Theorem hbtlem7 44070

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 3100	. . . . . . . 8 ⊢ (∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥)) → ∃𝑗 ∈ 𝐼 𝑦 = ((coe1‘𝑗)‘𝑥))
3	2	ss2abi 4013	. . . . . . 7 ⊢ {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))} ⊆ {𝑦 ∣ ∃𝑗 ∈ 𝐼 𝑦 = ((coe1‘𝑗)‘𝑥)}
4		abrexexg 7956	. . . . . . 7 ⊢ (𝐼 ∈ 𝑈 → {𝑦 ∣ ∃𝑗 ∈ 𝐼 𝑦 = ((coe1‘𝑗)‘𝑥)} ∈ V)
5		ssexg 5280	. . . . . . 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 3158	. . . . 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 2760	. . . . 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 6667	. . . 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 3471	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ V)
14		fveq2 6873	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
15		hbtlem.p	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Poly1‘𝑅)
16	14, 15	eqtr4di 2813	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
17	16	fveq2d 6877	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = (LIdeal‘𝑃))
18		hbtlem.u	. . . . . . . . . . 11 ⊢ 𝑈 = (LIdeal‘𝑃)
19	17, 18	eqtr4di 2813	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = 𝑈)
20		fveq2 6873	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = (deg1‘𝑅))
21	20	fveq1d 6875	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → ((deg1‘𝑟)‘𝑗) = ((deg1‘𝑅)‘𝑗))
22	21	breq1d 5112	. . . . . . . . . . . . . 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 3186	. . . . . . . . . . . 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 2826	. . . . . . . . . . 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 5198	. . . . . . . . . 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 5191	. . . . . . . . 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 44067	. . . . . . . . 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 7222	. . . . . . . 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 2807	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑆 = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))})))
32	31	fveq1d 6875	. . . . 5 ⊢ (𝑅 ∈ Ring → (𝑆‘𝐼) = ((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝑖 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}))‘𝐼))
33		rexeq 3315	. . . . . . . 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 2826	. . . . . . 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 5198	. . . . . 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 2760	. . . . . 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 12581	. . . . . . 7 ⊢ ℕ0 ∈ V
38	37	mptex 7217	. . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}) ∈ V
39	35, 36, 38	fvmpt 6981	. . . . 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 2815	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (𝑆‘𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑦 ∣ ∃𝑗 ∈ 𝐼 (((deg1‘𝑅)‘𝑗) ≤ 𝑥 ∧ 𝑦 = ((coe1‘𝑗)‘𝑥))}))
41	40	fneq1d 6620	. . 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 44069	. . . 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 3154	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ∀𝑥 ∈ ℕ0 ((𝑆‘𝐼)‘𝑥) ∈ 𝑇)
47		ffnfv 7107	. 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 2145  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3898   class class class wbr 5102   ↦ cmpt 5185   Fn wfn 6522  ⟶wf 6523  ‘cfv 6527   ≤ cle 11315  ℕ₀cn0 12575  Ringcrg 20420  LIdealclidl 21445  Poly₁cpl1 22456  coe₁cco1 22457  deg₁cdg1 26333  ldgIdlSeqcldgis 44066

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248  ax-pre-sup 11249  ax-addf 11250

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7676  df-ofr 7677  df-om 7861  df-1st 7984  df-2nd 7985  df-supp 8156  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-er 8695  df-map 8827  df-pm 8828  df-ixp 8904  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-fsupp 9332  df-sup 9412  df-oi 9482  df-card 9991  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-nn 12305  df-2 12374  df-3 12375  df-4 12376  df-5 12377  df-6 12378  df-7 12379  df-8 12380  df-9 12381  df-n0 12576  df-z 12663  df-dec 12784  df-uz 12935  df-fz 13609  df-fzo 13757  df-seq 14113  df-hash 14442  df-struct 17286  df-sets 17303  df-slot 17321  df-ndx 17333  df-base 17349  df-ress 17370  df-plusg 17402  df-mulr 17403  df-starv 17404  df-sca 17405  df-vsca 17406  df-ip 17407  df-tset 17408  df-ple 17409  df-ds 17411  df-unif 17412  df-hom 17413  df-cco 17414  df-0g 17573  df-gsum 17574  df-prds 17579  df-pws 17581  df-mre 17717  df-mrc 17718  df-acs 17720  df-mgm 18777  df-sgrp 18869  df-mnd 18885  df-mhm 18939  df-submnd 18940  df-grp 19108  df-minusg 19109  df-sbg 19110  df-mulg 19239  df-subg 19294  df-ghm 19389  df-cntz 19492  df-cmn 19957  df-abl 19958  df-mgp 20322  df-rng 20336  df-ur 20369  df-ring 20422  df-cring 20423  df-subrng 20759  df-subrg 20783  df-lmod 21098  df-lss 21168  df-sra 21409  df-rgmod 21410  df-lidl 21447  df-cnfld 21640  df-ascl 22124  df-psr 22178  df-mvr 22179  df-mpl 22180  df-opsr 22182  df-psr1 22459  df-vr1 22460  df-ply1 22461  df-coe1 22462  df-mdeg 26334  df-deg1 26335  df-ldgis 44067

This theorem is used by:  hbt  44075