Theorem hbtlem2 43112

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 2734	. . 3 ⊢ (deg1‘𝑅) = (deg1‘𝑅)
5	1, 2, 3, 4	hbtlem1 43111	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) = {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))})
6		eqid 2734	. . . . . . . . . . . 12 ⊢ (Base‘𝑃) = (Base‘𝑃)
7	6, 2	lidlss 21239	. . . . . . . . . . 11 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ (Base‘𝑃))
8	7	3ad2ant2 1133	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝐼 ⊆ (Base‘𝑃))
9	8	sselda 3994	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑃))
10		eqid 2734	. . . . . . . . . 10 ⊢ (coe1‘𝑏) = (coe1‘𝑏)
11		eqid 2734	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	10, 6, 1, 11	coe1f 22228	. . . . . . . . 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 1192	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → 𝑋 ∈ ℕ0)
15	13, 14	ffvelcdmd 7104	. . . . . . 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 490	. . . . 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 4077	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ⊆ (Base‘𝑅))
21	1	ply1ring 22264	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
22	21	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑃 ∈ Ring)
23		simp2 1136	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝐼 ∈ 𝑈)
24		eqid 2734	. . . . . . . 8 ⊢ (0g‘𝑃) = (0g‘𝑃)
25	2, 24	lidl0cl 21247	. . . . . . 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 26140	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → ((deg1‘𝑅)‘(0g‘𝑃)) = -∞)
28	27	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((deg1‘𝑅)‘(0g‘𝑃)) = -∞)
29		nn0ssre 12527	. . . . . . . . . 10 ⊢ ℕ0 ⊆ ℝ
30		ressxr 11302	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ*
31	29, 30	sstri 4004	. . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ*
32		simp3 1137	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
33	31, 32	sselid 3992	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈ ℝ*)
34		mnfle 13173	. . . . . . . 8 ⊢ (𝑋 ∈ ℝ* → -∞ ≤ 𝑋)
35	33, 34	syl 17	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → -∞ ≤ 𝑋)
36	28, 35	eqbrtrd 5169	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((deg1‘𝑅)‘(0g‘𝑃)) ≤ 𝑋)
37		eqid 2734	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
38	1, 24, 37	coe1z 22281	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
39	38	3ad2ant1 1132	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
40	39	fveq1d 6908	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((coe1‘(0g‘𝑃))‘𝑋) = ((ℕ0 × {(0g‘𝑅)})‘𝑋))
41		fvex 6919	. . . . . . . . 9 ⊢ (0g‘𝑅) ∈ V
42	41	fvconst2 7223	. . . . . . . 8 ⊢ (𝑋 ∈ ℕ0 → ((ℕ0 × {(0g‘𝑅)})‘𝑋) = (0g‘𝑅))
43	42	3ad2ant3 1134	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((ℕ0 × {(0g‘𝑅)})‘𝑋) = (0g‘𝑅))
44	40, 43	eqtr2d 2775	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (0g‘𝑅) = ((coe1‘(0g‘𝑃))‘𝑋))
45		fveq2 6906	. . . . . . . . 9 ⊢ (𝑏 = (0g‘𝑃) → ((deg1‘𝑅)‘𝑏) = ((deg1‘𝑅)‘(0g‘𝑃)))
46	45	breq1d 5157	. . . . . . . 8 ⊢ (𝑏 = (0g‘𝑃) → (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ↔ ((deg1‘𝑅)‘(0g‘𝑃)) ≤ 𝑋))
47		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑏 = (0g‘𝑃) → (coe1‘𝑏) = (coe1‘(0g‘𝑃)))
48	47	fveq1d 6908	. . . . . . . . 9 ⊢ (𝑏 = (0g‘𝑃) → ((coe1‘𝑏)‘𝑋) = ((coe1‘(0g‘𝑃))‘𝑋))
49	48	eqeq2d 2745	. . . . . . . 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 3621	. . . . . 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 837	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ (0g‘𝑅) = ((coe1‘𝑏)‘𝑋)))
53		eqeq1 2738	. . . . . . . 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 3176	. . . . . 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 3680	. . . . 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 4347	. . 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 1191	. . . . . . . . . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
62	1, 61, 11, 6	ply1sclf 22303	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑅 ∈ Ring → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
63	62	3ad2ant1 1132	. . . . . . . . . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . . . . . . . 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 7104	. . . . . . . . . . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (.r‘𝑃) = (.r‘𝑃)
70	2, 6, 69	lidlmcl 21252	. . . . . . . . . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (+g‘𝑃) = (+g‘𝑃)
75	2, 74	lidlacl 21248	. . . . . . . . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . . . . . . . . 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 1190	. . . . . . . . . . . . . . . . . . . . . 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 3995	. . . . . . . . . . . . . . . . . . . . . . 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 20267	. . . . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . . . 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 3995	. . . . . . . . . . . . . . . . . . . . . 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 1192	. . . . . . . . . . . . . . . . . . . . . . 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 3992	. . . . . . . . . . . . . . . . . . . . . 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 26135	. . . . . . . . . . . . . . . . . . . . . . . 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 26135	. . . . . . . . . . . . . . . . . . . . . . . 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 26170	. . . . . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . . . . . 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 13200	. . . . . . . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . . . . . . . . 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 26155	. . . . . . . . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (+g‘𝑅) = (+g‘𝑅)
