Theorem hbtlem2 41437

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 2736	. . 3 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅)
5	1, 2, 3, 4	hbtlem1 41436	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
6		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝑃) = (Base‘𝑃)
7	6, 2	lidlss 20680	. . . . . . . . . . 11 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ (Base‘𝑃))
8	7	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝐼 ⊆ (Base‘𝑃))
9	8	sselda 3944	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑃))
10		eqid 2736	. . . . . . . . . 10 ⊢ (coe1‘𝑏) = (coe1‘𝑏)
11		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	10, 6, 1, 11	coe1f 21582	. . . . . . . . 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 1193	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → 𝑋 ∈ ℕ0)
15	13, 14	ffvelcdmd 7036	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → ((coe1‘𝑏)‘𝑋) ∈ (Base‘𝑅))
16		eleq1a 2833	. . . . . . 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 491	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → (((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) → 𝑎 ∈ (Base‘𝑅)))
19	18	rexlimdva 3152	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) → 𝑎 ∈ (Base‘𝑅)))
20	19	abssdv 4025	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ⊆ (Base‘𝑅))
21	1	ply1ring 21619	. . . . . . . 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 2736	. . . . . . . 8 ⊢ (0g‘𝑃) = (0g‘𝑃)
25	2, 24	lidl0cl 20682	. . . . . . 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 25452	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (( deg1 ‘𝑅)‘(0g‘𝑃)) = -∞)
28	27	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (( deg1 ‘𝑅)‘(0g‘𝑃)) = -∞)
29		nn0ssre 12417	. . . . . . . . . 10 ⊢ ℕ0 ⊆ ℝ
30		ressxr 11199	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ*
31	29, 30	sstri 3953	. . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ*
32		simp3 1138	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
33	31, 32	sselid 3942	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈ ℝ*)
34		mnfle 13055	. . . . . . . 8 ⊢ (𝑋 ∈ ℝ* → -∞ ≤ 𝑋)
35	33, 34	syl 17	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → -∞ ≤ 𝑋)
36	28, 35	eqbrtrd 5127	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (( deg1 ‘𝑅)‘(0g‘𝑃)) ≤ 𝑋)
37		eqid 2736	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
38	1, 24, 37	coe1z 21634	. . . . . . . . 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 6844	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((coe1‘(0g‘𝑃))‘𝑋) = ((ℕ0 × {(0g‘𝑅)})‘𝑋))
41		fvex 6855	. . . . . . . . 9 ⊢ (0g‘𝑅) ∈ V
42	41	fvconst2 7153	. . . . . . . 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 2777	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (0g‘𝑅) = ((coe1‘(0g‘𝑃))‘𝑋))
45		fveq2 6842	. . . . . . . . 9 ⊢ (𝑏 = (0g‘𝑃) → (( deg1 ‘𝑅)‘𝑏) = (( deg1 ‘𝑅)‘(0g‘𝑃)))
46	45	breq1d 5115	. . . . . . . 8 ⊢ (𝑏 = (0g‘𝑃) → ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1 ‘𝑅)‘(0g‘𝑃)) ≤ 𝑋))
47		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑏 = (0g‘𝑃) → (coe1‘𝑏) = (coe1‘(0g‘𝑃)))
48	47	fveq1d 6844	. . . . . . . . 9 ⊢ (𝑏 = (0g‘𝑃) → ((coe1‘𝑏)‘𝑋) = ((coe1‘(0g‘𝑃))‘𝑋))
49	48	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑏 = (0g‘𝑃) → ((0g‘𝑅) = ((coe1‘𝑏)‘𝑋) ↔ (0g‘𝑅) = ((coe1‘(0g‘𝑃))‘𝑋)))
50	46, 49	anbi12d 631	. . . . . . 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 3581	. . . . . 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 835	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ (0g‘𝑅) = ((coe1‘𝑏)‘𝑋)))
53		eqeq1 2740	. . . . . . . 8 ⊢ (𝑎 = (0g‘𝑅) → (𝑎 = ((coe1‘𝑏)‘𝑋) ↔ (0g‘𝑅) = ((coe1‘𝑏)‘𝑋)))
54	53	anbi2d 629	. . . . . . 7 ⊢ (𝑎 = (0g‘𝑅) → (((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ (0g‘𝑅) = ((coe1‘𝑏)‘𝑋))))
55	54	rexbidv 3175	. . . . . 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 3630	. . . . 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 233	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (0g‘𝑅) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
58	57	ne0d 4295	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ≠ ∅)
59	22	adantr 481	. . . . . . . . . . . . . . . . . . . . . 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 1192	. . . . . . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
