Theorem hbtlem2 43136

Description: Leading coefficient ideals are ideals. (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)
hbtlem2.t	⊢ 𝑇 = (LIdeal‘𝑅)

Assertion

Ref	Expression
hbtlem2	⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) ∈ 𝑇)

Proof of Theorem hbtlem2

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

Step	Hyp	Ref	Expression
1		hbtlem.p	. . 3 ⊢ 𝑃 = (Poly1‘𝑅)
2		hbtlem.u	. . 3 ⊢ 𝑈 = (LIdeal‘𝑃)
3		hbtlem.s	. . 3 ⊢ 𝑆 = (ldgIdlSeq‘𝑅)
4		eqid 2730	. . 3 ⊢ (deg1‘𝑅) = (deg1‘𝑅)
5	1, 2, 3, 4	hbtlem1 43135	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
6		eqid 2730	. . . . . . . . . . . 12 ⊢ (Base‘𝑃) = (Base‘𝑃)
7	6, 2	lidlss 21142	. . . . . . . . . . 11 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ (Base‘𝑃))
8	7	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝐼 ⊆ (Base‘𝑃))
9	8	sselda 3932	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑃))
10		eqid 2730	. . . . . . . . . 10 ⊢ (coe1‘𝑏) = (coe1‘𝑏)
11		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	10, 6, 1, 11	coe1f 22117	. . . . . . . . 9 ⊢ (𝑏 ∈ (Base‘𝑃) → (coe1‘𝑏):ℕ0⟶(Base‘𝑅))
13	9, 12	syl 17	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → (coe1‘𝑏):ℕ0⟶(Base‘𝑅))
14		simpl3 1194	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → 𝑋 ∈ ℕ0)
15	13, 14	ffvelcdmd 7013	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → ((coe1‘𝑏)‘𝑋) ∈ (Base‘𝑅))
16		eleq1a 2824	. . . . . . 7 ⊢ (((coe1‘𝑏)‘𝑋) ∈ (Base‘𝑅) → (𝑎 = ((coe1‘𝑏)‘𝑋) → 𝑎 ∈ (Base‘𝑅)))
17	15, 16	syl 17	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → (𝑎 = ((coe1‘𝑏)‘𝑋) → 𝑎 ∈ (Base‘𝑅)))
18	17	adantld 490	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → ((((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) → 𝑎 ∈ (Base‘𝑅)))
19	18	rexlimdva 3131	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) → 𝑎 ∈ (Base‘𝑅)))
20	19	abssdv 4017	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ⊆ (Base‘𝑅))
21	1	ply1ring 22153	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
22	21	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑃 ∈ Ring)
23		simp2 1137	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝐼 ∈ 𝑈)
24		eqid 2730	. . . . . . . 8 ⊢ (0g‘𝑃) = (0g‘𝑃)
25	2, 24	lidl0cl 21150	. . . . . . 7 ⊢ ((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (0g‘𝑃) ∈ 𝐼)
26	22, 23, 25	syl2anc 584	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (0g‘𝑃) ∈ 𝐼)
27	4, 1, 24	deg1z 26012	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → ((deg1‘𝑅)‘(0g‘𝑃)) = -∞)
28	27	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((deg1‘𝑅)‘(0g‘𝑃)) = -∞)
29		nn0ssre 12377	. . . . . . . . . 10 ⊢ ℕ0 ⊆ ℝ
30		ressxr 11148	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ*
31	29, 30	sstri 3942	. . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ*
32		simp3 1138	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
33	31, 32	sselid 3930	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈ ℝ*)
34		mnfle 13026	. . . . . . . 8 ⊢ (𝑋 ∈ ℝ* → -∞ ≤ 𝑋)
35	33, 34	syl 17	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → -∞ ≤ 𝑋)
36	28, 35	eqbrtrd 5111	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((deg1‘𝑅)‘(0g‘𝑃)) ≤ 𝑋)
37		eqid 2730	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
38	1, 24, 37	coe1z 22170	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
39	38	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
40	39	fveq1d 6819	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((coe1‘(0g‘𝑃))‘𝑋) = ((ℕ0 × {(0g‘𝑅)})‘𝑋))
41		fvex 6830	. . . . . . . . 9 ⊢ (0g‘𝑅) ∈ V
42	41	fvconst2 7133	. . . . . . . 8 ⊢ (𝑋 ∈ ℕ0 → ((ℕ0 × {(0g‘𝑅)})‘𝑋) = (0g‘𝑅))
43	42	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((ℕ0 × {(0g‘𝑅)})‘𝑋) = (0g‘𝑅))
44	40, 43	eqtr2d 2766	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (0g‘𝑅) = ((coe1‘(0g‘𝑃))‘𝑋))
45		fveq2 6817	. . . . . . . . 9 ⊢ (𝑏 = (0g‘𝑃) → ((deg1‘𝑅)‘𝑏) = ((deg1‘𝑅)‘(0g‘𝑃)))
46	45	breq1d 5099	. . . . . . . 8 ⊢ (𝑏 = (0g‘𝑃) → (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ↔ ((deg1‘𝑅)‘(0g‘𝑃)) ≤ 𝑋))
