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Theorem hbtlem3 37523
 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 3665 . . . 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 3673 . 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 2621 . . . 4 ( deg1𝑅) = ( deg1𝑅)
128, 9, 10, 11hbtlem1 37519 . . 3 ((𝑅 ∈ Ring ∧ 𝐼𝑈𝑋 ∈ ℕ0) → ((𝑆𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
135, 6, 7, 12syl3anc 1325 . 2 (𝜑 → ((𝑆𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏𝐼 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
14 hbtlem3.j . . 3 (𝜑𝐽𝑈)
158, 9, 10, 11hbtlem1 37519 . . 3 ((𝑅 ∈ Ring ∧ 𝐽𝑈𝑋 ∈ ℕ0) → ((𝑆𝐽)‘𝑋) = {𝑎 ∣ ∃𝑏𝐽 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
165, 14, 7, 15syl3anc 1325 . 2 (𝜑 → ((𝑆𝐽)‘𝑋) = {𝑎 ∣ ∃𝑏𝐽 ((( deg1𝑅)‘𝑏) ≤ 𝑋𝑎 = ((coe1𝑏)‘𝑋))})
174, 13, 163sstr4d 3646 1 (𝜑 → ((𝑆𝐼)‘𝑋) ⊆ ((𝑆𝐽)‘𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1482   ∈ wcel 1989  {cab 2607  ∃wrex 2912   ⊆ wss 3572   class class class wbr 4651  ‘cfv 5886   ≤ cle 10072  ℕ0cn0 11289  Ringcrg 18541  LIdealclidl 19164  Poly1cpl1 19541  coe1cco1 19542   deg1 cdg1 23808  ldgIdlSeqcldgis 37517 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-i2m1 10001  ax-1ne0 10002  ax-rrecex 10005  ax-cnre 10006 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-om 7063  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-nn 11018  df-n0 11290  df-ldgis 37518 This theorem is referenced by:  hbt  37526
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