Theorem hbtlem2 43799

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 2761	. . 3 ⊢ (deg1‘𝑅) = (deg1‘𝑅)
5	1, 2, 3, 4	hbtlem1 43798	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
6		eqid 2761	. . . . . . . . . . . 12 ⊢ (Base‘𝑃) = (Base‘𝑃)
7	6, 2	lidlss 21315	. . . . . . . . . . 11 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ (Base‘𝑃))
8	7	3ad2ant2 1150	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝐼 ⊆ (Base‘𝑃))
9	8	sselda 3936	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑃))
10		eqid 2761	. . . . . . . . . 10 ⊢ (coe1‘𝑏) = (coe1‘𝑏)
11		eqid 2761	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	10, 6, 1, 11	coe1f 22350	. . . . . . . . 9 ⊢ (𝑏 ∈ (Base‘𝑃) → (coe1‘𝑏):ℕ0⟶(Base‘𝑅))
13	9, 12	syl 18	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → (coe1‘𝑏):ℕ0⟶(Base‘𝑅))
14		simpl3 1210	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → 𝑋 ∈ ℕ0)
15	13, 14	ffvelcdmd 7080	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → ((coe1‘𝑏)‘𝑋) ∈ (Base‘𝑅))
16		eleq1a 2856	. . . . . . 7 ⊢ (((coe1‘𝑏)‘𝑋) ∈ (Base‘𝑅) → (𝑎 = ((coe1‘𝑏)‘𝑋) → 𝑎 ∈ (Base‘𝑅)))
17	15, 16	syl 18	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → (𝑎 = ((coe1‘𝑏)‘𝑋) → 𝑎 ∈ (Base‘𝑅)))
18	17	adantld 495	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → ((((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) → 𝑎 ∈ (Base‘𝑅)))
19	18	rexlimdva 3164	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) → 𝑎 ∈ (Base‘𝑅)))
20	19	abssdv 4020	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ⊆ (Base‘𝑅))
21	1	ply1ring 22386	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
22	21	3ad2ant1 1149	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑃 ∈ Ring)
23		simp2 1153	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝐼 ∈ 𝑈)
24		eqid 2761	. . . . . . . 8 ⊢ (0g‘𝑃) = (0g‘𝑃)
25	2, 24	lidl0cl 21324	. . . . . . 7 ⊢ ((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (0g‘𝑃) ∈ 𝐼)
26	22, 23, 25	syl2anc 595	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (0g‘𝑃) ∈ 𝐼)
27	4, 1, 24	deg1z 26223	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → ((deg1‘𝑅)‘(0g‘𝑃)) = -∞)
28	27	3ad2ant1 1149	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((deg1‘𝑅)‘(0g‘𝑃)) = -∞)
29		nn0ssre 12507	. . . . . . . . . 10 ⊢ ℕ0 ⊆ ℝ
30		ressxr 11252	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ*
31	29, 30	sstri 3945	. . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ*
32		simp3 1154	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
33	31, 32	sselid 3934	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈ ℝ*)
34		mnfle 13159	. . . . . . . 8 ⊢ (𝑋 ∈ ℝ* → -∞ ≤ 𝑋)
35	33, 34	syl 18	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → -∞ ≤ 𝑋)
36	28, 35	eqbrtrd 5132	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((deg1‘𝑅)‘(0g‘𝑃)) ≤ 𝑋)
37		eqid 2761	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
38	1, 24, 37	coe1z 22403	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
39	38	3ad2ant1 1149	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
40	39	fveq1d 6883	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((coe1‘(0g‘𝑃))‘𝑋) = ((ℕ0 × {(0g‘𝑅)})‘𝑋))
41		fvex 6894	. . . . . . . . 9 ⊢ (0g‘𝑅) ∈ V
42	41	fvconst2 7202	. . . . . . . 8 ⊢ (𝑋 ∈ ℕ0 → ((ℕ0 × {(0g‘𝑅)})‘𝑋) = (0g‘𝑅))
43	42	3ad2ant3 1151	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((ℕ0 × {(0g‘𝑅)})‘𝑋) = (0g‘𝑅))
44	40, 43	eqtr2d 2797	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (0g‘𝑅) = ((coe1‘(0g‘𝑃))‘𝑋))
45		fveq2 6881	. . . . . . . . 9 ⊢ (𝑏 = (0g‘𝑃) → ((deg1‘𝑅)‘𝑏) = ((deg1‘𝑅)‘(0g‘𝑃)))
46	45	breq1d 5118	. . . . . . . 8 ⊢ (𝑏 = (0g‘𝑃) → (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ↔ ((deg1‘𝑅)‘(0g‘𝑃)) ≤ 𝑋))
47		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑏 = (0g‘𝑃) → (coe1‘𝑏) = (coe1‘(0g‘𝑃)))
48	47	fveq1d 6883	. . . . . . . . 9 ⊢ (𝑏 = (0g‘𝑃) → ((coe1‘𝑏)‘𝑋) = ((coe1‘(0g‘𝑃))‘𝑋))
49	48	eqeq2d 2772	. . . . . . . 8 ⊢ (𝑏 = (0g‘𝑃) → ((0g‘𝑅) = ((coe1‘𝑏)‘𝑋) ↔ (0g‘𝑅) = ((coe1‘(0g‘𝑃))‘𝑋)))
