Theorem hbtlem7 43114

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 3067	. . . . . . . 8 ⊢ (∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥)) → ∃𝑗 ∈ 𝐼 𝑦 = ((coe1‘𝑗)‘𝑥))
3	2	ss2abi 4030	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ⊆ {𝑦 ∣ ∃𝑗 ∈ 𝐼 𝑦 = ((coe1‘𝑗)‘𝑥)}
4		abrexexg 7939	. . . . . . 7 ⊢ (𝐼 ∈ 𝑈 → {𝑦 ∣ ∃𝑗 ∈ 𝐼 𝑦 = ((coe1‘𝑗)‘𝑥)} ∈ V)
5		ssexg 5278	. . . . . . 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 587	. . . . . 6 ⊢ (𝐼 ∈ 𝑈 → {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ∈ V)
7	6	ralrimivw 3129	. . . . 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 2729	. . . . 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 6658	. . . 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 3468	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
14		fveq2 6858	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
15		hbtlem.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Poly1‘𝑅)
16	14, 15	eqtr4di 2782	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
17	16	fveq2d 6862	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = (LIdeal‘𝑃))
18		hbtlem.u	. . . . . . . . . . 11 ⊢ 𝑈 = (LIdeal‘𝑃)
19	17, 18	eqtr4di 2782	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = 𝑈)
20		fveq2 6858	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = (deg1‘𝑅))
21	20	fveq1d 6860	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → ((deg1‘𝑟)‘𝑗) = ((deg1‘𝑅)‘𝑗))
22	21	breq1d 5117	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → (((deg1‘𝑟)‘𝑗) ≤ 𝑥 ↔ ((deg1‘𝑅)‘𝑗) ≤ 𝑥))
23	22	anbi1d 631	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → ((((deg1‘𝑟)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥)) ↔ (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))))
24	23	rexbidv 3157	. . . . . . . . . . . 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 2795	. . . . . . . . . . 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 5201	. . . . . . . . . 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 5194	. . . . . . . . 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 43111	. . . . . . . . 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 7202	. . . . . . . 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	eqtrid 2776	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑆 = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))})))
32	31	fveq1d 6860	. . . . 5 ⊢ (𝑅 ∈ Ring → (𝑆‘𝐼) = ((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}))‘𝐼))
33		rexeq 3295	. . . . . . . 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 2795	. . . . . . 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 5201	. . . . . 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 2729	. . . . . 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 12448	. . . . . . 7 ⊢ ℕ0 ∈ V
38	37	mptex 7197	. . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}) ∈ V
39	35, 36, 38	fvmpt 6968	. . . . 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 2784	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝑆‘𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}))
41	40	fneq1d 6611	. . 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 257	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝑆‘𝐼) Fn ℕ0)
43		hbtlem7.t	. . . . 5 ⊢ 𝑇 = (LIdeal‘𝑅)
44	15, 18, 12, 43	hbtlem2 43113	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑥 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑥) ∈ 𝑇)
45	44	3expa 1118	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑥) ∈ 𝑇)
46	45	ralrimiva 3125	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ∀𝑥 ∈ ℕ0 ((𝑆‘𝐼)‘𝑥) ∈ 𝑇)
47		ffnfv 7091	. 2 ⊢ ((𝑆‘𝐼):ℕ0⟶𝑇 ↔ ((𝑆‘𝐼) Fn ℕ0 ∧ ∀𝑥 ∈ ℕ0 ((𝑆‘𝐼)‘𝑥) ∈ 𝑇))
48	42, 46, 47	sylanbrc 583	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝑆‘𝐼):ℕ0⟶𝑇)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ⊆ wss 3914   class class class wbr 5107   ↦ cmpt 5188   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511   ≤ cle 11209  ℕ₀cn0 12442  Ringcrg 20142  LIdealclidl 21116  Poly₁cpl1 22061  coe₁cco1 22062  deg₁cdg1 25959  ldgIdlSeqcldgis 43110

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146  ax-addf 11147

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-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-ofr 7654  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-map 8801  df-pm 8802  df-ixp 8871  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-sup 9393  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-fzo 13616  df-seq 13967  df-hash 14296  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17547  df-mrc 17548  df-acs 17550  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-mulg 19000  df-subg 19055  df-ghm 19145  df-cntz 19249  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-ring 20144  df-cring 20145  df-subrng 20455  df-subrg 20479  df-lmod 20768  df-lss 20838  df-sra 21080  df-rgmod 21081  df-lidl 21118  df-cnfld 21265  df-ascl 21764  df-psr 21818  df-mvr 21819  df-mpl 21820  df-opsr 21822  df-psr1 22064  df-vr1 22065  df-ply1 22066  df-coe1 22067  df-mdeg 25960  df-deg1 25961  df-ldgis 43111

This theorem is referenced by:  hbt  43119