Theorem hbtlem7 44319

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 490	. . . . . . . . 9 ⊢ ((((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥)) → 𝑦 = ((coe1‘𝑗)‘𝑥))
2	1	reximi 3104	. . . . . . . 8 ⊢ (∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥)) → ∃𝑗 ∈ 𝐼 𝑦 = ((coe1‘𝑗)‘𝑥))
3	2	ss2abi 3994	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ⊆ {𝑦 ∣ ∃𝑗 ∈ 𝐼 𝑦 = ((coe1‘𝑗)‘𝑥)}
4		abrexexg 7997	. . . . . . 7 ⊢ (𝐼 ∈ 𝑈 → {𝑦 ∣ ∃𝑗 ∈ 𝐼 𝑦 = ((coe1‘𝑗)‘𝑥)} ∈ V)
5		ssexg 5285	. . . . . . 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 599	. . . . . 6 ⊢ (𝐼 ∈ 𝑈 → {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ∈ V)
7	6	ralrimivw 3162	. . . . 5 ⊢ (𝐼 ∈ 𝑈 → ∀𝑥 ∈ ℕ0 {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ∈ V)
8	7	adantl 487	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ∀𝑥 ∈ ℕ0 {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ∈ V)
9		eqid 2763	. . . . 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 6701	. . . 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 18	. . 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 3475	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
14		fveq2 6908	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
15		hbtlem.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Poly1‘𝑅)
16	14, 15	eqtr4di 2816	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
17	16	fveq2d 6912	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = (LIdeal‘𝑃))
18		hbtlem.u	. . . . . . . . . . 11 ⊢ 𝑈 = (LIdeal‘𝑃)
19	17, 18	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = 𝑈)
20		fveq2 6908	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = (deg1‘𝑅))
21	20	fveq1d 6910	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → ((deg1‘𝑟)‘𝑗) = ((deg1‘𝑅)‘𝑗))
22	21	breq1d 5116	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → (((deg1‘𝑟)‘𝑗) ≤ 𝑥 ↔ ((deg1‘𝑅)‘𝑗) ≤ 𝑥))
23	22	anbi1d 643	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → ((((deg1‘𝑟)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥)) ↔ (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))))
24	23	rexbidv 3190	. . . . . . . . . . . 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 2829	. . . . . . . . . . 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 5202	. . . . . . . . . 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 5195	. . . . . . . . 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 44316	. . . . . . . . 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 7257	. . . . . . . 8 ⊢ (𝑅 ∈ V → (ldgIdlSeq‘𝑅) = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))})))
30	13, 29	syl 18	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (ldgIdlSeq‘𝑅) = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))})))
31	12, 30	eqtrid 2810	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑆 = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))})))
32	31	fveq1d 6910	. . . . 5 ⊢ (𝑅 ∈ Ring → (𝑆‘𝐼) = ((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}))‘𝐼))
33		rexeq 3319	. . . . . . . 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 2829	. . . . . . 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 5202	. . . . . 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 2763	. . . . . 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 12639	. . . . . . 7 ⊢ ℕ0 ∈ V
38	37	mptex 7252	. . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}) ∈ V
39	35, 36, 38	fvmpt 7016	. . . . 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 2818	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝑆‘𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}))
41	40	fneq1d 6654	. . 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 260	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝑆‘𝐼) Fn ℕ0)
43		hbtlem7.t	. . . . 5 ⊢ 𝑇 = (LIdeal‘𝑅)
44	15, 18, 12, 43	hbtlem2 44318	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑥 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑥) ∈ 𝑇)
45	44	3expa 1136	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑥) ∈ 𝑇)
46	45	ralrimiva 3158	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ∀𝑥 ∈ ℕ0 ((𝑆‘𝐼)‘𝑥) ∈ 𝑇)
47		ffnfv 7143	. 2 ⊢ ((𝑆‘𝐼):ℕ0⟶𝑇 ↔ ((𝑆‘𝐼) Fn ℕ0 ∧ ∀𝑥 ∈ ℕ0 ((𝑆‘𝐼)‘𝑥) ∈ 𝑇))
48	42, 46, 47	sylanbrc 595	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝑆‘𝐼):ℕ0⟶𝑇)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   class class class wbr 5106   ↦ cmpt 5189   Fn wfn 6553  ⟶wf 6554  ‘cfv 6558   ≤ cle 11371  ℕ₀cn0 12633  Ringcrg 20489  LIdealclidl 21514  Poly₁cpl1 22525  coe₁cco1 22526  deg₁cdg1 26402  ldgIdlSeqcldgis 44315

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304  ax-pre-sup 11305  ax-addf 11306

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7717  df-ofr 7718  df-om 7902  df-1st 8025  df-2nd 8026  df-supp 8205  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-er 8745  df-map 8877  df-pm 8878  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-fsupp 9382  df-sup 9461  df-oi 9531  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-uz 12993  df-fz 13667  df-fzo 13816  df-seq 14172  df-hash 14502  df-struct 17354  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-plusg 17470  df-mulr 17471  df-starv 17472  df-sca 17473  df-vsca 17474  df-ip 17475  df-tset 17476  df-ple 17477  df-ds 17479  df-unif 17480  df-hom 17481  df-cco 17482  df-0g 17641  df-gsum 17642  df-prds 17647  df-pws 17649  df-mre 17785  df-mrc 17786  df-acs 17788  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-mhm 19007  df-submnd 19008  df-grp 19176  df-minusg 19177  df-sbg 19178  df-mulg 19307  df-subg 19362  df-ghm 19457  df-cntz 19561  df-cmn 20026  df-abl 20027  df-mgp 20391  df-rng 20405  df-ur 20438  df-ring 20491  df-cring 20492  df-subrng 20828  df-subrg 20852  df-lmod 21167  df-lss 21237  df-sra 21478  df-rgmod 21479  df-lidl 21516  df-cnfld 21709  df-ascl 22193  df-psr 22247  df-mvr 22248  df-mpl 22249  df-opsr 22251  df-psr1 22528  df-vr1 22529  df-ply1 22530  df-coe1 22531  df-mdeg 26403  df-deg1 26404  df-ldgis 44316

This theorem is used by:  hbt  44324