Theorem hbtlem6 39131

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 39114	. . . . 5 ⊢ (𝑅 ∈ LNoeR → 𝑅 ∈ Ring)
3	1, 2	syl 17	. . . 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 2778	. . . . 5 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
10	6, 7, 8, 9	hbtlem2 39126	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
11	3, 4, 5, 10	syl3anc 1351	. . 3 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
12		eqid 2778	. . . 4 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
13	9, 12	lnr2i 39118	. . 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 576	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝒫 ((𝑆‘𝐼)‘𝑋) ∩ Fin)((𝑆‘𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎))
15		elfpw 8621	. . . . 5 ⊢ (𝑎 ∈ (𝒫 ((𝑆‘𝐼)‘𝑋) ∩ Fin) ↔ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin))
16		fvex 6512	. . . . . . . . 9 ⊢ ((coe1‘𝑏)‘𝑋) ∈ V
17		eqid 2778	. . . . . . . . 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 6321	. . . . . . . 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 758	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ((𝑆‘𝐼)‘𝑋))
21		eqid 2778	. . . . . . . . . . . 12 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅)
22	6, 7, 8, 21	hbtlem1 39125	. . . . . . . . . . 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 1351	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) = {𝑑 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋))})
24	17	rnmpt 5670	. . . . . . . . . . 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 6499	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → (( deg1 ‘𝑅)‘𝑐) = (( deg1 ‘𝑅)‘𝑏))
26	25	breq1d 4939	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → ((( deg1 ‘𝑅)‘𝑐) ≤ 𝑋 ↔ (( deg1 ‘𝑅)‘𝑏) ≤ 𝑋))
27	26	rexrab 3603	. . . . . . . . . . . 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 2844	. . . . . . . . . . 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 2802	. . . . . . . . . 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	syl6eqr 2832	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) = ran (𝑏 ∈ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)))
31	30	adantr 473	. . . . . . . 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 3897	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ran (𝑏 ∈ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)))
33		simprr 760	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ∈ Fin)
34		fipreima 8625	. . . . . . 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 1351	. . . . . 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 8621	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝒫 {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∩ Fin) ↔ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin))
37		ssrab2 3946	. . . . . . . . . . . . . . . . 17 ⊢ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ⊆ 𝐼
38		sstr2 3865	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} → ({𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ⊆ 𝐼 → 𝑘 ⊆ 𝐼))
39	37, 38	mpi 20	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} → 𝑘 ⊆ 𝐼)
40	39	adantl 474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋}) → 𝑘 ⊆ 𝐼)
41		selpw 4429	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝒫 𝐼 ↔ 𝑘 ⊆ 𝐼)
42	40, 41	sylibr 226	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋}) → 𝑘 ∈ 𝒫 𝐼)
43	42	adantrr 704	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ 𝒫 𝐼)
44		simprr 760	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ Fin)
45	43, 44	elind 4059	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ (𝒫 𝐼 ∩ Fin))
46	3	adantr 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑅 ∈ Ring)
47	6	ply1ring 20119	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
48	3, 47	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ Ring)
49	48	adantr 473	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑃 ∈ Ring)
50		simprl 758	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋})
51	50, 37	syl6ss 3870	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ 𝐼)
52		eqid 2778	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑃) = (Base‘𝑃)
53	52, 7	lidlss 19704	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ (Base‘𝑃))
54	4, 53	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐼 ⊆ (Base‘𝑃))
55	54	adantr 473	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝐼 ⊆ (Base‘𝑃))
56	51, 55	sstrd 3868	. . . . . . . . . . . . . . 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 19716	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ Ring ∧ 𝑘 ⊆ (Base‘𝑃)) → (𝑁‘𝑘) ∈ 𝑈)
59	49, 56, 58	syl2anc 576	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → (𝑁‘𝑘) ∈ 𝑈)
60	5	adantr 473	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑋 ∈ ℕ0)
61	6, 7, 8, 9	hbtlem2 39126	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘𝑘) ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘(𝑁‘𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
62	46, 59, 60, 61	syl3anc 1351	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁‘𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
63		df-ima 5420	. . . . . . . . . . . . . . 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 19717	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃 ∈ Ring ∧ 𝑘 ⊆ (Base‘𝑃)) → 𝑘 ⊆ (𝑁‘𝑘))
65	49, 56, 64	syl2anc 576	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ (𝑁‘𝑘))
66		ssrab 3939	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ↔ (𝑘 ⊆ 𝐼 ∧ ∀𝑐 ∈ 𝑘 (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋))
67	66	simprbi 489	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} → ∀𝑐 ∈ 𝑘 (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋)
68	67	ad2antrl 715	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ∀𝑐 ∈ 𝑘 (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋)
69		ssrab 3939	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ⊆ {𝑐 ∈ (𝑁‘𝑘) ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ↔ (𝑘 ⊆ (𝑁‘𝑘) ∧ ∀𝑐 ∈ 𝑘 (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋))
70	65, 68, 69	sylanbrc 575	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ {𝑐 ∈ (𝑁‘𝑘) ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋})
71	70	resmptd 5753	. . . . . . . . . . . . . . . . . 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 5750	. . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . 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 5723	. . . . . . . . . . . . . . . . 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	syl6eqssr 3912	. . . . . . . . . . . . . . . 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 5652	. . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . 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	syl5eqss 3905	. . . . . . . . . . . . . 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 39125	. . . . . . . . . . . . . . . 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 1351	. . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . 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 5670	. . . . . . . . . . . . . . . 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 3603	. . . . . . . . . . . . . . . . 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 2844	. . . . . . . . . . . . . . . 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 2802	. . . . . . . . . . . . . . 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	syl6eqr 2832	. . . . . . . . . . . . . 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	sseqtr4d 3898	. . . . . . . . . . . . 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 19720	. . . . . . . . . . . . 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 1351	. . . . . . . . . . . 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 504	. . . . . . . . . . 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 6499	. . . . . . . . . . . . 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 3888	. . . . . . . . . . . 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 619	. . . . . . . . . . 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 237	. . . . . . . . . 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 584	. . . . . . . . 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 446	. . . . . . . 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 473	. . . . . . 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 3216	. . . . . 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 15	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))
101	15, 100	sylan2b 584	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 ((𝑆‘𝐼)‘𝑋) ∩ Fin)) → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))
102		sseq1 3882	. . . . 5 ⊢ (((𝑆‘𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎) → (((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋) ↔ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋)))
103	102	rexbidv 3242	. . . 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 239	. . 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 3229	. 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 15	1 ⊢ (𝜑 → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050  {cab 2758  ∀wral 3088  ∃wrex 3089  {crab 3092   ∩ cin 3828   ⊆ wss 3829  𝒫 cpw 4422   class class class wbr 4929   ↦ cmpt 5008  ran crn 5408   ↾ cres 5409   “ cima 5410   Fn wfn 6183  ‘cfv 6188  Fincfn 8306   ≤ cle 10475  ℕ₀cn0 11707  Basecbs 16339  Ringcrg 19020  LIdealclidl 19664  RSpancrsp 19665  Poly₁cpl1 20048  coe₁cco1 20049   deg₁ cdg1 24351  LNoeRclnr 39111  ldgIdlSeqcldgis 39123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412  ax-pre-sup 10413  ax-addf 10414  ax-mulf 10415