47		fveq2 6817	. . . . . . . . . 10 ⊢ (𝑏 = (0g‘𝑃) → (coe1‘𝑏) = (coe1‘(0g‘𝑃)))
48	47	fveq1d 6819	. . . . . . . . 9 ⊢ (𝑏 = (0g‘𝑃) → ((coe1‘𝑏)‘𝑋) = ((coe1‘(0g‘𝑃))‘𝑋))
49	48	eqeq2d 2741	. . . . . . . 8 ⊢ (𝑏 = (0g‘𝑃) → ((0g‘𝑅) = ((coe1‘𝑏)‘𝑋) ↔ (0g‘𝑅) = ((coe1‘(0g‘𝑃))‘𝑋)))
50	46, 49	anbi12d 632	. . . . . . 7 ⊢ (𝑏 = (0g‘𝑃) → ((((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ (0g‘𝑅) = ((coe1‘𝑏)‘𝑋)) ↔ (((deg1‘𝑅)‘(0g‘𝑃)) ≤ 𝑋 ∧ (0g‘𝑅) = ((coe1‘(0g‘𝑃))‘𝑋))))
51	50	rspcev 3575	. . . . . 6 ⊢ (((0g‘𝑃) ∈ 𝐼 ∧ (((deg1‘𝑅)‘(0g‘𝑃)) ≤ 𝑋 ∧ (0g‘𝑅) = ((coe1‘(0g‘𝑃))‘𝑋))) → ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ (0g‘𝑅) = ((coe1‘𝑏)‘𝑋)))
52	26, 36, 44, 51	syl12anc 836	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ (0g‘𝑅) = ((coe1‘𝑏)‘𝑋)))
53		eqeq1 2734	. . . . . . . 8 ⊢ (𝑎 = (0g‘𝑅) → (𝑎 = ((coe1‘𝑏)‘𝑋) ↔ (0g‘𝑅) = ((coe1‘𝑏)‘𝑋)))
54	53	anbi2d 630	. . . . . . 7 ⊢ (𝑎 = (0g‘𝑅) → ((((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ (0g‘𝑅) = ((coe1‘𝑏)‘𝑋))))
55	54	rexbidv 3154	. . . . . 6 ⊢ (𝑎 = (0g‘𝑅) → (∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ (0g‘𝑅) = ((coe1‘𝑏)‘𝑋))))
56	41, 55	elab 3633	. . . . 5 ⊢ ((0g‘𝑅) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ↔ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ (0g‘𝑅) = ((coe1‘𝑏)‘𝑋)))
57	52, 56	sylibr 234	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (0g‘𝑅) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
58	57	ne0d 4290	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ≠ ∅)
59	22	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑃 ∈ Ring)
60		simpl2 1193	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝐼 ∈ 𝑈)
61		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
62	1, 61, 11, 6	ply1sclf 22192	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑅 ∈ Ring → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
63	62	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
64	63	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
65		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑐 ∈ (Base‘𝑅))
66	64, 65	ffvelcdmd 7013	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → ((algSc‘𝑃)‘𝑐) ∈ (Base‘𝑃))
67		simprll 778	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋))) → 𝑓 ∈ 𝐼)
68	67	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑓 ∈ 𝐼)
69		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (.r‘𝑃) = (.r‘𝑃)
70	2, 6, 69	lidlmcl 21155	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (((algSc‘𝑃)‘𝑐) ∈ (Base‘𝑃) ∧ 𝑓 ∈ 𝐼)) → (((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓) ∈ 𝐼)
71	59, 60, 66, 68, 70	syl22anc 838	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → (((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓) ∈ 𝐼)
72		simprrl 780	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋))) → 𝑔 ∈ 𝐼)
73	72	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑔 ∈ 𝐼)
74		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (+g‘𝑃) = (+g‘𝑃)
75	2, 74	lidlacl 21151	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓) ∈ 𝐼 ∧ 𝑔 ∈ 𝐼)) → ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) ∈ 𝐼)
76	59, 60, 71, 73, 75	syl22anc 838	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) ∈ 𝐼)
77		simpl1 1192	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑅 ∈ Ring)
78	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝐼 ⊆ (Base‘𝑃))
79	78, 68	sseldd 3933	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑓 ∈ (Base‘𝑃))
80	6, 69	ringcl 20161	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃 ∈ Ring ∧ ((algSc‘𝑃)‘𝑐) ∈ (Base‘𝑃) ∧ 𝑓 ∈ (Base‘𝑃)) → (((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓) ∈ (Base‘𝑃))
81	59, 66, 79, 80	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → (((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓) ∈ (Base‘𝑃))
82	78, 73	sseldd 3933	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑔 ∈ (Base‘𝑃))
83		simpl3 1194	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑋 ∈ ℕ0)