98	1, 6, 74, 97	coe1addfv 22283	. . . . . . . . . . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (.r‘𝑅) = (.r‘𝑅)
101	1, 6, 11, 61, 69, 100	coe1sclmulfv 22301	. . . . . . . . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . . . . . . . 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 7445	. . . . . . . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . . . . . . . . . . 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 5157	. . . . . . . . . . . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . . . . . . . . . . . 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 6908	. . . . . . . . . . . . . . . . . . . . . . . 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 2745	. . . . . . . . . . . . . . . . . . . . . . 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 3621	. . . . . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . . . . 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 7463	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ V
114		eqeq1 2738	. . . . . . . . . . . . . . . . . . . . . . 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 3176	. . . . . . . . . . . . . . . . . . . . 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 3680	. . . . . . . . . . . . . . . . . . . 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 7438	. . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . 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 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 1924	. . . . . . . . . 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 2738	. . . . . . . . . . . . . 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 3176	. . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑔 → ((deg1‘𝑅)‘𝑏) = ((deg1‘𝑅)‘𝑔))
134	133	breq1d 5157	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑔 → (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ↔ ((deg1‘𝑅)‘𝑔) ≤ 𝑋))
135		fveq2 6906	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑔 → (coe1‘𝑏) = (coe1‘𝑔))
136	135	fveq1d 6908	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑔 → ((coe1‘𝑏)‘𝑋) = ((coe1‘𝑔)‘𝑋))
137	136	eqeq2d 2745	. . . . . . . . . . . . . 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 3235	. . . . . . . . . . . 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 3699	. . . . . . . . . 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 7438	. . . . . . . . . . . 12 ⊢ (𝑑 = ((coe1‘𝑓)‘𝑋) → (𝑐(.r‘𝑅)𝑑) = (𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋)))
144	143	oveq1d 7445	. . . . . . . . . . 11 ⊢ (𝑑 = ((coe1‘𝑓)‘𝑋) → ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) = ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒))
145	144	eleq1d 2823	. . . . . . . . . 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 3175	. . . . . . . . 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 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 1924	. . . . 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 2738	. . . . . . . . 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 3176	. . . . . . 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 6906	. . . . . . . . . 10 ⊢ (𝑏 = 𝑓 → ((deg1‘𝑅)‘𝑏) = ((deg1‘𝑅)‘𝑓))
155	154	breq1d 5157	. . . . . . . . 9 ⊢ (𝑏 = 𝑓 → (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ↔ ((deg1‘𝑅)‘𝑓) ≤ 𝑋))
156		fveq2 6906	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑓 → (coe1‘𝑏) = (coe1‘𝑓))
157	156	fveq1d 6908	. . . . . . . . . 10 ⊢ (𝑏 = 𝑓 → ((coe1‘𝑏)‘𝑋) = ((coe1‘𝑓)‘𝑋))
158	157	eqeq2d 2745	. . . . . . . . 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 3235	. . . . . . 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 3699	. . . . 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 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 21242	. . 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 1342	. 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 395   ∧ w3a 1086  ∀wal 1534   = wceq 1536   ∈ wcel 2105  {cab 2711   ≠ wne 2937  ∀wral 3058  ∃wrex 3067   ⊆ wss 3962  ∅c0 4338  {csn 4630   class class class wbr 5147   × cxp 5686  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430  ℝcr 11151  -∞cmnf 11290  ℝ^*cxr 11291   ≤ cle 11293  ℕ₀cn0 12523  Basecbs 17244  +_gcplusg 17297  ._rcmulr 17298  0_gc0g 17485  Ringcrg 20250  LIdealclidl 21233  algSccascl 21889  Poly₁cpl1 22193  coe₁cco1 22194  deg₁cdg1 26107  ldgIdlSeqcldgis 43109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-mulg 19098  df-subg 19153  df-ghm 19243  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-cring 20253  df-subrng 20562  df-subrg 20586  df-lmod 20876  df-lss 20947  df-sra 21189  df-rgmod 21190  df-lidl 21235  df-cnfld 21382  df-ascl 21892  df-psr 21946  df-mvr 21947  df-mpl 21948  df-opsr 21950  df-psr1 22196  df-vr1 22197  df-ply1 22198  df-coe1 22199  df-mdeg 26108  df-deg1 26109  df-ldgis 43110

This theorem is referenced by:  hbtlem7  43113  hbtlem6  43117