Theorem hbtlem3 43100

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 4007	. . . 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 4020	. 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 2729	. . . 4 ⊢ (deg1‘𝑅) = (deg1‘𝑅)
12	8, 9, 10, 11	hbtlem1 43096	. . 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 1373	. 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 43096	. . 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 1373	. 2 ⊢ (𝜑 → ((𝑆‘𝐽)‘𝑋) = {𝑎 ∣ ∃𝑏 ∈ 𝐽 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
17	4, 13, 16	3sstr4d 3993	1 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘𝐽)‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053   ⊆ wss 3905   class class class wbr 5095  ‘cfv 6486   ≤ cle 11169  ℕ₀cn0 12402  Ringcrg 20136  LIdealclidl 21131  Poly₁cpl1 22077  coe₁cco1 22078  deg₁cdg1 25975  ldgIdlSeqcldgis 43094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-1cn 11086  ax-addcl 11088

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-nn 12147  df-n0 12403  df-ldgis 43095

This theorem is referenced by:  hbt  43103