Theorem hbtlem1 40592

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 3416	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		fveq2 6695	. . . . . . . . . . . 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 6699	. . . . . . . . . 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 6695	. . . . . . . . . . . . . . . 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 6697	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → (( deg1 ‘𝑟)‘𝑘) = (𝐷‘𝑘))
13	12	breq1d 5049	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → ((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ↔ (𝐷‘𝑘) ≤ 𝑥))
14	13	anbi1d 633	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥)) ↔ ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))))
15	14	rexbidv 3206	. . . . . . . . . . 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 5136	. . . . . . . . 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 5125	. . . . . . . 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 40591	. . . . . . . 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 7022	. . . . . . 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	syl5eq 2783	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})))
23	22	fveq1d 6697	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑆‘𝐼) = ((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))‘𝐼))
24	23	fveq1d 6697	. . 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 1135	. 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 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 5136	. . . . 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 2736	. . . . 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 12061	. . . . . 6 ⊢ ℕ0 ∈ V
31	30	mptex 7017	. . . . 5 ⊢ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}) ∈ V
32	28, 29, 31	fvmpt 6796	. . . 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 6697	. . 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 1136	. 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 2736	. . 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 5043	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐷‘𝑘) ≤ 𝑥 ↔ (𝐷‘𝑘) ≤ 𝑋))
37		fveq2 6695	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((coe1‘𝑘)‘𝑥) = ((coe1‘𝑘)‘𝑋))
38	37	eqeq2d 2747	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑗 = ((coe1‘𝑘)‘𝑥) ↔ 𝑗 = ((coe1‘𝑘)‘𝑋)))
39	36, 38	anbi12d 634	. . . . 5 ⊢ (𝑥 = 𝑋 → (((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥)) ↔ ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))))
40	39	rexbidv 3206	. . . 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 1140	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
43		simpr 488	. . . . . 6 ⊢ (((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋)) → 𝑗 = ((coe1‘𝑘)‘𝑋))
44	43	reximi 3156	. . . . 5 ⊢ (∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋)) → ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋))
45	44	ss2abi 3966	. . . 4 ⊢ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))} ⊆ {𝑗 ∣ ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋)}
46		abrexexg 7712	. . . . 5 ⊢ (𝐼 ∈ 𝑈 → {𝑗 ∣ ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋)} ∈ V)
47	46	3ad2ant2 1136	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑗 ∣ ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋)} ∈ V)
48		ssexg 5201	. . . 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 590	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))} ∈ V)
50	35, 41, 42, 49	fvmptd3 6819	. 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 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  {cab 2714  ∃wrex 3052  Vcvv 3398   ⊆ wss 3853   class class class wbr 5039   ↦ cmpt 5120  ‘cfv 6358   ≤ cle 10833  ℕ₀cn0 12055  LIdealclidl 20161  Poly₁cpl1 21052  coe₁cco1 21053   deg₁ cdg1 24903  ldgIdlSeqcldgis 40590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501  ax-cnex 10750  ax-1cn 10752  ax-addcl 10754

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-om 7623  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-nn 11796  df-n0 12056  df-ldgis 40591

This theorem is referenced by:  hbtlem2  40593  hbtlem4  40595  hbtlem3  40596  hbtlem5  40597  hbtlem6  40598