Theorem hbtlem1 42686

Description: Value of the leading coefficient sequence function. (Contributed by Stefan O'Rear, 31-Mar-2015.)

Hypotheses

Ref	Expression
hbtlem.p	⊢ 𝑃 = (Poly1‘𝑅)
hbtlem.u	⊢ 𝑈 = (LIdeal‘𝑃)
hbtlem.s	⊢ 𝑆 = (ldgIdlSeq‘𝑅)
hbtlem.d	⊢ 𝐷 = ( deg1 ‘𝑅)

Assertion

Ref	Expression
hbtlem1	⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) = {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))})

Distinct variable groups:   𝑗,𝐼,𝑘   𝑅,𝑗,𝑘   𝑗,𝑋,𝑘

Allowed substitution hints:   𝐷(𝑗,𝑘)   𝑃(𝑗,𝑘)   𝑆(𝑗,𝑘)   𝑈(𝑗,𝑘)   𝑉(𝑗,𝑘)

Proof of Theorem hbtlem1

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

Step	Hyp	Ref	Expression
1		hbtlem.s	. . . . . 6 ⊢ 𝑆 = (ldgIdlSeq‘𝑅)
2		elex 3480	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		fveq2 6896	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
4		hbtlem.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘𝑅)
5	3, 4	eqtr4di 2783	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
6	5	fveq2d 6900	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = (LIdeal‘𝑃))
7		hbtlem.u	. . . . . . . . . 10 ⊢ 𝑈 = (LIdeal‘𝑃)
8	6, 7	eqtr4di 2783	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = 𝑈)
9		fveq2 6896	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑅 → ( deg1 ‘𝑟) = ( deg1 ‘𝑅))
10		hbtlem.d	. . . . . . . . . . . . . . . 16 ⊢ 𝐷 = ( deg1 ‘𝑅)
11	9, 10	eqtr4di 2783	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → ( deg1 ‘𝑟) = 𝐷)
12	11	fveq1d 6898	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → (( deg1 ‘𝑟)‘𝑘) = (𝐷‘𝑘))
13	12	breq1d 5159	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → ((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ↔ (𝐷‘𝑘) ≤ 𝑥))
14	13	anbi1d 629	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥)) ↔ ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))))
15	14	rexbidv 3168	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (∃𝑘 ∈ 𝑖 ((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥)) ↔ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))))
16	15	abbidv 2794	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})
17	16	mpteq2dv 5251	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))
18	8, 17	mpteq12dv 5240	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑖 ∈ (LIdeal‘(Poly1‘𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})) = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})))
19		df-ldgis 42685	. . . . . . . 8 ⊢ ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1‘𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})))
20	18, 19, 7	mptfvmpt 7240	. . . . . . 7 ⊢ (𝑅 ∈ V → (ldgIdlSeq‘𝑅) = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})))
21	2, 20	syl 17	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (ldgIdlSeq‘𝑅) = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})))
22	1, 21	eqtrid 2777	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})))
23	22	fveq1d 6898	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑆‘𝐼) = ((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))‘𝐼))
24	23	fveq1d 6898	. . 3 ⊢ (𝑅 ∈ 𝑉 → ((𝑆‘𝐼)‘𝑋) = (((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))‘𝐼)‘𝑋))
25	24	3ad2ant1 1130	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) = (((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))‘𝐼)‘𝑋))
26		rexeq 3310	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥)) ↔ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))))
27	26	abbidv 2794	. . . . . 6 ⊢ (𝑖 = 𝐼 → {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})
28	27	mpteq2dv 5251	. . . . 5 ⊢ (𝑖 = 𝐼 → (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))
29		eqid 2725	. . . . 5 ⊢ (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})) = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))
30		nn0ex 12511	. . . . . 6 ⊢ ℕ0 ∈ V
31	30	mptex 7235	. . . . 5 ⊢ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}) ∈ V
32	28, 29, 31	fvmpt 7004	. . . 4 ⊢ (𝐼 ∈ 𝑈 → ((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))‘𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))
33	32	fveq1d 6898	. . 3 ⊢ (𝐼 ∈ 𝑈 → (((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))‘𝐼)‘𝑋) = ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})‘𝑋))
34	33	3ad2ant2 1131	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))‘𝐼)‘𝑋) = ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})‘𝑋))
35		eqid 2725	. . 3 ⊢ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})
36		breq2 5153	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐷‘𝑘) ≤ 𝑥 ↔ (𝐷‘𝑘) ≤ 𝑋))
37		fveq2 6896	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((coe1‘𝑘)‘𝑥) = ((coe1‘𝑘)‘𝑋))
38	37	eqeq2d 2736	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑗 = ((coe1‘𝑘)‘𝑥) ↔ 𝑗 = ((coe1‘𝑘)‘𝑋)))
39	36, 38	anbi12d 630	. . . . 5 ⊢ (𝑥 = 𝑋 → (((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥)) ↔ ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))))
40	39	rexbidv 3168	. . . 4 ⊢ (𝑥 = 𝑋 → (∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥)) ↔ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))))
41	40	abbidv 2794	. . 3 ⊢ (𝑥 = 𝑋 → {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))})
42		simp3 1135	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
43		simpr 483	. . . . . 6 ⊢ (((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋)) → 𝑗 = ((coe1‘𝑘)‘𝑋))
44	43	reximi 3073	. . . . 5 ⊢ (∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋)) → ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋))
45	44	ss2abi 4059	. . . 4 ⊢ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))} ⊆ {𝑗 ∣ ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋)}
46		abrexexg 7965	. . . . 5 ⊢ (𝐼 ∈ 𝑈 → {𝑗 ∣ ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋)} ∈ V)
47	46	3ad2ant2 1131	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑗 ∣ ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋)} ∈ V)
48		ssexg 5324	. . . 4 ⊢ (({𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))} ⊆ {𝑗 ∣ ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋)} ∧ {𝑗 ∣ ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋)} ∈ V) → {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))} ∈ V)
49	45, 47, 48	sylancr 585	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))} ∈ V)
50	35, 41, 42, 49	fvmptd3 7027	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})‘𝑋) = {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))})
51	25, 34, 50	3eqtrd 2769	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) = {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))})

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2702  ∃wrex 3059  Vcvv 3461   ⊆ wss 3944   class class class wbr 5149   ↦ cmpt 5232  ‘cfv 6549   ≤ cle 11281  ℕ₀cn0 12505  LIdealclidl 21114  Poly₁cpl1 22119  coe₁cco1 22120   deg₁ cdg1 26031  ldgIdlSeqcldgis 42684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-1cn 11198  ax-addcl 11200

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-nn 12246  df-n0 12506  df-ldgis 42685

This theorem is referenced by:  hbtlem2  42687  hbtlem4  42689  hbtlem3  42690  hbtlem5  42691  hbtlem6  42692