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-iin 4795  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-of 7227  df-ofr 7228  df-om 7397  df-1st 7501  df-2nd 7502  df-supp 7634  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-2o 7906  df-oadd 7909  df-er 8089  df-map 8208  df-pm 8209  df-ixp 8260  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-fsupp 8629  df-sup 8701  df-oi 8769  df-card 9162  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-nn 11440  df-2 11503  df-3 11504  df-4 11505  df-5 11506  df-6 11507  df-7 11508  df-8 11509  df-9 11510  df-n0 11708  df-z 11794  df-dec 11912  df-uz 12059  df-fz 12709  df-fzo 12850  df-seq 13185  df-hash 13506  df-struct 16341  df-ndx 16342  df-slot 16343  df-base 16345  df-sets 16346  df-ress 16347  df-plusg 16434  df-mulr 16435  df-starv 16436  df-sca 16437  df-vsca 16438  df-ip 16439  df-tset 16440  df-ple 16441  df-ds 16443  df-unif 16444  df-0g 16571  df-gsum 16572  df-mre 16715  df-mrc 16716  df-acs 16718  df-mgm 17710  df-sgrp 17752  df-mnd 17763  df-mhm 17803  df-submnd 17804  df-grp 17894  df-minusg 17895  df-sbg 17896  df-mulg 18012  df-subg 18060  df-ghm 18127  df-cntz 18218  df-cmn 18668  df-abl 18669  df-mgp 18963  df-ur 18975  df-ring 19022  df-cring 19023  df-subrg 19256  df-lmod 19358  df-lss 19426  df-lsp 19466  df-sra 19666  df-rgmod 19667  df-lidl 19668  df-rsp 19669  df-ascl 19808  df-psr 19850  df-mvr 19851  df-mpl 19852  df-opsr 19854  df-psr1 20051  df-vr1 20052  df-ply1 20053  df-coe1 20054  df-cnfld 20248  df-mdeg 24352  df-deg1 24353  df-lfig 39070  df-lnm 39078  df-lnr 39112  df-ldgis 39124

This theorem is referenced by:  hbt  39132