Theorem hbtlem6 43575

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 43558	. . . . 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 2737	. . . . 5 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
10	6, 7, 8, 9	hbtlem2 43570	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
11	3, 4, 5, 10	syl3anc 1374	. . 3 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
12		eqid 2737	. . . 4 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
13	9, 12	lnr2i 43562	. . 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 585	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝒫 ((𝑆‘𝐼)‘𝑋) ∩ Fin)((𝑆‘𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎))
15		elfpw 9257	. . . . 5 ⊢ (𝑎 ∈ (𝒫 ((𝑆‘𝐼)‘𝑋) ∩ Fin) ↔ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin))
16		fvex 6847	. . . . . . . . 9 ⊢ ((coe1‘𝑏)‘𝑋) ∈ V
17		eqid 2737	. . . . . . . . 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 6635	. . . . . . . 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 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ((𝑆‘𝐼)‘𝑋))
21		eqid 2737	. . . . . . . . . . . 12 ⊢ (deg1‘𝑅) = (deg1‘𝑅)
22	6, 7, 8, 21	hbtlem1 43569	. . . . . . . . . . 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 1374	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) = {𝑑 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋))})
24	17	rnmpt 5906	. . . . . . . . . . 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 6834	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → ((deg1‘𝑅)‘𝑐) = ((deg1‘𝑅)‘𝑏))
26	25	breq1d 5096	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (((deg1‘𝑅)‘𝑐) ≤ 𝑋 ↔ ((deg1‘𝑅)‘𝑏) ≤ 𝑋))
27	26	rexrab 3643	. . . . . . . . . . . 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 2804	. . . . . . . . . . 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 480	. . . . . . . 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 3959	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ran (𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)))
33		simprr 773	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ∈ Fin)
34		fipreima 9261	. . . . . . 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 1374	. . . . . 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 9257	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝒫 {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∩ Fin) ↔ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin))
37		ssrab2 4021	. . . . . . . . . . . . . . . . 17 ⊢ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ⊆ 𝐼
38		sstr2 3929	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋}) → 𝑘 ⊆ 𝐼)
41		velpw 4547	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝒫 𝐼 ↔ 𝑘 ⊆ 𝐼)
42	40, 41	sylibr 234	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋}) → 𝑘 ∈ 𝒫 𝐼)
43	42	adantrr 718	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ 𝒫 𝐼)
44		simprr 773	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ Fin)
45	43, 44	elind 4141	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ (𝒫 𝐼 ∩ Fin))
46	3	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑅 ∈ Ring)
47	6	ply1ring 22221	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
48	3, 47	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ Ring)
49	48	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑃 ∈ Ring)
50		simprl 771	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋})
51	50, 37	sstrdi 3935	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ 𝐼)
52		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑃) = (Base‘𝑃)
53	52, 7	lidlss 21202	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ (Base‘𝑃))
54	4, 53	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐼 ⊆ (Base‘𝑃))
55	54	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝐼 ⊆ (Base‘𝑃))
56	51, 55	sstrd 3933	. . . . . . . . . . . . . . 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 21225	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ Ring ∧ 𝑘 ⊆ (Base‘𝑃)) → (𝑁‘𝑘) ∈ 𝑈)
59	49, 56, 58	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → (𝑁‘𝑘) ∈ 𝑈)
60	5	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑋 ∈ ℕ0)
61	6, 7, 8, 9	hbtlem2 43570	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘𝑘) ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘(𝑁‘𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
62	46, 59, 60, 61	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁‘𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
63		df-ima 5637	. . . . . . . . . . . . . . 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 21226	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃 ∈ Ring ∧ 𝑘 ⊆ (Base‘𝑃)) → 𝑘 ⊆ (𝑁‘𝑘))
65	49, 56, 64	syl2anc 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ (𝑁‘𝑘))
66		ssrab 4012	. . . . . . . . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ∀𝑐 ∈ 𝑘 ((deg1‘𝑅)‘𝑐) ≤ 𝑋)
69		ssrab 4012	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ⊆ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↔ (𝑘 ⊆ (𝑁‘𝑘) ∧ ∀𝑐 ∈ 𝑘 ((deg1‘𝑅)‘𝑐) ≤ 𝑋))
70	65, 68, 69	sylanbrc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ {𝑐 ∈ (𝑁‘𝑘) ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋})
71	70	resmptd 5999	. . . . . . . . . . . . . . . . . 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 5996	. . . . . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . . . 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 5960	. . . . . . . . . . . . . . . . 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 3968	. . . . . . . . . . . . . . . 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 5888	. . . . . . . . . . . . . . . 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 3961	. . . . . . . . . . . . . 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 43569	. . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . 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 5906	. . . . . . . . . . . . . . . 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 3643	. . . . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . . . 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 3960	. . . . . . . . . . . . 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 21229	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 511	. . . . . . . . . . 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 6834	. . . . . . . . . . . . 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 3954	. . . . . . . . . . . 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 631	. . . . . . . . . . 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 245	. . . . . . . . . 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 595	. . . . . . . . 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 453	. . . . . . . 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 480	. . . . . . 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 3148	. . . . . 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 595	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝒫 ((𝑆‘𝐼)‘𝑋) ∩ Fin)) → ∃𝑘 ∈ (𝒫 𝐼 ∩ Fin)((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋))
102		sseq1 3948	. . . . 5 ⊢ (((𝑆‘𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎) → (((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋) ↔ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋)))
103	102	rexbidv 3162	. . . 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 247	. . 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 3139	. 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 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062  {crab 3390   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542   class class class wbr 5086   ↦ cmpt 5167  ran crn 5625   ↾ cres 5626   “ cima 5627   Fn wfn 6487  ‘cfv 6492  Fincfn 8886   ≤ cle 11171  ℕ₀cn0 12428  Basecbs 17170  Ringcrg 20205  LIdealclidl 21196  RSpancrsp 21197  Poly₁cpl1 22150  coe₁cco1 22151  deg₁cdg1 26029  LNoeRclnr 43555  ldgIdlSeqcldgis 43567

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107  ax-addf 11108

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-ofr 7625  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8104  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-er 8636  df-map 8768  df-pm 8769  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9268  df-sup 9348  df-oi 9418  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-fzo 13600  df-seq 13955  df-hash 14284  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-starv 17226  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-unif 17234  df-hom 17235  df-cco 17236  df-0g 17395  df-gsum 17396  df-prds 17401  df-pws 17403  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-submnd 18743  df-grp 18903  df-minusg 18904  df-sbg 18905  df-mulg 19035  df-subg 19090  df-ghm 19179  df-cntz 19283  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-cring 20208  df-subrng 20514  df-subrg 20538  df-lmod 20848  df-lss 20918  df-lsp 20958  df-sra 21160  df-rgmod 21161  df-lidl 21198  df-rsp 21199  df-cnfld 21345  df-ascl 21845  df-psr 21899  df-mvr 21900  df-mpl 21901  df-opsr 21903  df-psr1 22153  df-vr1 22154  df-ply1 22155  df-coe1 22156  df-mdeg 26030  df-deg1 26031  df-lfig 43514  df-lnm 43522  df-lnr 43556  df-ldgis 43568

This theorem is referenced by:  hbt  43576