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Theorem hbtlem3 39882
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 4010 . . . 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 4020 . 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 2821 . . . 4 ( deg1𝑅) = ( deg1𝑅)
128, 9, 10, 11hbtlem1 39878 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
135, 6, 7, 12syl3anc 1368 . 2 (𝜑 → ((𝑆𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
14 hbtlem3.j . . 3 (𝜑𝐽𝑈)
158, 9, 10, 11hbtlem1 39878 . . 3 ((𝑅 ∈ Ring ∧ 𝐽𝑈𝑋 ∈ ℕ0) → ((𝑆𝐽)‘𝑋) = {𝑎 ∣ ∃𝑏𝐽 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
165, 14, 7, 15syl3anc 1368 . 2 (𝜑 → ((𝑆𝐽)‘𝑋) = {𝑎 ∣ ∃𝑏𝐽 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
174, 13, 163sstr4d 3990 1 (𝜑 → ((𝑆𝐼)‘𝑋) ⊆ ((𝑆𝐽)‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {cab 2799  wrex 3127  wss 3910   class class class wbr 5039  cfv 6328  cle 10653  0cn0 11875  Ringcrg 19276  LIdealclidl 19918  Poly1cpl1 20321  coe1cco1 20322   deg1 cdg1 24634  ldgIdlSeqcldgis 39876
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-1cn 10572  ax-addcl 10574
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-om 7556  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-nn 11616  df-n0 11876  df-ldgis 39877
This theorem is referenced by:  hbt  39885
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