Theorem hbtlem6 44174

Description: There is a finite set of polynomials matching any single stage of the image. (Contributed by Stefan O'Rear, 1-Apr-2015.)

Hypotheses

Ref	Expression
hbtlem.p	⊢ 𝑃 = (Poly1‘𝑅)
hbtlem.u	⊢ 𝑈 = (LIdeal‘𝑃)
hbtlem.s	⊢ 𝑆 = (ldgIdlSeq‘𝑅)
hbtlem6.n	⊢ 𝑁 = (RSpan‘𝑃)
hbtlem6.r	⊢ (𝜑 → 𝑅 ∈ LNoeR)
hbtlem6.i	⊢ (𝜑 → 𝐼 ∈ 𝑈)
hbtlem6.x	⊢ (𝜑 → 𝑋 ∈ ℕ0)

Assertion

Ref	Expression
hbtlem6	⊢ (𝜑 → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))

Distinct variable groups:   𝜑,𝑘   𝑘,𝐼   𝑅,𝑘   𝑆,𝑘   𝑘,𝑋

Allowed substitution hints:   𝑃(𝑘)   𝑈(𝑘)   𝑁(𝑘)

Proof of Theorem hbtlem6

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hbtlem6.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ LNoeR)
2		lnrring 44157	. . . . 5 ⊢ (𝑅 ∈ LNoeR → 𝑅 ∈ Ring)
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
4		hbtlem6.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑈)
5		hbtlem6.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℕ0)
6		hbtlem.p	. . . . 5 ⊢ 𝑃 = (Poly1‘𝑅)
7		hbtlem.u	. . . . 5 ⊢ 𝑈 = (LIdeal‘𝑃)
8		hbtlem.s	. . . . 5 ⊢ 𝑆 = (ldgIdlSeq‘𝑅)
9		eqid 2760	. . . . 5 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
10	6, 7, 8, 9	hbtlem2 44169	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
11	3, 4, 5, 10	syl3anc 1398	. . 3 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
12		eqid 2760	. . . 4 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
13	9, 12	lnr2i 44161	. . 3 ⊢ ((𝑅 ∈ LNoeR ∧ ((𝑆‘𝐼)‘𝑋) ∈ (LIdeal‘𝑅)) → ∃𝑎 ∈ (𝒫 ((𝑆‘𝐼)‘𝑋) ∩ Fin)((𝑆‘𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎))
14	1, 11, 13	syl2anc 596	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝒫 ((𝑆‘𝐼)‘𝑋) ∩ Fin)((𝑆‘𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎))
15		elfpw 9343	. . . . 5 ⊢ (𝑎 ∈ (𝒫 ((𝑆‘𝐼)‘𝑋) ∩ Fin) ↔ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin))
16		fvex 6898	. . . . . . . . 9 ⊢ ((coe1‘𝑏)‘𝑋) ∈ V
17		eqid 2760	. . . . . . . . 9 ⊢ (𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) = (𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋))
18	16, 17	fnmpti 6682	. . . . . . . 8 ⊢ (𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) Fn {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋}
19	18	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → (𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) Fn {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋})
20		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ((𝑆‘𝐼)‘𝑋))
21		eqid 2760	. . . . . . . . . . . 12 ⊢ (deg1‘𝑅) = (deg1‘𝑅)
22	6, 7, 8, 21	hbtlem1 44168	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ LNoeR ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) = {𝑑 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋))})
23	1, 4, 5, 22	syl3anc 1398	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) = {𝑑 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋))})
24	17	rnmpt 5938	. . . . . . . . . . 11 ⊢ ran (𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) = {𝑑 ∣ ∃𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋}𝑑 = ((coe1‘𝑏)‘𝑋)}
25		fveq2 6885	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → ((deg1‘𝑅)‘𝑐) = ((deg1‘𝑅)‘𝑏))
26	25	breq1d 5112	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (((deg1‘𝑅)‘𝑐) ≤ 𝑋 ↔ ((deg1‘𝑅)‘𝑏) ≤ 𝑋))
27	26	rexrab 3653	. . . . . . . . . . . 12 ⊢ (∃𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋}𝑑 = ((coe1‘𝑏)‘𝑋) ↔ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋)))
28	27	abbii 2827	. . . . . . . . . . 11 ⊢ {𝑑 ∣ ∃𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋}𝑑 = ((coe1‘𝑏)‘𝑋)} = {𝑑 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋))}
29	24, 28	eqtri 2783	. . . . . . . . . 10 ⊢ ran (𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) = {𝑑 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋))}
30	23, 29	eqtr4di 2813	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) = ran (𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)))
31	30	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → ((𝑆‘𝐼)‘𝑋) = ran (𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)))
32	20, 31	sseqtrd 3944	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ran (𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)))
33		simprr 785	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ∈ Fin)
34		fipreima 9347	. . . . . . 7 ⊢ (((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) Fn {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑎 ⊆ ran (𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) ∧ 𝑎 ∈ Fin) → ∃𝑘 ∈ (𝒫 {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∩ Fin)((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘) = 𝑎)
35	19, 32, 33, 34	syl3anc 1398	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → ∃𝑘 ∈ (𝒫 {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∩ Fin)((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘) = 𝑎)
36		elfpw 9343	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝒫 {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∩ Fin) ↔ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin))
37		ssrab2 4005	. . . . . . . . . . . . . . . . 17 ⊢ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ⊆ 𝐼
