Theorem hbtlem1 41508

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 3464	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		fveq2 6847	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
4		hbtlem.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘𝑅)
5	3, 4	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
6	5	fveq2d 6851	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = (LIdeal‘𝑃))
7		hbtlem.u	. . . . . . . . . 10 ⊢ 𝑈 = (LIdeal‘𝑃)
8	6, 7	eqtr4di 2789	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = 𝑈)
9		fveq2 6847	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑅 → ( deg1 ‘𝑟) = ( deg1 ‘𝑅))
10		hbtlem.d	. . . . . . . . . . . . . . . 16 ⊢ 𝐷 = ( deg1 ‘𝑅)
11	9, 10	eqtr4di 2789	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → ( deg1 ‘𝑟) = 𝐷)
12	11	fveq1d 6849	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → (( deg1 ‘𝑟)‘𝑘) = (𝐷‘𝑘))
13	12	breq1d 5120	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → ((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ↔ (𝐷‘𝑘) ≤ 𝑥))
14	13	anbi1d 630	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥)) ↔ ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))))
15	14	rexbidv 3171	. . . . . . . . . . 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 2800	. . . . . . . . . 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 5212	. . . . . . . . 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 5201	. . . . . . . 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 41507	. . . . . . . 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 7183	. . . . . . 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 2783	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})))
23	22	fveq1d 6849	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑆‘𝐼) = ((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))‘𝐼))
24	23	fveq1d 6849	. . 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 1133	. 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 3308	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥)) ↔ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))))
27	26	abbidv 2800	. . . . . 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 5212	. . . . 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 2731	. . . . 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 12428	. . . . . 6 ⊢ ℕ0 ∈ V
31	30	mptex 7178	. . . . 5 ⊢ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}) ∈ V
32	28, 29, 31	fvmpt 6953	. . . 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 6849	. . 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 1134	. 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 2731	. . 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 5114	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐷‘𝑘) ≤ 𝑥 ↔ (𝐷‘𝑘) ≤ 𝑋))
37		fveq2 6847	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((coe1‘𝑘)‘𝑥) = ((coe1‘𝑘)‘𝑋))
38	37	eqeq2d 2742	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑗 = ((coe1‘𝑘)‘𝑥) ↔ 𝑗 = ((coe1‘𝑘)‘𝑋)))
39	36, 38	anbi12d 631	. . . . 5 ⊢ (𝑥 = 𝑋 → (((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥)) ↔ ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))))
40	39	rexbidv 3171	. . . 4 ⊢ (𝑥 = 𝑋 → (∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥)) ↔ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))))
41	40	abbidv 2800	. . 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 1138	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
43		simpr 485	. . . . . 6 ⊢ (((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋)) → 𝑗 = ((coe1‘𝑘)‘𝑋))
44	43	reximi 3083	. . . . 5 ⊢ (∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋)) → ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋))
45	44	ss2abi 4028	. . . 4 ⊢ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))} ⊆ {𝑗 ∣ ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋)}
46		abrexexg 7898	. . . . 5 ⊢ (𝐼 ∈ 𝑈 → {𝑗 ∣ ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋)} ∈ V)
47	46	3ad2ant2 1134	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑗 ∣ ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋)} ∈ V)
48		ssexg 5285	. . . 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 587	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))} ∈ V)
50	35, 41, 42, 49	fvmptd3 6976	. 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 2775	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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2708  ∃wrex 3069  Vcvv 3446   ⊆ wss 3913   class class class wbr 5110   ↦ cmpt 5193  ‘cfv 6501   ≤ cle 11199  ℕ₀cn0 12422  LIdealclidl 20690  Poly₁cpl1 21585  coe₁cco1 21586   deg₁ cdg1 25453  ldgIdlSeqcldgis 41506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-1cn 11118  ax-addcl 11120

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  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 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-nn 12163  df-n0 12423  df-ldgis 41507

This theorem is referenced by:  hbtlem2  41509  hbtlem4  41511  hbtlem3  41512  hbtlem5  41513  hbtlem6  41514