62	1, 61, 11, 6	ply1sclf 21656	. . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . . . . 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 7036	. . . . . . . . . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋))) → 𝑓 ∈ 𝐼)
68	67	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (.r‘𝑃) = (.r‘𝑃)
70	2, 6, 69	lidlmcl 20687	. . . . . . . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋))) → 𝑔 ∈ 𝐼)
73	72	adantl 482	. . . . . . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (+g‘𝑃) = (+g‘𝑃)
75	2, 74	lidlacl 20683	. . . . . . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . . . . . 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 1191	. . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . 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 3945	. . . . . . . . . . . . . . . . . . . . . . 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 19981	. . . . . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . . . . 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 3945	. . . . . . . . . . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . . . . . . . . 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 3942	. . . . . . . . . . . . . . . . . . . . . 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 25447	. . . . . . . . . . . . . . . . . . . . . . . 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 25447	. . . . . . . . . . . . . . . . . . . . . . . 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 25481	. . . . . . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋))) → (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋)
92	91	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 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 13081	. . . . . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋))) → (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋)
95	94	adantl 482	. . . . . . . . . . . . . . . . . . . . . 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 25467	. . . . . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (+g‘𝑅) = (+g‘𝑅)
98	1, 6, 74, 97	coe1addfv 21636	. . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (.r‘𝑅) = (.r‘𝑅)
101	1, 6, 11, 61, 69, 100	coe1sclmulfv 21654	. . . . . . . . . . . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . . . . . . . . . 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 5115	. . . . . . . . . . . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . . . . . . . . . . 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 2747	. . . . . . . . . . . . . . . . . . . . . . 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 631	. . . . . . . . . . . . . . . . . . . . . 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 3581	. . . . . . . . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . . . . . . . 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 7390	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ V
114		eqeq1 2740	. . . . . . . . . . . . . . . . . . . . . . 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 629	. . . . . . . . . . . . . . . . . . . . . 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 3175	. . . . . . . . . . . . . . . . . . . . 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 3630	. . . . . . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . . . . . . 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 439	. . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . 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 445	. . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . 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 426	. . . . . . . . . . . . . 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 7365	. . . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . . 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 246	. . . . . . . . . . . . 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 454	. . . . . . . . . . . 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 3152	. . . . . . . . . . 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 1930	. . . . . . . . . 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 2740	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑒 → (𝑎 = ((coe1‘𝑏)‘𝑋) ↔ 𝑒 = ((coe1‘𝑏)‘𝑋)))
131	130	anbi2d 629	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑒 → (((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑏)‘𝑋))))
