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Theorem hbtlem3 42582
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 4051 . . . 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 4060 . 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 2728 . . . 4 ( deg1 β€˜π‘…) = ( deg1 β€˜π‘…)
128, 9, 10, 11hbtlem1 42578 . . 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 1368 . 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 42578 . . 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 1368 . 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 4029 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 394   = wceq 1533   ∈ wcel 2098  {cab 2705  βˆƒwrex 3067   βŠ† wss 3949   class class class wbr 5152  β€˜cfv 6553   ≀ cle 11287  β„•0cn0 12510  Ringcrg 20180  LIdealclidl 21109  Poly1cpl1 22103  coe1cco1 22104   deg1 cdg1 26007  ldgIdlSeqcldgis 42576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-1cn 11204  ax-addcl 11206
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-nn 12251  df-n0 12511  df-ldgis 42577
This theorem is referenced by:  hbt  42585
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