Theorem hbtlem7 40866

Description: Functionality of leading coefficient ideal sequence. (Contributed by Stefan O'Rear, 4-Apr-2015.)

Hypotheses

Ref	Expression
hbtlem.p	⊢ 𝑃 = (Poly1‘𝑅)
hbtlem.u	⊢ 𝑈 = (LIdeal‘𝑃)
hbtlem.s	⊢ 𝑆 = (ldgIdlSeq‘𝑅)
hbtlem7.t	⊢ 𝑇 = (LIdeal‘𝑅)

Assertion

Ref	Expression
hbtlem7	⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝑆‘𝐼):ℕ0⟶𝑇)

Proof of Theorem hbtlem7

Dummy variables 𝑖 𝑗 𝑥 𝑦 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . . . . 9 ⊢ (((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥)) → 𝑦 = ((coe1‘𝑗)‘𝑥))
2	1	reximi 3174	. . . . . . . 8 ⊢ (∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥)) → ∃𝑗 ∈ 𝐼 𝑦 = ((coe1‘𝑗)‘𝑥))
3	2	ss2abi 3996	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ⊆ {𝑦 ∣ ∃𝑗 ∈ 𝐼 𝑦 = ((coe1‘𝑗)‘𝑥)}
4		abrexexg 7777	. . . . . . 7 ⊢ (𝐼 ∈ 𝑈 → {𝑦 ∣ ∃𝑗 ∈ 𝐼 𝑦 = ((coe1‘𝑗)‘𝑥)} ∈ V)
5		ssexg 5242	. . . . . . 7 ⊢ (({𝑦 ∣ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ⊆ {𝑦 ∣ ∃𝑗 ∈ 𝐼 𝑦 = ((coe1‘𝑗)‘𝑥)} ∧ {𝑦 ∣ ∃𝑗 ∈ 𝐼 𝑦 = ((coe1‘𝑗)‘𝑥)} ∈ V) → {𝑦 ∣ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ∈ V)
6	3, 4, 5	sylancr 586	. . . . . 6 ⊢ (𝐼 ∈ 𝑈 → {𝑦 ∣ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ∈ V)
7	6	ralrimivw 3108	. . . . 5 ⊢ (𝐼 ∈ 𝑈 → ∀𝑥 ∈ ℕ0 {𝑦 ∣ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ∈ V)
8	7	adantl 481	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ∀𝑥 ∈ ℕ0 {𝑦 ∣ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ∈ V)
9		eqid 2738	. . . . 5 ⊢ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))})
10	9	fnmpt 6557	. . . 4 ⊢ (∀𝑥 ∈ ℕ0 {𝑦 ∣ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ∈ V → (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}) Fn ℕ0)
11	8, 10	syl 17	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}) Fn ℕ0)
12		hbtlem.s	. . . . . . 7 ⊢ 𝑆 = (ldgIdlSeq‘𝑅)
13		elex 3440	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
14		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
15		hbtlem.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Poly1‘𝑅)
16	14, 15	eqtr4di 2797	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
17	16	fveq2d 6760	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = (LIdeal‘𝑃))
18		hbtlem.u	. . . . . . . . . . 11 ⊢ 𝑈 = (LIdeal‘𝑃)
19	17, 18	eqtr4di 2797	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = 𝑈)
20		fveq2 6756	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑅 → ( deg1 ‘𝑟) = ( deg1 ‘𝑅))
21	20	fveq1d 6758	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → (( deg1 ‘𝑟)‘𝑗) = (( deg1 ‘𝑅)‘𝑗))
22	21	breq1d 5080	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → ((( deg1 ‘𝑟)‘𝑗) ≤ 𝑥 ↔ (( deg1 ‘𝑅)‘𝑗) ≤ 𝑥))
23	22	anbi1d 629	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (((( deg1 ‘𝑟)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥)) ↔ ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))))
24	23	rexbidv 3225	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑟)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥)) ↔ ∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))))
25	24	abbidv 2808	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → {𝑦 ∣ ∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑟)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} = {𝑦 ∣ ∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))})
26	25	mpteq2dv 5172	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑟)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}))
27	19, 26	mpteq12dv 5161	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑖 ∈ (LIdeal‘(Poly1‘𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑟)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))})) = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))})))
