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Theorem hbtlem3 41483
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 4016 . . . 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 4025 . 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 2737 . . . 4 ( deg1 β€˜π‘…) = ( deg1 β€˜π‘…)
128, 9, 10, 11hbtlem1 41479 . . 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 1372 . 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 41479 . . 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 1372 . 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 3996 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 397   = wceq 1542   ∈ wcel 2107  {cab 2714  βˆƒwrex 3074   βŠ† wss 3915   class class class wbr 5110  β€˜cfv 6501   ≀ cle 11197  β„•0cn0 12420  Ringcrg 19971  LIdealclidl 20647  Poly1cpl1 21564  coe1cco1 21565   deg1 cdg1 25432  ldgIdlSeqcldgis 41477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-1cn 11116  ax-addcl 11118
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-nn 12161  df-n0 12421  df-ldgis 41478
This theorem is referenced by:  hbt  41486
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