38		sstr2 3915	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} → ({𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ⊆ 𝐼 → 𝑘 ⊆ 𝐼))
39	37, 38	mpi 21	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} → 𝑘 ⊆ 𝐼)
40	39	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋}) → 𝑘 ⊆ 𝐼)
41		velpw 4561	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝒫 𝐼 ↔ 𝑘 ⊆ 𝐼)
42	40, 41	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋}) → 𝑘 ∈ 𝒫 𝐼)
43	42	adantrr 730	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ 𝒫 𝐼)
44		simprr 785	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ Fin)
45	43, 44	elind 4097	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ (𝒫 𝐼 ∩ Fin))
46	3	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑅 ∈ Ring)
47	6	ply1ring 22565	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
48	3, 47	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ Ring)
49	48	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑃 ∈ Ring)
50		simprl 783	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋})
51	50, 37	sstrdi 3920	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ 𝐼)
52		eqid 2760	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑃) = (Base‘𝑃)
53	52, 7	lidlss 21490	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ (Base‘𝑃))
54	4, 53	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐼 ⊆ (Base‘𝑃))
55	54	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝐼 ⊆ (Base‘𝑃))
56	51, 55	sstrd 3918	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ (Base‘𝑃))
57		hbtlem6.n	. . . . . . . . . . . . . . . 16 ⊢ 𝑁 = (RSpan‘𝑃)
58	57, 52, 7	rspcl 21518	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ Ring ∧ 𝑘 ⊆ (Base‘𝑃)) → (𝑁‘𝑘) ∈ 𝑈)
59	49, 56, 58	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → (𝑁‘𝑘) ∈ 𝑈)
60	5	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑋 ∈ ℕ0)
61	6, 7, 8, 9	hbtlem2 44169	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘𝑘) ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘(𝑁‘𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
62	46, 59, 60, 61	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁‘𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
63		df-ima 5663	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘) = ran ((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) ↾ 𝑘)
64	57, 52	rspssid 21519	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃 ∈ Ring ∧ 𝑘 ⊆ (Base‘𝑃)) → 𝑘 ⊆ (𝑁‘𝑘))
65	49, 56, 64	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ (𝑁‘𝑘))
66		ssrab 3996	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↔ (𝑘 ⊆ 𝐼 ∧ ∀𝑐 ∈ 𝑘 ((deg1‘𝑅)‘𝑐) ≤ 𝑋))
67	66	simprbi 503	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} → ∀𝑐 ∈ 𝑘 ((deg1‘𝑅)‘𝑐) ≤ 𝑋)
68	67	ad2antrl 741	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ∀𝑐 ∈ 𝑘 ((deg1‘𝑅)‘𝑐) ≤ 𝑋)
69		ssrab 3996	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ⊆ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↔ (𝑘 ⊆ (𝑁‘𝑘) ∧ ∀𝑐 ∈ 𝑘 ((deg1‘𝑅)‘𝑐) ≤ 𝑋))
70	65, 68, 69	sylanbrc 595	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋})
71	70	resmptd 6031	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) ↾ 𝑘) = (𝑏 ∈ 𝑘 ↦ ((coe1‘𝑏)‘𝑋)))
72		resmpt 6028	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} → ((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) ↾ 𝑘) = (𝑏 ∈ 𝑘 ↦ ((coe1‘𝑏)‘𝑋)))
73	72	ad2antrl 741	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) ↾ 𝑘) = (𝑏 ∈ 𝑘 ↦ ((coe1‘𝑏)‘𝑋)))
74	71, 73	eqtr4d 2798	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) ↾ 𝑘) = ((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) ↾ 𝑘))
75		resss 5991	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) ↾ 𝑘) ⊆ (𝑏 ∈ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋))
76	74, 75	eqsstrrdi 3953	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) ↾ 𝑘) ⊆ (𝑏 ∈ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)))
77		rnss 5920	. . . . . . . . . . . . . . . 16 ⊢ (((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) ↾ 𝑘) ⊆ (𝑏 ∈ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) → ran ((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) ↾ 𝑘) ⊆ ran (𝑏 ∈ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)))
78	76, 77	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ran ((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) ↾ 𝑘) ⊆ ran (𝑏 ∈ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)))
79	63, 78	eqsstrid 3946	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘) ⊆ ran (𝑏 ∈ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)))
80	6, 7, 8, 21	hbtlem1 44168	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘𝑘) ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘(𝑁‘𝑘))‘𝑋) = {𝑒 ∣ ∃𝑏 ∈ (𝑁‘𝑘)(((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑏)‘𝑋))})
81	46, 59, 60, 80	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁‘𝑘))‘𝑋) = {𝑒 ∣ ∃𝑏 ∈ (𝑁‘𝑘)(((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑏)‘𝑋))})
82		eqid 2760	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) = (𝑏 ∈ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋))
83	82	rnmpt 5938	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑏 ∈ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) = {𝑒 ∣ ∃𝑏 ∈ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋}𝑒 = ((coe1‘𝑏)‘𝑋)}
