Theorem hbtlem6 41856

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 41839	. . . . 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 2732	. . . . 5 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
10	6, 7, 8, 9	hbtlem2 41851	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
11	3, 4, 5, 10	syl3anc 1371	. . 3 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
12		eqid 2732	. . . 4 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
13	9, 12	lnr2i 41843	. . 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 584	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝒫 ((𝑆‘𝐼)‘𝑋) ∩ Fin)((𝑆‘𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎))
15		elfpw 9350	. . . . 5 ⊢ (𝑎 ∈ (𝒫 ((𝑆‘𝐼)‘𝑋) ∩ Fin) ↔ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin))
16		fvex 6901	. . . . . . . . 9 ⊢ ((coe1‘𝑏)‘𝑋) ∈ V
17		eqid 2732	. . . . . . . . 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 6690	. . . . . . . 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 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ((𝑆‘𝐼)‘𝑋))
21		eqid 2732	. . . . . . . . . . . 12 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅)
22	6, 7, 8, 21	hbtlem1 41850	. . . . . . . . . . 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 1371	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) = {𝑑 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋))})
24	17	rnmpt 5952	. . . . . . . . . . 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 6888	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → (( deg1 ‘𝑅)‘𝑐) = (( deg1 ‘𝑅)‘𝑏))
26	25	breq1d 5157	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → ((( deg1 ‘𝑅)‘𝑐) ≤ 𝑋 ↔ (( deg1 ‘𝑅)‘𝑏) ≤ 𝑋))
27	26	rexrab 3691	. . . . . . . . . . . 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 2802	. . . . . . . . . . 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 2760	. . . . . . . . . 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 2790	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) = ran (𝑏 ∈ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)))
31	30	adantr 481	. . . . . . . 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 4021	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ran (𝑏 ∈ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)))
33		simprr 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ∈ Fin)
34		fipreima 9354	. . . . . . 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 1371	. . . . . 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 9350	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝒫 {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∩ Fin) ↔ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin))
37		ssrab2 4076	. . . . . . . . . . . . . . . . 17 ⊢ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ⊆ 𝐼
38		sstr2 3988	. . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋}) → 𝑘 ⊆ 𝐼)
41		velpw 4606	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝒫 𝐼 ↔ 𝑘 ⊆ 𝐼)
42	40, 41	sylibr 233	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋}) → 𝑘 ∈ 𝒫 𝐼)
43	42	adantrr 715	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ 𝒫 𝐼)
44		simprr 771	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ Fin)
45	43, 44	elind 4193	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ (𝒫 𝐼 ∩ Fin))
46	3	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑅 ∈ Ring)
47	6	ply1ring 21761	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
48	3, 47	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ Ring)
49	48	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑃 ∈ Ring)
50		simprl 769	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋})
51	50, 37	sstrdi 3993	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ 𝐼)
52		eqid 2732	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑃) = (Base‘𝑃)
53	52, 7	lidlss 20825	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ (Base‘𝑃))
54	4, 53	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐼 ⊆ (Base‘𝑃))
55	54	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝐼 ⊆ (Base‘𝑃))
56	51, 55	sstrd 3991	. . . . . . . . . . . . . . 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 20839	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ Ring ∧ 𝑘 ⊆ (Base‘𝑃)) → (𝑁‘𝑘) ∈ 𝑈)
59	49, 56, 58	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → (𝑁‘𝑘) ∈ 𝑈)
60	5	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑋 ∈ ℕ0)
61	6, 7, 8, 9	hbtlem2 41851	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘𝑘) ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘(𝑁‘𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
62	46, 59, 60, 61	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁‘𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
63		df-ima 5688	. . . . . . . . . . . . . . 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 20840	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃 ∈ Ring ∧ 𝑘 ⊆ (Base‘𝑃)) → 𝑘 ⊆ (𝑁‘𝑘))
65	49, 56, 64	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ (𝑁‘𝑘))
66		ssrab 4069	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ↔ (𝑘 ⊆ 𝐼 ∧ ∀𝑐 ∈ 𝑘 (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋))
67	66	simprbi 497	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} → ∀𝑐 ∈ 𝑘 (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋)
68	67	ad2antrl 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ∀𝑐 ∈ 𝑘 (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋)
69		ssrab 4069	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ⊆ {𝑐 ∈ (𝑁‘𝑘) ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ↔ (𝑘 ⊆ (𝑁‘𝑘) ∧ ∀𝑐 ∈ 𝑘 (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋))
70	65, 68, 69	sylanbrc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ {𝑐 ∈ (𝑁‘𝑘) ∣ (( deg1 ‘𝑅)‘𝑐) ≤ 𝑋})
71	70	resmptd 6038	. . . . . . . . . . . . . . . . . 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 6035	. . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . . 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 6004	. . . . . . . . . . . . . . . . 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 4036	. . . . . . . . . . . . . . . 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 5936	. . . . . . . . . . . . . . . 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	eqsstrid 4029	. . . . . . . . . . . . . 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 41850	. . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . . 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 5952	. . . . . . . . . . . . . . . 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 3691	. . . . . . . . . . . . . . . . 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 2802	. . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . 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 2790	. . . . . . . . . . . . . 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 4022	. . . . . . . . . . . . 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 20843	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 512	. . . . . . . . . . 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 6888	. . . . . . . . . . . . 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 4012	. . . . . . . . . . . 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 629	. . . . . . . . . . 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 244	. . . . . . . . . 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 594	. . . . . . . . 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 454	. . . . . . . 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 481	. . . . . . 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 3164	. . . . . 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 594	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 ((𝑆‘𝐼)‘𝑋) ∩ Fin)) → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))
102		sseq1 4006	. . . . 5 ⊢ (((𝑆‘𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎) → (((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋) ↔ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋)))
103	102	rexbidv 3178	. . . 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 246	. . 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 3155	. 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 396   = wceq 1541   ∈ wcel 2106  {cab 2709  ∀wral 3061  ∃wrex 3070  {crab 3432   ∩ cin 3946   ⊆ wss 3947  𝒫 cpw 4601   class class class wbr 5147   ↦ cmpt 5230  ran crn 5676   ↾ cres 5677   “ cima 5678   Fn wfn 6535  ‘cfv 6540  Fincfn 8935   ≤ cle 11245  ℕ₀cn0 12468  Basecbs 17140  Ringcrg 20049  LIdealclidl 20775  RSpancrsp 20776  Poly₁cpl1 21692  coe₁cco1 21693   deg₁ cdg1 25560  LNoeRclnr 41836  ldgIdlSeqcldgis 41848

