Theorem hbtlem7 42786

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 483	. . . . . . . . 9 ⊢ ((((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥)) → 𝑦 = ((coe1‘𝑗)‘𝑥))
2	1	reximi 3074	. . . . . . . 8 ⊢ (∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥)) → ∃𝑗 ∈ 𝐼 𝑦 = ((coe1‘𝑗)‘𝑥))
3	2	ss2abi 4062	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ⊆ {𝑦 ∣ ∃𝑗 ∈ 𝐼 𝑦 = ((coe1‘𝑗)‘𝑥)}
4		abrexexg 7974	. . . . . . 7 ⊢ (𝐼 ∈ 𝑈 → {𝑦 ∣ ∃𝑗 ∈ 𝐼 𝑦 = ((coe1‘𝑗)‘𝑥)} ∈ V)
5		ssexg 5328	. . . . . . 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 585	. . . . . 6 ⊢ (𝐼 ∈ 𝑈 → {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ∈ V)
7	6	ralrimivw 3140	. . . . 5 ⊢ (𝐼 ∈ 𝑈 → ∀𝑥 ∈ ℕ0 {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ∈ V)
8	7	adantl 480	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ∀𝑥 ∈ ℕ0 {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ∈ V)
9		eqid 2726	. . . . 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 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 3482	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
14		fveq2 6901	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
15		hbtlem.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Poly1‘𝑅)
16	14, 15	eqtr4di 2784	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
17	16	fveq2d 6905	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = (LIdeal‘𝑃))
18		hbtlem.u	. . . . . . . . . . 11 ⊢ 𝑈 = (LIdeal‘𝑃)
19	17, 18	eqtr4di 2784	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = 𝑈)
20		fveq2 6901	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = (deg1‘𝑅))
21	20	fveq1d 6903	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → ((deg1‘𝑟)‘𝑗) = ((deg1‘𝑅)‘𝑗))
22	21	breq1d 5163	. . . . . . . . . . . . . 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 3169	. . . . . . . . . . . 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 5255	. . . . . . . . . 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 5244	. . . . . . . . 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 42783	. . . . . . . . 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 7245	. . . . . . . 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 2778	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑆 = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))})))
32	31	fveq1d 6903	. . . . 5 ⊢ (𝑅 ∈ Ring → (𝑆‘𝐼) = ((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}))‘𝐼))
33		rexeq 3311	. . . . . . . 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 5255	. . . . . 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 2726	. . . . . 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 12530	. . . . . . 7 ⊢ ℕ0 ∈ V
38	37	mptex 7240	. . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}) ∈ V
39	35, 36, 38	fvmpt 7009	. . . . 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 2786	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝑆‘𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}))
41	40	fneq1d 6653	. . 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 42785	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑥 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑥) ∈ 𝑇)
45	44	3expa 1115	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ 𝑥 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑥) ∈ 𝑇)
46	45	ralrimiva 3136	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ∀𝑥 ∈ ℕ0 ((𝑆‘𝐼)‘𝑥) ∈ 𝑇)
47		ffnfv 7133	. 2 ⊢ ((𝑆‘𝐼):ℕ0⟶𝑇 ↔ ((𝑆‘𝐼) Fn ℕ0 ∧ ∀𝑥 ∈ ℕ0 ((𝑆‘𝐼)‘𝑥) ∈ 𝑇))
48	42, 46, 47	sylanbrc 581	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝑆‘𝐼):ℕ0⟶𝑇)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099  {cab 2703  ∀wral 3051  ∃wrex 3060  Vcvv 3462   ⊆ wss 3947   class class class wbr 5153   ↦ cmpt 5236   Fn wfn 6549  ⟶wf 6550  ‘cfv 6554   ≤ cle 11299  ℕ₀cn0 12524  Ringcrg 20216  LIdealclidl 21195  Poly₁cpl1 22166  coe₁cco1 22167  deg₁cdg1 26078  ldgIdlSeqcldgis 42782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236  ax-addf 11237

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7690  df-ofr 7691  df-om 7877  df-1st 8003  df-2nd 8004  df-supp 8175  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-er 8734  df-map 8857  df-pm 8858  df-ixp 8927  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-fsupp 9406  df-sup 9485  df-oi 9553  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12611  df-dec 12730  df-uz 12875  df-fz 13539  df-fzo 13682  df-seq 14022  df-hash 14348  df-struct 17149  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-ress 17243  df-plusg 17279  df-mulr 17280  df-starv 17281  df-sca 17282  df-vsca 17283  df-ip 17284  df-tset 17285  df-ple 17286  df-ds 17288  df-unif 17289  df-hom 17290  df-cco 17291  df-0g 17456  df-gsum 17457  df-prds 17462  df-pws 17464  df-mre 17599  df-mrc 17600  df-acs 17602  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-mhm 18773  df-submnd 18774  df-grp 18931  df-minusg 18932  df-sbg 18933  df-mulg 19062  df-subg 19117  df-ghm 19207  df-cntz 19311  df-cmn 19780  df-abl 19781  df-mgp 20118  df-rng 20136  df-ur 20165  df-ring 20218  df-cring 20219  df-subrng 20528  df-subrg 20553  df-lmod 20838  df-lss 20909  df-sra 21151  df-rgmod 21152  df-lidl 21197  df-cnfld 21344  df-ascl 21853  df-psr 21906  df-mvr 21907  df-mpl 21908  df-opsr 21910  df-psr1 22169  df-vr1 22170  df-ply1 22171  df-coe1 22172  df-mdeg 26079  df-deg1 26080  df-ldgis 42783

This theorem is referenced by:  hbt  42791