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Theorem hbtlem3 42429
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 4046 . . . 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 4055 . 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 2726 . . . 4 ( deg1 β€˜π‘…) = ( deg1 β€˜π‘…)
128, 9, 10, 11hbtlem1 42425 . . 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 42425 . . 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 4024 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 395   = wceq 1533   ∈ wcel 2098  {cab 2703  βˆƒwrex 3064   βŠ† wss 3943   class class class wbr 5141  β€˜cfv 6536   ≀ cle 11250  β„•0cn0 12473  Ringcrg 20135  LIdealclidl 21062  Poly1cpl1 22046  coe1cco1 22047   deg1 cdg1 25937  ldgIdlSeqcldgis 42423
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-1cn 11167  ax-addcl 11169
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-nn 12214  df-n0 12474  df-ldgis 42424
This theorem is referenced by:  hbt  42432
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