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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184  ax-addf 11185  ax-mulf 11186

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-ofr 7667  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-fzo 13624  df-seq 13963  df-hash 14287  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-starv 17208  df-sca 17209  df-vsca 17210  df-ip 17211  df-tset 17212  df-ple 17213  df-ds 17215  df-unif 17216  df-hom 17217  df-cco 17218  df-0g 17383  df-gsum 17384  df-prds 17389  df-pws 17391  df-mre 17526  df-mrc 17527  df-acs 17529  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-submnd 18668  df-grp 18818  df-minusg 18819  df-sbg 18820  df-mulg 18945  df-subg 18997  df-ghm 19084  df-cntz 19175  df-cmn 19644  df-abl 19645  df-mgp 19982  df-ur 19999  df-ring 20051  df-cring 20052  df-subrg 20353  df-lmod 20465  df-lss 20535  df-lsp 20575  df-sra 20777  df-rgmod 20778  df-lidl 20779  df-rsp 20780  df-cnfld 20937  df-ascl 21401  df-psr 21453  df-mvr 21454  df-mpl 21455  df-opsr 21457  df-psr1 21695  df-vr1 21696  df-ply1 21697  df-coe1 21698  df-mdeg 25561  df-deg1 25562  df-lfig 41795  df-lnm 41803  df-lnr 41837  df-ldgis 41849

This theorem is referenced by:  hbt  41857