132	131	rexbidv 3175	. . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑔 → (( deg1 ‘𝑅)‘𝑏) = (( deg1 ‘𝑅)‘𝑔))
134	133	breq1d 5115	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑔 → ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1 ‘𝑅)‘𝑔) ≤ 𝑋))
135		fveq2 6842	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑔 → (coe1‘𝑏) = (coe1‘𝑔))
136	135	fveq1d 6844	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑔 → ((coe1‘𝑏)‘𝑋) = ((coe1‘𝑔)‘𝑋))
137	136	eqeq2d 2747	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑔 → (𝑒 = ((coe1‘𝑏)‘𝑋) ↔ 𝑒 = ((coe1‘𝑔)‘𝑋)))
138	134, 137	anbi12d 631	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑔 → (((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑏)‘𝑋)) ↔ ((( deg1 ‘𝑅)‘𝑔) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑔)‘𝑋))))
139	138	cbvrexvw 3226	. . . . . . . . . . . 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 286	. . . . . . . . . . 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 3649	. . . . . . . . . 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 233	. . . . . . . . 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 7365	. . . . . . . . . . . 12 ⊢ (𝑑 = ((coe1‘𝑓)‘𝑋) → (𝑐(.r‘𝑅)𝑑) = (𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋)))
144	143	oveq1d 7372	. . . . . . . . . . 11 ⊢ (𝑑 = ((coe1‘𝑓)‘𝑋) → ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) = ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒))
145	144	eleq1d 2822	. . . . . . . . . 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 3174	. . . . . . . . 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 246	. . . . . . . 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 454	. . . . . . 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 3152	. . . . . 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 1930	. . . . 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 2740	. . . . . . . . 9 ⊢ (𝑎 = 𝑑 → (𝑎 = ((coe1‘𝑏)‘𝑋) ↔ 𝑑 = ((coe1‘𝑏)‘𝑋)))
152	151	anbi2d 629	. . . . . . . 8 ⊢ (𝑎 = 𝑑 → (((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋))))
153	152	rexbidv 3175	. . . . . . 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 6842	. . . . . . . . . 10 ⊢ (𝑏 = 𝑓 → (( deg1 ‘𝑅)‘𝑏) = (( deg1 ‘𝑅)‘𝑓))
155	154	breq1d 5115	. . . . . . . . 9 ⊢ (𝑏 = 𝑓 → ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ↔ (( deg1 ‘𝑅)‘𝑓) ≤ 𝑋))
156		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑓 → (coe1‘𝑏) = (coe1‘𝑓))
157	156	fveq1d 6844	. . . . . . . . . 10 ⊢ (𝑏 = 𝑓 → ((coe1‘𝑏)‘𝑋) = ((coe1‘𝑓)‘𝑋))
158	157	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑏 = 𝑓 → (𝑑 = ((coe1‘𝑏)‘𝑋) ↔ 𝑑 = ((coe1‘𝑓)‘𝑋)))
159	155, 158	anbi12d 631	. . . . . . . 8 ⊢ (𝑏 = 𝑓 → (((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋)) ↔ ((( deg1 ‘𝑅)‘𝑓) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑓)‘𝑋))))
160	159	cbvrexvw 3226	. . . . . . 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 286	. . . . . 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 3649	. . . . 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 233	. . . 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 3143	. . 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 20681	. . 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 1343	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ∈ 𝑇)
168	5, 167	eqeltrd 2838	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087  ∀wal 1539   = wceq 1541   ∈ wcel 2106  {cab 2713   ≠ wne 2943  ∀wral 3064  ∃wrex 3073   ⊆ wss 3910  ∅c0 4282  {csn 4586   class class class wbr 5105   × cxp 5631  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  ℝcr 11050  -∞cmnf 11187  ℝ^*cxr 11188   ≤ cle 11190  ℕ₀cn0 12413  Basecbs 17083  +_gcplusg 17133  ._rcmulr 17134  0_gc0g 17321  Ringcrg 19964  LIdealclidl 20631  algSccascl 21258  Poly₁cpl1 21548  coe₁cco1 21549   deg₁ cdg1 25416  ldgIdlSeqcldgis 41434

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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-subrg 20220  df-lmod 20324  df-lss 20393  df-sra 20633  df-rgmod 20634  df-lidl 20635  df-cnfld 20797  df-ascl 21261  df-psr 21311  df-mvr 21312  df-mpl 21313  df-opsr 21315  df-psr1 21551  df-vr1 21552  df-ply1 21553  df-coe1 21554  df-mdeg 25417  df-deg1 25418  df-ldgis 41435

This theorem is referenced by:  hbtlem7  41438  hbtlem6  41442