Theorem hbtlem3 43642

Description: The leading ideal function is monotone. (Contributed by Stefan O'Rear, 31-Mar-2015.)

Hypotheses

Ref	Expression
hbtlem.p	⊢ 𝑃 = (Poly1‘𝑅)
hbtlem.u	⊢ 𝑈 = (LIdeal‘𝑃)
hbtlem.s	⊢ 𝑆 = (ldgIdlSeq‘𝑅)
hbtlem3.r	⊢ (𝜑 → 𝑅 ∈ Ring)
hbtlem3.i	⊢ (𝜑 → 𝐼 ∈ 𝑈)
hbtlem3.j	⊢ (𝜑 → 𝐽 ∈ 𝑈)
hbtlem3.ij	⊢ (𝜑 → 𝐼 ⊆ 𝐽)
hbtlem3.x	⊢ (𝜑 → 𝑋 ∈ ℕ0)

Assertion

Ref	Expression
hbtlem3	⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘𝐽)‘𝑋))

Proof of Theorem hbtlem3

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hbtlem3.ij	. . . 4 ⊢ (𝜑 → 𝐼 ⊆ 𝐽)
2		ssrexv 3997	. . . 4 ⊢ (𝐼 ⊆ 𝐽 → (∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) → ∃𝑏 ∈ 𝐽 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))))
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → (∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) → ∃𝑏 ∈ 𝐽 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))))
4	3	ss2abdv 4009	. 2 ⊢ (𝜑 → {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ⊆ {𝑎 ∣ ∃𝑏 ∈ 𝐽 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
5		hbtlem3.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
6		hbtlem3.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑈)
7		hbtlem3.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℕ0)
8		hbtlem.p	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
9		hbtlem.u	. . . 4 ⊢ 𝑈 = (LIdeal‘𝑃)
10		hbtlem.s	. . . 4 ⊢ 𝑆 = (ldgIdlSeq‘𝑅)
11		eqid 2752	. . . 4 ⊢ (deg1‘𝑅) = (deg1‘𝑅)
12	8, 9, 10, 11	hbtlem1 43638	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
13	5, 6, 7, 12	syl3anc 1382	. 2 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
14		hbtlem3.j	. . 3 ⊢ (𝜑 → 𝐽 ∈ 𝑈)
15	8, 9, 10, 11	hbtlem1 43638	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐽 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐽)‘𝑋) = {𝑎 ∣ ∃𝑏 ∈ 𝐽 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
16	5, 14, 7, 15	syl3anc 1382	. 2 ⊢ (𝜑 → ((𝑆‘𝐽)‘𝑋) = {𝑎 ∣ ∃𝑏 ∈ 𝐽 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
17	4, 13, 16	3sstr4d 3982	1 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘𝐽)‘𝑋))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1550   ∈ wcel 2132  {cab 2730  ∃wrex 3076   ⊆ wss 3895   class class class wbr 5090  ‘cfv 6506   ≤ cle 11203  ℕ₀cn0 12467  Ringcrg 20251  LIdealclidl 21245  Poly₁cpl1 22208  coe₁cco1 22209  deg₁cdg1 26083  ldgIdlSeqcldgis 43636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pr 5380  ax-un 7703  ax-cnex 11115  ax-1cn 11117  ax-addcl 11119

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-ov 7384  df-om 7832  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-nn 12197  df-n0 12468  df-ldgis 43637

This theorem is referenced by:  hbt  43645