50	46, 49	anbi12d 643	. . . . . . 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 3580	. . . . . 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 849	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ (0g‘𝑅) = ((coe1‘𝑏)‘𝑋)))
53		eqeq1 2765	. . . . . . . 8 ⊢ (𝑎 = (0g‘𝑅) → (𝑎 = ((coe1‘𝑏)‘𝑋) ↔ (0g‘𝑅) = ((coe1‘𝑏)‘𝑋)))
54	53	anbi2d 641	. . . . . . 7 ⊢ (𝑎 = (0g‘𝑅) → ((((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ (0g‘𝑅) = ((coe1‘𝑏)‘𝑋))))
55	54	rexbidv 3187	. . . . . 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 3637	. . . . 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 237	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (0g‘𝑅) ∈ {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
58	57	ne0d 4294	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ≠ ∅)
59	22	adantr 485	. . . . . . . . . . . . . . . . . . . . . 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 1209	. . . . . . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
62	1, 61, 11, 6	ply1sclf 22425	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑅 ∈ Ring → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
63	62	3ad2ant1 1149	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
64	63	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . . . . . . 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 7080	. . . . . . . . . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋))) → 𝑓 ∈ 𝐼)
68	67	adantl 486	. . . . . . . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (.r‘𝑃) = (.r‘𝑃)
70	2, 6, 69	lidlmcl 21329	. . . . . . . . . . . . . . . . . . . . . . 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 851	. . . . . . . . . . . . . . . . . . . . . 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 792	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋))) → 𝑔 ∈ 𝐼)
73	72	adantl 486	. . . . . . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (+g‘𝑃) = (+g‘𝑃)
75	2, 74	lidlacl 21325	. . . . . . . . . . . . . . . . . . . . . 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 851	. . . . . . . . . . . . . . . . . . . . 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 1208	. . . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . . . 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 3937	. . . . . . . . . . . . . . . . . . . . . . 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 20331	. . . . . . . . . . . . . . . . . . . . . . 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 1396	. . . . . . . . . . . . . . . . . . . . . 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 3937	. . . . . . . . . . . . . . . . . . . . . 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 1210	. . . . . . . . . . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . . . . . . . . . . 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 26218	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓) ∈ (Base‘𝑃) → ((deg1‘𝑅)‘(((algSc‘𝑃)‘𝑐)(.r‘𝑃)𝑓)) ∈ ℝ*)
86	81, 85	syl 18	. . . . . . . . . . . . . . . . . . . . . . 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 26218	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 ∈ (Base‘𝑃) → ((deg1‘𝑅)‘𝑓) ∈ ℝ*)
88	79, 87	syl 18	. . . . . . . . . . . . . . . . . . . . . . 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 26253	. . . . . . . . . . . . . . . . . . . . . . . 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 1396	. . . . . . . . . . . . . . . . . . . . . . 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 791	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋))) → ((deg1‘𝑅)‘𝑓) ≤ 𝑋)
92	91	adantl 486	. . . . . . . . . . . . . . . . . . . . . . 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 13186	. . . . . . . . . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑐 ∈ (Base‘𝑅) ∧ ((𝑓 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑓) ≤ 𝑋) ∧ (𝑔 ∈ 𝐼 ∧ ((deg1‘𝑅)‘𝑔) ≤ 𝑋))) → ((deg1‘𝑅)‘𝑔) ≤ 𝑋)
95	94	adantl 486	. . . . . . . . . . . . . . . . . . . . . 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 26238	. . . . . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (+g‘𝑅) = (+g‘𝑅)
