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Theorem hbtlem1 38302
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 3365 . . . . . . 7 (𝑅𝑉𝑅 ∈ V)
3 fveq2 6375 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
4 hbtlem.p . . . . . . . . . . . 12 𝑃 = (Poly1𝑅)
53, 4syl6eqr 2817 . . . . . . . . . . 11 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
65fveq2d 6379 . . . . . . . . . 10 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = (LIdeal‘𝑃))
7 hbtlem.u . . . . . . . . . 10 𝑈 = (LIdeal‘𝑃)
86, 7syl6eqr 2817 . . . . . . . . 9 (𝑟 = 𝑅 → (LIdeal‘(Poly1𝑟)) = 𝑈)
9 fveq2 6375 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
10 hbtlem.d . . . . . . . . . . . . . . . 16 𝐷 = ( deg1𝑅)
119, 10syl6eqr 2817 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅 → ( deg1𝑟) = 𝐷)
1211fveq1d 6377 . . . . . . . . . . . . . 14 (𝑟 = 𝑅 → (( deg1𝑟)‘𝑘) = (𝐷𝑘))
1312breq1d 4819 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → ((( deg1𝑟)‘𝑘) ≤ 𝑥 ↔ (𝐷𝑘) ≤ 𝑥))
1413anbi1d 623 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))))
1514rexbidv 3199 . . . . . . . . . . 11 (𝑟 = 𝑅 → (∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))))
1615abbidv 2884 . . . . . . . . . 10 (𝑟 = 𝑅 → {𝑗 ∣ ∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})
1716mpteq2dv 4904 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
188, 17mpteq12dv 4892 . . . . . . . 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 38301 . . . . . . . 8 ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((( deg1𝑟)‘𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))
2018, 19, 7mptfvmpt 6683 . . . . . . 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, 21syl5eq 2811 . . . . 5 (𝑅𝑉𝑆 = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})))
2322fveq1d 6377 . . . 4 (𝑅𝑉 → (𝑆𝐼) = ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼))
2423fveq1d 6377 . . 3 (𝑅𝑉 → ((𝑆𝐼)‘𝑋) = (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋))
25243ad2ant1 1163 . 2 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋))
26 rexeq 3287 . . . . . . 7 (𝑖 = 𝐼 → (∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))))
2726abbidv 2884 . . . . . 6 (𝑖 = 𝐼 → {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})
2827mpteq2dv 4904 . . . . 5 (𝑖 = 𝐼 → (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
29 eqid 2765 . . . . 5 (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})) = (𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
30 nn0ex 11545 . . . . . 6 0 ∈ V
3130mptex 6679 . . . . 5 (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) ∈ V
3228, 29, 31fvmpt 6471 . . . 4 (𝐼𝑈 → ((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))
3332fveq1d 6377 . . 3 (𝐼𝑈 → (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋) = ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})‘𝑋))
34333ad2ant2 1164 . 2 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → (((𝑖𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝑖 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}))‘𝐼)‘𝑋) = ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})‘𝑋))
35 simp3 1168 . . 3 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
36 simpr 477 . . . . . 6 (((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋)) → 𝑗 = ((coe1𝑘)‘𝑋))
3736reximi 3157 . . . . 5 (∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋)) → ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋))
3837ss2abi 3834 . . . 4 {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))} ⊆ {𝑗 ∣ ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋)}
39 abrexexg 7338 . . . . 5 (𝐼𝑈 → {𝑗 ∣ ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋)} ∈ V)
40393ad2ant2 1164 . . . 4 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → {𝑗 ∣ ∃𝑘𝐼 𝑗 = ((coe1𝑘)‘𝑋)} ∈ V)
41 ssexg 4965 . . . 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)
4238, 40, 41sylancr 581 . . 3 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))} ∈ V)
43 breq2 4813 . . . . . . 7 (𝑥 = 𝑋 → ((𝐷𝑘) ≤ 𝑥 ↔ (𝐷𝑘) ≤ 𝑋))
44 fveq2 6375 . . . . . . . 8 (𝑥 = 𝑋 → ((coe1𝑘)‘𝑥) = ((coe1𝑘)‘𝑋))
4544eqeq2d 2775 . . . . . . 7 (𝑥 = 𝑋 → (𝑗 = ((coe1𝑘)‘𝑥) ↔ 𝑗 = ((coe1𝑘)‘𝑋)))
4643, 45anbi12d 624 . . . . . 6 (𝑥 = 𝑋 → (((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))))
4746rexbidv 3199 . . . . 5 (𝑥 = 𝑋 → (∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥)) ↔ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))))
4847abbidv 2884 . . . 4 (𝑥 = 𝑋 → {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})
49 eqid 2765 . . . 4 (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})
5048, 49fvmptg 6469 . . 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))})
5135, 42, 50syl2anc 579 . 2 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑥𝑗 = ((coe1𝑘)‘𝑥))})‘𝑋) = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})
5225, 34, 513eqtrd 2803 1 ((𝑅𝑉𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑗 ∣ ∃𝑘𝐼 ((𝐷𝑘) ≤ 𝑋𝑗 = ((coe1𝑘)‘𝑋))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  {cab 2751  wrex 3056  Vcvv 3350  wss 3732   class class class wbr 4809  cmpt 4888  cfv 6068  cle 10329  0cn0 11538  LIdealclidl 19444  Poly1cpl1 19820  coe1cco1 19821   deg1 cdg1 24105  ldgIdlSeqcldgis 38300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-1cn 10247  ax-addcl 10249
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-nn 11275  df-n0 11539  df-ldgis 38301
This theorem is referenced by:  hbtlem2  38303  hbtlem4  38305  hbtlem3  38306  hbtlem5  38307  hbtlem6  38308
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