84	31, 83	sselid 3930	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → 𝑋 ∈ ℝ*)
85	4, 1, 6	deg1xrcl 26007	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓) ∈ (Base‘𝑃) → ((deg1‘𝑅)‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)) ∈ ℝ*)
86	81, 85	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → ((deg1‘𝑅)‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)) ∈ ℝ*)
87	4, 1, 6	deg1xrcl 26007	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 ∈ (Base‘𝑃) → ((deg1‘𝑅)‘𝑓) ∈ ℝ*)
88	79, 87	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → ((deg1‘𝑅)‘𝑓) ∈ ℝ*)
89	4, 1, 11, 6, 69, 61	deg1mul3le 26042	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ∈ Ring ∧ 𝑐 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑃)) → ((deg1‘𝑅)‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)) ≤ ((deg1‘𝑅)‘𝑓))
90	77, 65, 79, 89	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → ((deg1‘𝑅)‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)) ≤ ((deg1‘𝑅)‘𝑓))
91		simprlr 779	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋))) → ((deg1‘𝑅)‘𝑓) ≤ 𝑋)
92	91	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → ((deg1‘𝑅)‘𝑓) ≤ 𝑋)
93	86, 88, 84, 90, 92	xrletrd 13053	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → ((deg1‘𝑅)‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)) ≤ 𝑋)
94		simprrr 781	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋))) → ((deg1‘𝑅)‘𝑔) ≤ 𝑋)
95	94	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → ((deg1‘𝑅)‘𝑔) ≤ 𝑋)
96	1, 4, 77, 6, 74, 81, 82, 84, 93, 95	deg1addle2 26027	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → ((deg1‘𝑅)‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔)) ≤ 𝑋)
97		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (+g‘𝑅) = (+g‘𝑅)
98	1, 6, 74, 97	coe1addfv 22172	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑅 ∈ Ring ∧ (((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓) ∈ (Base‘𝑃) ∧ 𝑔 ∈ (Base‘𝑃)) ∧ 𝑋 ∈ ℕ0) → ((coe1‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔))‘𝑋) = (((coe1‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓))‘𝑋)(+g‘𝑅)((coe1‘𝑔)‘𝑋)))
99	77, 81, 82, 83, 98	syl31anc 1375	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → ((coe1‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔))‘𝑋) = (((coe1‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓))‘𝑋)(+g‘𝑅)((coe1‘𝑔)‘𝑋)))
100		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (.r‘𝑅) = (.r‘𝑅)
101	1, 6, 11, 61, 69, 100	coe1sclmulfv 22190	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ∈ Ring ∧ (𝑐 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑃)) ∧ 𝑋 ∈ ℕ0) → ((coe1‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓))‘𝑋) = (𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋)))
102	77, 65, 79, 83, 101	syl121anc 1377	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → ((coe1‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓))‘𝑋) = (𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋)))
103	102	oveq1d 7356	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → (((coe1‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓))‘𝑋)(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)))
104	99, 103	eqtr2d 2766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔))‘𝑋))
105		fveq2 6817	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) → ((deg1‘𝑅)‘𝑏) = ((deg1‘𝑅)‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔)))
106	105	breq1d 5099	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) → (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ↔ ((deg1‘𝑅)‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔)) ≤ 𝑋))
107		fveq2 6817	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) → (coe1‘𝑏) = (coe1‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔)))
108	107	fveq1d 6819	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) → ((coe1‘𝑏)‘𝑋) = ((coe1‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔))‘𝑋))