98	1, 6, 74, 97	coe1addfv 22405	. . . . . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (.r‘𝑅) = (.r‘𝑅)
101	1, 6, 11, 61, 69, 100	coe1sclmulfv 22423	. . . . . . . . . . . . . . . . . . . . . . . 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 1400	. . . . . . . . . . . . . . . . . . . . . . 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 7425	. . . . . . . . . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . . . . . . . . . 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 5118	. . . . . . . . . . . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . . . . . . . . . . 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 6883	. . . . . . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . . . . . . . . . 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 3580	. . . . . . . . . . . . . . . . . . . . 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 849	. . . . . . . . . . . . . . . . . . . 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 7443	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ V
114		eqeq1 2765	. . . . . . . . . . . . . . . . . . . . . . 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 641	. . . . . . . . . . . . . . . . . . . . . 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 3187	. . . . . . . . . . . . . . . . . . . . 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 3637	. . . . . . . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . . . . . . 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 443	. . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . 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 449	. . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . 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 430	. . . . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . 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 2846	. . . . . . . . . . . . . 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 250	. . . . . . . . . . . . 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 458	. . . . . . . . . . . 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 3164	. . . . . . . . . . 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 1955	. . . . . . . . . 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 2765	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑒 → (𝑎 = ((coe1‘𝑏)‘𝑋) ↔ 𝑒 = ((coe1‘𝑏)‘𝑋)))
131	130	anbi2d 641	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑒 → ((((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑏)‘𝑋))))
132	131	rexbidv 3187	. . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑔 → ((deg1‘𝑅)‘𝑏) = ((deg1‘𝑅)‘𝑔))
134	133	breq1d 5118	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑔 → (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ↔ ((deg1‘𝑅)‘𝑔) ≤ 𝑋))
135		fveq2 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑔 → (coe1‘𝑏) = (coe1‘𝑔))
136	135	fveq1d 6883	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑔 → ((coe1‘𝑏)‘𝑋) = ((coe1‘𝑔)‘𝑋))
137	136	eqeq2d 2772	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑔 → (𝑒 = ((coe1‘𝑏)‘𝑋) ↔ 𝑒 = ((coe1‘𝑔)‘𝑋)))
138	134, 137	anbi12d 643	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑔 → ((((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑏)‘𝑋)) ↔ (((deg1‘𝑅)‘𝑔) ≤ 𝑋 ∧ 𝑒 = ((coe1‘𝑔)‘𝑋))))
139	138	cbvrexvw 3242	. . . . . . . . . . . 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 290	. . . . . . . . . . 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 3655	. . . . . . . . . 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 237	. . . . . . . . 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 7418	. . . . . . . . . . . 12 ⊢ (𝑑 = ((coe1‘𝑓)‘𝑋) → (𝑐(.r‘𝑅)𝑑) = (𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋)))
144	143	oveq1d 7425	. . . . . . . . . . 11 ⊢ (𝑑 = ((coe1‘𝑓)‘𝑋) → ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) = ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒))
