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Theorem hbtlem3 36716
Description: The leading ideal function is monotone. (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)
hbtlem3.r (𝜑𝑅 ∈ Ring)
hbtlem3.i (𝜑𝐼𝑈)
hbtlem3.j (𝜑𝐽𝑈)
hbtlem3.ij (𝜑𝐼𝐽)
hbtlem3.x (𝜑𝑋 ∈ ℕ0)
Assertion
Ref Expression
hbtlem3 (𝜑 → ((𝑆𝐼)‘𝑋) ⊆ ((𝑆𝐽)‘𝑋))

Proof of Theorem hbtlem3
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hbtlem3.ij . . . 4 (𝜑𝐼𝐽)
2 ssrexv 3630 . . . 4 (𝐼𝐽 → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) → ∃𝑏𝐽 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))))
31, 2syl 17 . . 3 (𝜑 → (∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋)) → ∃𝑏𝐽 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))))
43ss2abdv 3638 . 2 (𝜑 → {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))} ⊆ {𝑎 ∣ ∃𝑏𝐽 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
5 hbtlem3.r . . 3 (𝜑𝑅 ∈ Ring)
6 hbtlem3.i . . 3 (𝜑𝐼𝑈)
7 hbtlem3.x . . 3 (𝜑𝑋 ∈ ℕ0)
8 hbtlem.p . . . 4 𝑃 = (Poly1𝑅)
9 hbtlem.u . . . 4 𝑈 = (LIdeal‘𝑃)
10 hbtlem.s . . . 4 𝑆 = (ldgIdlSeq‘𝑅)
11 eqid 2610 . . . 4 ( deg1𝑅) = ( deg1𝑅)
128, 9, 10, 11hbtlem1 36712 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
135, 6, 7, 12syl3anc 1318 . 2 (𝜑 → ((𝑆𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
14 hbtlem3.j . . 3 (𝜑𝐽𝑈)
158, 9, 10, 11hbtlem1 36712 . . 3 ((𝑅 ∈ Ring ∧ 𝐽𝑈𝑋 ∈ ℕ0) → ((𝑆𝐽)‘𝑋) = {𝑎 ∣ ∃𝑏𝐽 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
165, 14, 7, 15syl3anc 1318 . 2 (𝜑 → ((𝑆𝐽)‘𝑋) = {𝑎 ∣ ∃𝑏𝐽 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
174, 13, 163sstr4d 3611 1 (𝜑 → ((𝑆𝐼)‘𝑋) ⊆ ((𝑆𝐽)‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  {cab 2596  wrex 2897  wss 3540   class class class wbr 4583  cfv 5804  cle 9954  0cn0 11169  Ringcrg 18370  LIdealclidl 18991  Poly1cpl1 19368  coe1cco1 19369   deg1 cdg1 23618  ldgIdlSeqcldgis 36710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rrecex 9887  ax-cnre 9888
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-nn 10898  df-n0 11170  df-ldgis 36711
This theorem is referenced by:  hbt  36719
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