109	108	eqeq2d 2741	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) → (((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘𝑏)‘𝑋) ↔ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔))‘𝑋)))
110	106, 109	anbi12d 632	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) → ((((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘𝑏)‘𝑋)) ↔ (((deg1‘𝑅)‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔)) ≤ 𝑋 ∧ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔))‘𝑋))))
111	110	rspcev 3575	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔) ∈ 𝐼 ∧ (((deg1‘𝑅)‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔)) ≤ 𝑋 ∧ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)(+g‘𝑃)𝑔))‘𝑋))) → ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘𝑏)‘𝑋)))
112	76, 96, 104, 111	syl12anc 836	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘𝑏)‘𝑋)))
113		ovex 7374	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ V
114		eqeq1 2734	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) → (𝑎 = ((coe1‘𝑏)‘𝑋) ↔ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘𝑏)‘𝑋)))
115	114	anbi2d 630	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) → ((((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘𝑏)‘𝑋))))
116	115	rexbidv 3154	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) → (∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘𝑏)‘𝑋))))
117	113, 116	elab 3633	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ↔ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) = ((coe1‘𝑏)‘𝑋)))
118	112, 117	sylibr 234	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ (𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋)))) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
119	118	exp45 438	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (𝑐 ∈ (Base‘𝑅) → ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) → ((𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))))
120	119	imp 406	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) → ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) → ((𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})))
121	120	exp5c 444	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) → (𝑓 ∈ 𝐼 → (((deg1‘𝑅)‘𝑓) ≤ 𝑋 → (𝑔 ∈ 𝐼 → (((deg1‘𝑅)‘𝑔) ≤ 𝑋 → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})))))
122	121	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) → (((deg1‘𝑅)‘𝑓) ≤ 𝑋 → (𝑔 ∈ 𝐼 → (((deg1‘𝑅)‘𝑔) ≤ 𝑋 → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))))
123	122	imp41 425	. . . . . . . . . . . . . 14 ⊢ (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ 𝑔 ∈ 𝐼) ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
124		oveq2 7349	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = ((coe1‘𝑔)‘𝑋) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) = ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)))
125	124	eleq1d 2814	. . . . . . . . . . . . . 14 ⊢ (𝑒 = ((coe1‘𝑔)‘𝑋) → (((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ↔ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))
126	123, 125	syl5ibrcom 247	. . . . . . . . . . . . 13 ⊢ (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ 𝑔 ∈ 𝐼) ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋) → (𝑒 = ((coe1‘𝑔)‘𝑋) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))
127	126	expimpd 453	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ 𝑔 ∈ 𝐼) → ((((deg1‘𝑅)‘𝑔) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑔)‘𝑋)) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))
128	127	rexlimdva 3131	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) → (∃𝑔 ∈ 𝐼 (((deg1‘𝑅)‘𝑔) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑔)‘𝑋)) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))
129	128	alrimiv 1928	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) → ∀𝑒(∃𝑔 ∈ 𝐼 (((deg1‘𝑅)‘𝑔) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑔)‘𝑋)) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))
130		eqeq1 2734	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑒 → (𝑎 = ((coe1‘𝑏)‘𝑋) ↔ 𝑒 = ((coe1‘𝑏)‘𝑋)))