84	26	rexrab 3653	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑏 ∈ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋}𝑒 = ((coe1‘𝑏)‘𝑋) ↔ ∃𝑏 ∈ (𝑁‘𝑘)(((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑏)‘𝑋)))
85	84	abbii 2827	. . . . . . . . . . . . . . . 16 ⊢ {𝑒 ∣ ∃𝑏 ∈ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋}𝑒 = ((coe1‘𝑏)‘𝑋)} = {𝑒 ∣ ∃𝑏 ∈ (𝑁‘𝑘)(((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑏)‘𝑋))}
86	83, 85	eqtri 2783	. . . . . . . . . . . . . . 15 ⊢ ran (𝑏 ∈ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) = {𝑒 ∣ ∃𝑏 ∈ (𝑁‘𝑘)(((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑏)‘𝑋))}
87	81, 86	eqtr4di 2813	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁‘𝑘))‘𝑋) = ran (𝑏 ∈ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)))
88	79, 87	sseqtrrd 3945	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))
89	12, 9	rspssp 21522	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ ((𝑆‘(𝑁‘𝑘))‘𝑋) ∈ (LIdeal‘𝑅) ∧ ((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋)) → ((RSpan‘𝑅)‘((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘)) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))
90	46, 62, 88, 89	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((RSpan‘𝑅)‘((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘)) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))
91	45, 90	jca 521	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → (𝑘 ∈ (𝒫 𝐼 ∩ Fin) ∧ ((RSpan‘𝑅)‘((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘)) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋)))
92		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘) = 𝑎 → ((RSpan‘𝑅)‘((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘)) = ((RSpan‘𝑅)‘𝑎))
93	92	sseq1d 3939	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘) = 𝑎 → (((RSpan‘𝑅)‘((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘)) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋) ↔ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋)))
94	93	anbi2d 642	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘) = 𝑎 → ((𝑘 ∈ (𝒫 𝐼 ∩ Fin) ∧ ((RSpan‘𝑅)‘((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘)) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋)) ↔ (𝑘 ∈ (𝒫 𝐼 ∩ Fin) ∧ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))))
95	91, 94	syl5ibcom 248	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → (((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘) = 𝑎 → (𝑘 ∈ (𝒫 𝐼 ∩ Fin) ∧ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))))
96	36, 95	sylan2b 606	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝒫 {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∩ Fin)) → (((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘) = 𝑎 → (𝑘 ∈ (𝒫 𝐼 ∩ Fin) ∧ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))))
97	96	expimpd 459	. . . . . . . 8 ⊢ (𝜑 → ((𝑘 ∈ (𝒫 {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∩ Fin) ∧ ((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘) = 𝑎) → (𝑘 ∈ (𝒫 𝐼 ∩ Fin) ∧ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))))
98	97	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → ((𝑘 ∈ (𝒫 {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∩ Fin) ∧ ((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘) = 𝑎) → (𝑘 ∈ (𝒫 𝐼 ∩ Fin) ∧ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))))
99	98	reximdv2 3172	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → (∃𝑘 ∈ (𝒫 {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∩ Fin)((𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)) “ 𝑘) = 𝑎 → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋)))
100	35, 99	mpd 16	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))
101	15, 100	sylan2b 606	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 ((𝑆‘𝐼)‘𝑋) ∩ Fin)) → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))
102		sseq1 3933	. . . . 5 ⊢ (((𝑆‘𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎) → (((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋) ↔ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋)))
103	102	rexbidv 3186	. . . 4 ⊢ (((𝑆‘𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎) → (∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋) ↔ ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋)))
104	101, 103	syl5ibrcom 250	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 ((𝑆‘𝐼)‘𝑋) ∩ Fin)) → (((𝑆‘𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎) → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋)))
105	104	rexlimdva 3163	. 2 ⊢ (𝜑 → (∃𝑎 ∈ (𝒫 ((𝑆‘𝐼)‘𝑋) ∩ Fin)((𝑆‘𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎) → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋)))
106	14, 105	mpd 16	1 ⊢ (𝜑 → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086  {crab 3412   ⊆ wss 3895   ∩ cin 4089  𝒫 cpw 4556   class class class wbr 5102   ↦ cmpt 5185  ran crn 5651   ↾ cres 5652   “ cima 5653   Fn wfn 6533  ‘cfv 6538  Fincfn 8973   ≤ cle 11344  ℕ₀cn0 12606  Basecbs 17387  Ringcrg 20459  LIdealclidl 21484  RSpancrsp 21485  Poly₁cpl1 22495  coe₁cco1 22496  deg₁cdg1 26372  LNoeRclnr 44154  ldgIdlSeqcldgis 44166