28		df-ldgis 40863	. . . . . . . . 9 ⊢ ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1‘𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑟)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))})))
29	27, 28, 18	mptfvmpt 7086	. . . . . . . 8 ⊢ (𝑅 ∈ V → (ldgIdlSeq‘𝑅) = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))})))
30	13, 29	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (ldgIdlSeq‘𝑅) = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))})))
31	12, 30	syl5eq 2791	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑆 = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))})))
32	31	fveq1d 6758	. . . . 5 ⊢ (𝑅 ∈ Ring → (𝑆‘𝐼) = ((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}))‘𝐼))
33		rexeq 3334	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥)) ↔ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))))
34	33	abbidv 2808	. . . . . . 7 ⊢ (𝑖 = 𝐼 → {𝑦 ∣ ∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} = {𝑦 ∣ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))})
35	34	mpteq2dv 5172	. . . . . 6 ⊢ (𝑖 = 𝐼 → (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}))
36		eqid 2738	. . . . . 6 ⊢ (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))})) = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}))
37		nn0ex 12169	. . . . . . 7 ⊢ ℕ0 ∈ V
38	37	mptex 7081	. . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}) ∈ V
39	35, 36, 38	fvmpt 6857	. . . . 5 ⊢ (𝐼 ∈ 𝑈 → ((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}))‘𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}))
40	32, 39	sylan9eq 2799	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝑆‘𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}))
41	40	fneq1d 6510	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ((𝑆‘𝐼) Fn ℕ0 ↔ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}) Fn ℕ0))
42	11, 41	mpbird 256	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝑆‘𝐼) Fn ℕ0)
43		hbtlem7.t	. . . . 5 ⊢ 𝑇 = (LIdeal‘𝑅)
44	15, 18, 12, 43	hbtlem2 40865	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑥 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑥) ∈ 𝑇)
45	44	3expa 1116	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑥) ∈ 𝑇)
46	45	ralrimiva 3107	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ∀𝑥 ∈ ℕ0 ((𝑆‘𝐼)‘𝑥) ∈ 𝑇)
47		ffnfv 6974	. 2 ⊢ ((𝑆‘𝐼):ℕ0⟶𝑇 ↔ ((𝑆‘𝐼) Fn ℕ0 ∧ ∀𝑥 ∈ ℕ0 ((𝑆‘𝐼)‘𝑥) ∈ 𝑇))
48	42, 46, 47	sylanbrc 582	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝑆‘𝐼):ℕ0⟶𝑇)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883   class class class wbr 5070   ↦ cmpt 5153   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418   ≤ cle 10941  ℕ₀cn0 12163  Ringcrg 19698  LIdealclidl 20347  Poly₁cpl1 21258  coe₁cco1 21259   deg₁ cdg1 25121  ldgIdlSeqcldgis 40862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880  ax-addf 10881  ax-mulf 10882

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-ofr 7512  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-sup 9131  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-fzo 13312  df-seq 13650  df-hash 13973  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-starv 16903  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-unif 16911  df-0g 17069  df-gsum 17070  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-mhm 18345  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-mulg 18616  df-subg 18667  df-ghm 18747  df-cntz 18838  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-cring 19701  df-subrg 19937  df-lmod 20040  df-lss 20109  df-sra 20349  df-rgmod 20350  df-lidl 20351  df-cnfld 20511  df-ascl 20972  df-psr 21022  df-mvr 21023  df-mpl 21024  df-opsr 21026  df-psr1 21261  df-vr1 21262  df-ply1 21263  df-coe1 21264  df-mdeg 25122  df-deg1 25123  df-ldgis 40863

This theorem is referenced by:  hbt  40871