131	130	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑒 → ((((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑏)‘𝑋))))
132	131	rexbidv 3154	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑒 → (∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑏)‘𝑋))))
133		fveq2 6817	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑔 → ((deg1‘𝑅)‘𝑏) = ((deg1‘𝑅)‘𝑔))
134	133	breq1d 5099	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑔 → (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ↔ ((deg1‘𝑅)‘𝑔) ≤ 𝑋))
135		fveq2 6817	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑔 → (coe1‘𝑏) = (coe1‘𝑔))
136	135	fveq1d 6819	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑔 → ((coe1‘𝑏)‘𝑋) = ((coe1‘𝑔)‘𝑋))
137	136	eqeq2d 2741	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑔 → (𝑒 = ((coe1‘𝑏)‘𝑋) ↔ 𝑒 = ((coe1‘𝑔)‘𝑋)))
138	134, 137	anbi12d 632	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑔 → ((((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑏)‘𝑋)) ↔ (((deg1‘𝑅)‘𝑔) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑔)‘𝑋))))
139	138	cbvrexvw 3209	. . . . . . . . . . . 12 ⊢ (∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑏)‘𝑋)) ↔ ∃𝑔 ∈ 𝐼 (((deg1‘𝑅)‘𝑔) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑔)‘𝑋)))
140	132, 139	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑒 → (∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ∃𝑔 ∈ 𝐼 (((deg1‘𝑅)‘𝑔) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑔)‘𝑋))))
141	140	ralab 3650	. . . . . . . . . 10 ⊢ (∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ↔ ∀𝑒(∃𝑔 ∈ 𝐼 (((deg1‘𝑅)‘𝑔) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑔)‘𝑋)) → ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))
142	129, 141	sylibr 234	. . . . . . . . 9 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
143		oveq2 7349	. . . . . . . . . . . 12 ⊢ (𝑑 = ((coe1‘𝑓)‘𝑋) → (𝑐(.r‘𝑅)𝑑) = (𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋)))
144	143	oveq1d 7356	. . . . . . . . . . 11 ⊢ (𝑑 = ((coe1‘𝑓)‘𝑋) → ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) = ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒))
145	144	eleq1d 2814	. . . . . . . . . 10 ⊢ (𝑑 = ((coe1‘𝑓)‘𝑋) → (((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ↔ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))
146	145	ralbidv 3153	. . . . . . . . 9 ⊢ (𝑑 = ((coe1‘𝑓)‘𝑋) → (∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ↔ ∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))
147	142, 146	syl5ibrcom 247	. . . . . . . 8 ⊢ (((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) → (𝑑 = ((coe1‘𝑓)‘𝑋) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))
148	147	expimpd 453	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) ∧ 𝑓 ∈ 𝐼) → ((((deg1‘𝑅)‘𝑓) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑓)‘𝑋)) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))
149	148	rexlimdva 3131	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) → (∃𝑓 ∈ 𝐼 (((deg1‘𝑅)‘𝑓) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑓)‘𝑋)) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))
150	149	alrimiv 1928	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) → ∀𝑑(∃𝑓 ∈ 𝐼 (((deg1‘𝑅)‘𝑓) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑓)‘𝑋)) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))
151		eqeq1 2734	. . . . . . . . 9 ⊢ (𝑎 = 𝑑 → (𝑎 = ((coe1‘𝑏)‘𝑋) ↔ 𝑑 = ((coe1‘𝑏)‘𝑋)))
152	151	anbi2d 630	. . . . . . . 8 ⊢ (𝑎 = 𝑑 → ((((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋))))
153	152	rexbidv 3154	. . . . . . 7 ⊢ (𝑎 = 𝑑 → (∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋))))
154		fveq2 6817	. . . . . . . . . 10 ⊢ (𝑏 = 𝑓 → ((deg1‘𝑅)‘𝑏) = ((deg1‘𝑅)‘𝑓))
155	154	breq1d 5099	. . . . . . . . 9 ⊢ (𝑏 = 𝑓 → (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ↔ ((deg1‘𝑅)‘𝑓) ≤ 𝑋))
156		fveq2 6817	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑓 → (coe1‘𝑏) = (coe1‘𝑓))
157	156	fveq1d 6819	. . . . . . . . . 10 ⊢ (𝑏 = 𝑓 → ((coe1‘𝑏)‘𝑋) = ((coe1‘𝑓)‘𝑋))
