = (Poly1𝑅)
53, 4eqtr4di 2794 . . . . . . . . . . 11 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
65fveq2d 6846 . . . . . . . . . 10 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = (LIdeal‘𝑃))
7 hbtlem.u . . . . . . . . . 10 𝑈 = (LIdeal‘𝑃)
86, 7eqtr4di 2794 . . . . . . . . 9 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = 𝑈)
9 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
10 hbtlem.d . . . . . . . . . . . . . . . 16 𝐷 = ( deg1𝑅)
119, 10eqtr4di 2794 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅 → ( deg1𝑟) = 𝐷)
1211fveq1d 6844 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → (( deg1𝑟)‘𝑘) = (𝐷𝑘))
1312breq1d 5115 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → ((( deg1𝑟)‘𝑘) ≤ 𝑥 ↔ (𝐷𝑘) ≤ 𝑥))
1413anbi1d 630 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))))
1514rexbidv 3175 . . . . . . . . . . 11 (𝑟 = 𝑅 → (∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))))
1615abbidv 2805 . . . . . . . . . 10 (𝑟 = 𝑅 → {𝑗 ∣ ∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})
1716mpteq2dv 5207 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
188, 17mpteq12dv 5196 . . . . . . . 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 41435 . . . . . . . 8 ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))
2018, 19, 7mptfvmpt 7178 . . . . . . 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 2788 . . . . 5 (𝑅𝑉𝑆 = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))
2322fveq1d 6844 . . . 4 (𝑅𝑉 → (𝑆𝐼) = ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼))
2423fveq1d 6844 . . 3 (𝑅𝑉 → ((𝑆𝐼)‘𝑋) = (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋))
25243ad2ant1 1133 . 2 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋))
26 rexeq 3310 . . . . . . 7 (𝑖 = 𝐼 → (∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))))
2726abbidv 2805 . . . . . 6 (𝑖 = 𝐼 → {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})
2827mpteq2dv 5207 . . . . 5 (𝑖 = 𝐼 → (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
29 eqid 2736 . . . . 5 (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
30 nn0ex 12419 . . . . . 6 0 ∈ V
3130mptex 7173 . . . . 5 (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) ∈ V
3228, 29, 31fvmpt 6948 . . . 4 (𝐼𝑈 → ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
3332fveq1d 6844 . . 3 (𝐼𝑈 → (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋) = ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})‘𝑋))
34333ad2ant2 1134 . 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 2736 . . 3 (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})
36 breq2 5109 . . . . . 6 (𝑥 = 𝑋 → ((𝐷𝑘) ≤ 𝑥 ↔ (𝐷𝑘) ≤ 𝑋))
37 fveq2 6842 . . . . . . 7 (𝑥 = 𝑋 → ((coe1𝑘)‘𝑥) = ((coe1𝑘)‘𝑋))
3837eqeq2d 2747 . . . . . 6 (𝑥 = 𝑋 → (𝑗 = ((coe1𝑘)‘𝑥) ↔ 𝑗 = ((coe1𝑘)‘𝑋)))
3936, 38anbi12d 631 . . . . 5 (𝑥 = 𝑋 → (((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))))
4039rexbidv 3175 . . . 4 (𝑥 = 𝑋 → (∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))))
4140abbidv 2805 . . 3 (𝑥 = 𝑋 → {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})
42 simp3 1138 . . 3 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
43 simpr 485 . . . . . 6 (((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋)) → 𝑗 = ((coe1𝑘)‘𝑋))
4443reximi 3087 . . . . 5 (∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋)) → ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋))
4544ss2abi 4023 . . . 4 {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))} ⊆ {𝑗 ∣ ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋)}
46 abrexexg 7893 . . . . 5 (𝐼𝑈 → {𝑗 ∣ ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋)} ∈ V)
47463ad2ant2 1134 . . . 4 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → {𝑗 ∣ ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋)} ∈ V)
48 ssexg 5280 . . . 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 6971 . 2 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})‘𝑋) = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})
5125, 34, 503eqtrd 2780 1 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wrex 3073  Vcvv 3445  wss 3910   class class class wbr 5105  cmpt 5188  cfv 6496  cle 11190  0cn0 12413  LIdealclidl 20631  Poly1cpl1 21548  coe1cco1 21549   deg1 cdg1 25416  ldgIdlSeqcldgis 41434
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-1cn 11109  ax-addcl 11111
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-nn 12154  df-n0 12414  df-ldgis 41435
This theorem is referenced by:  hbtlem2  41437  hbtlem4  41439  hbtlem3  41440  hbtlem5  41441  hbtlem6  41442
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