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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  0cn0 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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