158	157	eqeq2d 2741	. . . . . . . . 9 ⊢ (𝑏 = 𝑓 → (𝑑 = ((coe1‘𝑏)‘𝑋) ↔ 𝑑 = ((coe1‘𝑓)‘𝑋)))
159	155, 158	anbi12d 632	. . . . . . . 8 ⊢ (𝑏 = 𝑓 → ((((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋)) ↔ (((deg1‘𝑅)‘𝑓) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑓)‘𝑋))))
160	159	cbvrexvw 3209	. . . . . . 7 ⊢ (∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋)) ↔ ∃𝑓 ∈ 𝐼 (((deg1‘𝑅)‘𝑓) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑓)‘𝑋)))
161	153, 160	bitrdi 287	. . . . . 6 ⊢ (𝑎 = 𝑑 → (∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ∃𝑓 ∈ 𝐼 (((deg1‘𝑅)‘𝑓) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑓)‘𝑋))))
162	161	ralab 3650	. . . . 5 ⊢ (∀𝑑 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ↔ ∀𝑑(∃𝑓 ∈ 𝐼 (((deg1‘𝑅)‘𝑓) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑓)‘𝑋)) → ∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))
163	150, 162	sylibr 234	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑐 ∈ (Base‘𝑅)) → ∀𝑑 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
164	163	ralrimiva 3122	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ∀𝑐 ∈ (Base‘𝑅)∀𝑑 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
165		hbtlem2.t	. . . 4 ⊢ 𝑇 = (LIdeal‘𝑅)
166	165, 11, 97, 100	islidl 21145	. . 3 ⊢ ({𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ∈ 𝑇 ↔ ({𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ⊆ (Base‘𝑅) ∧ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ≠ ∅ ∧ ∀𝑐 ∈ (Base‘𝑅)∀𝑑 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}∀𝑒 ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))}))
167	20, 58, 164, 166	syl3anbrc 1344	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ∈ 𝑇)
168	5, 167	eqeltrd 2829	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   = wceq 1541   ∈ wcel 2110  {cab 2708   ≠ wne 2926  ∀wral 3045  ∃wrex 3054   ⊆ wss 3900  ∅c0 4281  {csn 4574   class class class wbr 5089   × cxp 5612  ⟶wf 6473  ‘cfv 6477  (class class class)co 7341  ℝcr 10997  -∞cmnf 11136  ℝ^*cxr 11137   ≤ cle 11139  ℕ₀cn0 12373  Basecbs 17112  +_gcplusg 17153  ._rcmulr 17154  0_gc0g 17335  Ringcrg 20144  LIdealclidl 21136  algSccascl 21782  Poly₁cpl1 22082  coe₁cco1 22083  deg₁cdg1 25979  ldgIdlSeqcldgis 43133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075  ax-pre-sup 11076  ax-addf 11077

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-isom 6486  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-ofr 7606  df-om 7792  df-1st 7916  df-2nd 7917  df-supp 8086  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-er 8617  df-map 8747  df-pm 8748  df-ixp 8817  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-fsupp 9241  df-sup 9321  df-oi 9391  df-card 9824  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-2 12180  df-3 12181  df-4 12182  df-5 12183  df-6 12184  df-7 12185  df-8 12186  df-9 12187  df-n0 12374  df-z 12461  df-dec 12581  df-uz 12725  df-fz 13400  df-fzo 13547  df-seq 13901  df-hash 14230  df-struct 17050  df-sets 17067  df-slot 17085  df-ndx 17097  df-base 17113  df-ress 17134  df-plusg 17166  df-mulr 17167  df-starv 17168  df-sca 17169  df-vsca 17170  df-ip 17171  df-tset 17172  df-ple 17173  df-ds 17175  df-unif 17176  df-hom 17177  df-cco 17178  df-0g 17337  df-gsum 17338  df-prds 17343  df-pws 17345  df-mre 17480  df-mrc 17481  df-acs 17483  df-mgm 18540  df-sgrp 18619  df-mnd 18635  df-mhm 18683  df-submnd 18684  df-grp 18841  df-minusg 18842  df-sbg 18843  df-mulg 18973  df-subg 19028  df-ghm 19118  df-cntz 19222  df-cmn 19687  df-abl 19688  df-mgp 20052  df-rng 20064  df-ur 20093  df-ring 20146  df-cring 20147  df-subrng 20454  df-subrg 20478  df-lmod 20788  df-lss 20858  df-sra 21100  df-rgmod 21101  df-lidl 21138  df-cnfld 21285  df-ascl 21785  df-psr 21839  df-mvr 21840  df-mpl 21841  df-opsr 21843  df-psr1 22085  df-vr1 22086  df-ply1 22087  df-coe1 22088  df-mdeg 25980  df-deg1 25981  df-ldgis 43134

This theorem is referenced by:  hbtlem7  43137  hbtlem6  43141