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Theorem hbtlem3 41854
Description: The leading ideal function is monotone. (Contributed by Stefan O'Rear, 31-Mar-2015.)
Hypotheses
Ref Expression
hbtlem.p 𝑃 = (Poly1β€˜π‘…)
hbtlem.u π‘ˆ = (LIdealβ€˜π‘ƒ)
hbtlem.s 𝑆 = (ldgIdlSeqβ€˜π‘…)
hbtlem3.r (πœ‘ β†’ 𝑅 ∈ Ring)
hbtlem3.i (πœ‘ β†’ 𝐼 ∈ π‘ˆ)
hbtlem3.j (πœ‘ β†’ 𝐽 ∈ π‘ˆ)
hbtlem3.ij (πœ‘ β†’ 𝐼 βŠ† 𝐽)
hbtlem3.x (πœ‘ β†’ 𝑋 ∈ β„•0)
Assertion
Ref Expression
hbtlem3 (πœ‘ β†’ ((π‘†β€˜πΌ)β€˜π‘‹) βŠ† ((π‘†β€˜π½)β€˜π‘‹))

Proof of Theorem hbtlem3
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbtlem3.ij . . . 4 (πœ‘ β†’ 𝐼 βŠ† 𝐽)
2 ssrexv 4050 . . . 4 (𝐼 βŠ† 𝐽 β†’ (βˆƒπ‘ ∈ 𝐼 ((( deg1 β€˜π‘…)β€˜π‘) ≀ 𝑋 ∧ π‘Ž = ((coe1β€˜π‘)β€˜π‘‹)) β†’ βˆƒπ‘ ∈ 𝐽 ((( deg1 β€˜π‘…)β€˜π‘) ≀ 𝑋 ∧ π‘Ž = ((coe1β€˜π‘)β€˜π‘‹))))
31, 2syl 17 . . 3 (πœ‘ β†’ (βˆƒπ‘ ∈ 𝐼 ((( deg1 β€˜π‘…)β€˜π‘) ≀ 𝑋 ∧ π‘Ž = ((coe1β€˜π‘)β€˜π‘‹)) β†’ βˆƒπ‘ ∈ 𝐽 ((( deg1 β€˜π‘…)β€˜π‘) ≀ 𝑋 ∧ π‘Ž = ((coe1β€˜π‘)β€˜π‘‹))))
43ss2abdv 4059 . 2 (πœ‘ β†’ {π‘Ž ∣ βˆƒπ‘ ∈ 𝐼 ((( deg1 β€˜π‘…)β€˜π‘) ≀ 𝑋 ∧ π‘Ž = ((coe1β€˜π‘)β€˜π‘‹))} βŠ† {π‘Ž ∣ βˆƒπ‘ ∈ 𝐽 ((( deg1 β€˜π‘…)β€˜π‘) ≀ 𝑋 ∧ π‘Ž = ((coe1β€˜π‘)β€˜π‘‹))})
5 hbtlem3.r . . 3 (πœ‘ β†’ 𝑅 ∈ Ring)
6 hbtlem3.i . . 3 (πœ‘ β†’ 𝐼 ∈ π‘ˆ)
7 hbtlem3.x . . 3 (πœ‘ β†’ 𝑋 ∈ β„•0)
8 hbtlem.p . . . 4 𝑃 = (Poly1β€˜π‘…)
9 hbtlem.u . . . 4 π‘ˆ = (LIdealβ€˜π‘ƒ)
10 hbtlem.s . . . 4 𝑆 = (ldgIdlSeqβ€˜π‘…)
11 eqid 2732 . . . 4 ( deg1 β€˜π‘…) = ( deg1 β€˜π‘…)
128, 9, 10, 11hbtlem1 41850 . . 3 ((𝑅 ∈ Ring ∧ 𝐼 ∈ π‘ˆ ∧ 𝑋 ∈ β„•0) β†’ ((π‘†β€˜πΌ)β€˜π‘‹) = {π‘Ž ∣ βˆƒπ‘ ∈ 𝐼 ((( deg1 β€˜π‘…)β€˜π‘) ≀ 𝑋 ∧ π‘Ž = ((coe1β€˜π‘)β€˜π‘‹))})
135, 6, 7, 12syl3anc 1371 . 2 (πœ‘ β†’ ((π‘†β€˜πΌ)β€˜π‘‹) = {π‘Ž ∣ βˆƒπ‘ ∈ 𝐼 ((( deg1 β€˜π‘…)β€˜π‘) ≀ 𝑋 ∧ π‘Ž = ((coe1β€˜π‘)β€˜π‘‹))})
14 hbtlem3.j . . 3 (πœ‘ β†’ 𝐽 ∈ π‘ˆ)
158, 9, 10, 11hbtlem1 41850 . . 3 ((𝑅 ∈ Ring ∧ 𝐽 ∈ π‘ˆ ∧ 𝑋 ∈ β„•0) β†’ ((π‘†β€˜π½)β€˜π‘‹) = {π‘Ž ∣ βˆƒπ‘ ∈ 𝐽 ((( deg1 β€˜π‘…)β€˜π‘) ≀ 𝑋 ∧ π‘Ž = ((coe1β€˜π‘)β€˜π‘‹))})
165, 14, 7, 15syl3anc 1371 . 2 (πœ‘ β†’ ((π‘†β€˜π½)β€˜π‘‹) = {π‘Ž ∣ βˆƒπ‘ ∈ 𝐽 ((( deg1 β€˜π‘…)β€˜π‘) ≀ 𝑋 ∧ π‘Ž = ((coe1β€˜π‘)β€˜π‘‹))})
174, 13, 163sstr4d 4028 1 (πœ‘ β†’ ((π‘†β€˜πΌ)β€˜π‘‹) βŠ† ((π‘†β€˜π½)β€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  βˆƒwrex 3070   βŠ† wss 3947   class class class wbr 5147  β€˜cfv 6540   ≀ cle 11245  β„•0cn0 12468  Ringcrg 20049  LIdealclidl 20775  Poly1cpl1 21692  coe1cco1 21693   deg1 cdg1 25560  ldgIdlSeqcldgis 41848
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-1cn 11164  ax-addcl 11166
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-nn 12209  df-n0 12469  df-ldgis 41849
This theorem is referenced by:  hbt  41857
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