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278  ax-addf 11279

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  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 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-ofr 7694  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8178  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-map 8849  df-pm 8850  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fsupp 9354  df-sup 9434  df-oi 9504  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-fzo 13789  df-seq 14145  df-hash 14475  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-starv 17443  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-unif 17451  df-hom 17452  df-cco 17453  df-0g 17612  df-gsum 17613  df-prds 17618  df-pws 17620  df-mre 17756  df-mrc 17757  df-acs 17759  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-mhm 18978  df-submnd 18979  df-grp 19147  df-minusg 19148  df-sbg 19149  df-mulg 19278  df-subg 19333  df-ghm 19428  df-cntz 19531  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-ring 20461  df-cring 20462  df-subrng 20798  df-subrg 20822  df-lmod 21137  df-lss 21207  df-lsp 21247  df-sra 21448  df-rgmod 21449  df-lidl 21486  df-rsp 21487  df-cnfld 21679  df-ascl 22163  df-psr 22217  df-mvr 22218  df-mpl 22219  df-opsr 22221  df-psr1 22498  df-vr1 22499  df-ply1 22500  df-coe1 22501  df-mdeg 26373  df-deg1 26374  df-lfig 44113  df-lnm 44121  df-lnr 44155  df-ldgis 44167

This theorem is used by:  hbt  44175