145	144	eleq1d 2846	. . . . . . . . . 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 3186	. . . . . . . . 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 250	. . . . . . . 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 458	. . . . . . 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 3164	. . . . . 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 1955	. . . . 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 2765	. . . . . . . . 9 ⊢ (𝑎 = 𝑑 → (𝑎 = ((coe1‘𝑏)‘𝑋) ↔ 𝑑 = ((coe1‘𝑏)‘𝑋)))
152	151	anbi2d 641	. . . . . . . 8 ⊢ (𝑎 = 𝑑 → ((((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) ↔ (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋))))
153	152	rexbidv 3187	. . . . . . 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 6881	. . . . . . . . . 10 ⊢ (𝑏 = 𝑓 → ((deg1‘𝑅)‘𝑏) = ((deg1‘𝑅)‘𝑓))
155	154	breq1d 5118	. . . . . . . . 9 ⊢ (𝑏 = 𝑓 → (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ↔ ((deg1‘𝑅)‘𝑓) ≤ 𝑋))
156		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑓 → (coe1‘𝑏) = (coe1‘𝑓))
157	156	fveq1d 6883	. . . . . . . . . 10 ⊢ (𝑏 = 𝑓 → ((coe1‘𝑏)‘𝑋) = ((coe1‘𝑓)‘𝑋))
158	157	eqeq2d 2772	. . . . . . . . 9 ⊢ (𝑏 = 𝑓 → (𝑑 = ((coe1‘𝑏)‘𝑋) ↔ 𝑑 = ((coe1‘𝑓)‘𝑋)))
159	155, 158	anbi12d 643	. . . . . . . 8 ⊢ (𝑏 = 𝑓 → ((((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑏)‘𝑋)) ↔ (((deg1‘𝑅)‘𝑓) ≤ 𝑋 ∧ 𝑑 = ((coe1‘𝑓)‘𝑋))))
160	159	cbvrexvw 3242	. . . . . . 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 290	. . . . . 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 3655	. . . . 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 237	. . . 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 3155	. . 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 21319	. . 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 1360	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ∈ 𝑇)
168	5, 167	eqeltrd 2861	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101  ∀wal 1566   = wceq 1568   ∈ wcel 2141  {cab 2739   ≠ wne 2956  ∀wral 3077  ∃wrex 3087   ⊆ wss 3904  ∅c0 4285  {csn 4588   class class class wbr 5108   × cxp 5659  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ℝcr 11098  -∞cmnf 11240  ℝ^*cxr 11241   ≤ cle 11243  ℕ₀cn0 12503  Basecbs 17268  +_gcplusg 17309  ._rcmulr 17310  0_gc0g 17491  Ringcrg 20314  LIdealclidl 21309  algSccascl 21981  Poly₁cpl1 22316  coe₁cco1 22317  deg₁cdg1 26190  ldgIdlSeqcldgis 43796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177  ax-addf 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  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 7674  df-ofr 7675  df-om 7862  df-1st 7985  df-2nd 7986  df-supp 8156  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-er 8693  df-map 8825  df-pm 8826  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fsupp 9321  df-sup 9401  df-oi 9471  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-z 12591  df-dec 12711  df-uz 12862  df-fz 13535  df-fzo 13682  df-seq 14037  df-hash 14366  df-struct 17206  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-plusg 17322  df-mulr 17323  df-starv 17324  df-sca 17325  df-vsca 17326  df-ip 17327  df-tset 17328  df-ple 17329  df-ds 17331  df-unif 17332  df-hom 17333  df-cco 17334  df-0g 17493  df-gsum 17494  df-prds 17499  df-pws 17501  df-mre 17637  df-mrc 17638  df-acs 17640  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-mhm 18840  df-submnd 18841  df-grp 19002  df-minusg 19003  df-sbg 19004  df-mulg 19133  df-subg 19188  df-ghm 19283  df-cntz 19386  df-cmn 19851  df-abl 19852  df-mgp 20216  df-rng 20230  df-ur 20263  df-ring 20316  df-cring 20317  df-subrng 20630  df-subrg 20654  df-lmod 20962  df-lss 21032  df-sra 21273  df-rgmod 21274  df-lidl 21311  df-cnfld 21502  df-ascl 21984  df-psr 22038  df-mvr 22039  df-mpl 22040  df-opsr 22042  df-psr1 22319  df-vr1 22320  df-ply1 22321  df-coe1 22322  df-mdeg 26191  df-deg1 26192  df-ldgis 43797

This theorem is referenced by:  hbtlem7  43800  hbtlem6  43804