Theorem hbtlem6 43128

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 43111	. . . . 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 2736	. . . . 5 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
10	6, 7, 8, 9	hbtlem2 43123	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
11	3, 4, 5, 10	syl3anc 1373	. . 3 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) ∈ (LIdeal‘𝑅))
12		eqid 2736	. . . 4 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
13	9, 12	lnr2i 43115	. . 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 9371	. . . . 5 ⊢ (𝑎 ∈ (𝒫 ((𝑆‘𝐼)‘𝑋) ∩ Fin) ↔ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin))
16		fvex 6894	. . . . . . . . 9 ⊢ ((coe1‘𝑏)‘𝑋) ∈ V
17		eqid 2736	. . . . . . . . 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 6686	. . . . . . . 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 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ((𝑆‘𝐼)‘𝑋))
21		eqid 2736	. . . . . . . . . . . 12 ⊢ (deg1‘𝑅) = (deg1‘𝑅)
22	6, 7, 8, 21	hbtlem1 43122	. . . . . . . . . . 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 1373	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑆‘𝐼)‘𝑋) = {𝑑 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋))})
24	17	rnmpt 5942	. . . . . . . . . . 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 6881	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → ((deg1‘𝑅)‘𝑐) = ((deg1‘𝑅)‘𝑏))
26	25	breq1d 5134	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (((deg1‘𝑅)‘𝑐) ≤ 𝑋 ↔ ((deg1‘𝑅)‘𝑏) ≤ 𝑋))
27	26	rexrab 3684	. . . . . . . . . . . 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 2803	. . . . . . . . . . 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 2759	. . . . . . . . . 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 2789	. . . . . . . . 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 4000	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ⊆ ran (𝑏 ∈ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↦ ((coe1‘𝑏)‘𝑋)))
33		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ⊆ ((𝑆‘𝐼)‘𝑋) ∧ 𝑎 ∈ Fin)) → 𝑎 ∈ Fin)
34		fipreima 9375	. . . . . . 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 1373	. . . . . 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 9371	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝒫 {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∩ Fin) ↔ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin))
37		ssrab2 4060	. . . . . . . . . . . . . . . . 17 ⊢ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ⊆ 𝐼
38		sstr2 3970	. . . . . . . . . . . . . . . . 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 4585	. . . . . . . . . . . . . . 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 717	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ 𝒫 𝐼)
44		simprr 772	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ∈ Fin)
45	43, 44	elind 4180	. . . . . . . . . . . 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 22188	. . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋})
51	50, 37	sstrdi 3976	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑘 ⊆ 𝐼)
52		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑃) = (Base‘𝑃)
53	52, 7	lidlss 21178	. . . . . . . . . . . . . . . . . 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 3974	. . . . . . . . . . . . . . 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 21201	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → 𝑋 ∈ ℕ0)
61	6, 7, 8, 9	hbtlem2 43123	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘𝑘) ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘(𝑁‘𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
62	46, 59, 60, 61	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ((𝑆‘(𝑁‘𝑘))‘𝑋) ∈ (LIdeal‘𝑅))
63		df-ima 5672	. . . . . . . . . . . . . . 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 21202	. . . . . . . . . . . . . . . . . . . . 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 4053	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ↔ (𝑘 ⊆ 𝐼 ∧ ∀𝑐 ∈ 𝑘 ((deg1‘𝑅)‘𝑐) ≤ 𝑋))
67	66	simprbi 496	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} → ∀𝑐 ∈ 𝑘 ((deg1‘𝑅)‘𝑐) ≤ 𝑋)
68	67	ad2antrl 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑘 ⊆ {𝑐 ∈ 𝐼 ∣ ((deg1‘𝑅)‘𝑐) ≤ 𝑋} ∧ 𝑘 ∈ Fin)) → ∀𝑐 ∈ 𝑘 ((deg1‘𝑅)‘𝑐) ≤ 𝑋)
69		ssrab 4053	. . . . . . . . . . . . . . . . . . . 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 6032	. . . . . . . . . . . . . . . . . 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 6029	. . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . 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 5993	. . . . . . . . . . . . . . . . 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 4009	. . . . . . . . . . . . . . . 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 5924	. . . . . . . . . . . . . . . 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 4002	. . . . . . . . . . . . . 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 43122	. . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . 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 5942	. . . . . . . . . . . . . . . 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 3684	. . . . . . . . . . . . . . . . 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 2803	. . . . . . . . . . . . . . . 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 2759	. . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . 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 4001	. . . . . . . . . . . . 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 21205	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 6881	. . . . . . . . . . . . 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 3995	. . . . . . . . . . . 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 630	. . . . . . . . . . 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 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 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 3151	. . . . . 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 3989	. . . . 5 ⊢ (((𝑆‘𝐼)‘𝑋) = ((RSpan‘𝑅)‘𝑎) → (((𝑆‘𝐼)‘𝑋) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋) ↔ ((RSpan‘𝑅)‘𝑎) ⊆ ((𝑆‘(𝑁‘𝑘))‘𝑋)))
103	102	rexbidv 3165	. . . 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 3142	. 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 1540   ∈ wcel 2109  {cab 2714  ∀wral 3052  ∃wrex 3061  {crab 3420   ∩ cin 3930   ⊆ wss 3931  𝒫 cpw 4580   class class class wbr 5124   ↦ cmpt 5206  ran crn 5660   ↾ cres 5661   “ cima 5662   Fn wfn 6531  ‘cfv 6536  Fincfn 8964   ≤ cle 11275  ℕ₀cn0 12506  Basecbs 17233  Ringcrg 20198  LIdealclidl 21172  RSpancrsp 21173  Poly₁cpl1 22117  coe₁cco1 22118  deg₁cdg1 26016  LNoeRclnr 43108  ldgIdlSeqcldgis 43120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212  ax-addf 11213

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7676  df-ofr 7677  df-om 7867  df-1st 7993  df-2nd 7994  df-supp 8165  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-er 8724  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9379  df-sup 9459  df-oi 9529  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-fz 13530  df-fzo 13677  df-seq 14025  df-hash 14354  df-struct 17171  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-ress 17257  df-plusg 17289  df-mulr 17290  df-starv 17291  df-sca 17292  df-vsca 17293  df-ip 17294  df-tset 17295  df-ple 17296  df-ds 17298  df-unif 17299  df-hom 17300  df-cco 17301  df-0g 17460  df-gsum 17461  df-prds 17466  df-pws 17468  df-mre 17603  df-mrc 17604  df-acs 17606  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-mhm 18766  df-submnd 18767  df-grp 18924  df-minusg 18925  df-sbg 18926  df-mulg 19056  df-subg 19111  df-ghm 19201  df-cntz 19305  df-cmn 19768  df-abl 19769  df-mgp 20106  df-rng 20118  df-ur 20147  df-ring 20200  df-cring 20201  df-subrng 20511  df-subrg 20535  df-lmod 20824  df-lss 20894  df-lsp 20934  df-sra 21136  df-rgmod 21137  df-lidl 21174  df-rsp 21175  df-cnfld 21321  df-ascl 21820  df-psr 21874  df-mvr 21875  df-mpl 21876  df-opsr 21878  df-psr1 22120  df-vr1 22121  df-ply1 22122  df-coe1 22123  df-mdeg 26017  df-deg1 26018  df-lfig 43067  df-lnm 43075  df-lnr 43109  df-ldgis 43121

This theorem is referenced by:  hbt  43129