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 2737	. . 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 2737	. . . . . . . . . . . 12 ⊢ (Base‘𝑃) = (Base‘𝑃)
7	6, 2	lidlss 21222	. . . . . . . . . . 11 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ (Base‘𝑃))
8	7	3ad2ant2 1135	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝐼 ⊆ (Base‘𝑃))
9	8	sselda 3983	. . . . . . . . 9 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → 𝑏 ∈ (Base‘𝑃))
10		eqid 2737	. . . . . . . . . 10 ⊢ (coe1‘𝑏) = (coe1‘𝑏)
11		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
12	10, 6, 1, 11	coe1f 22213	. . . . . . . . 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 7105	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) ∧ 𝑏 ∈ 𝐼) → ((coe1‘𝑏)‘𝑋) ∈ (Base‘𝑅))
16		eleq1a 2836	. . . . . . 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 3155	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋)) → 𝑎 ∈ (Base‘𝑅)))
20	19	abssdv 4068	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑎 ∣ ∃𝑏 ∈ 𝐼 (((deg1‘𝑅)‘𝑏) ≤ 𝑋 ∧ 𝑎 = ((coe1‘𝑏)‘𝑋))} ⊆ (Base‘𝑅))
21	1	ply1ring 22249	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
22	21	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑃 ∈ Ring)
23		simp2 1138	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝐼 ∈ 𝑈)
24		eqid 2737	. . . . . . . 8 ⊢ (0g‘𝑃) = (0g‘𝑃)
25	2, 24	lidl0cl 21230	. . . . . . 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 26126	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → ((deg1‘𝑅)‘(0g‘𝑃)) = -∞)
28	27	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((deg1‘𝑅)‘(0g‘𝑃)) = -∞)
29		nn0ssre 12530	. . . . . . . . . 10 ⊢ ℕ0 ⊆ ℝ
30		ressxr 11305	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ*
31	29, 30	sstri 3993	. . . . . . . . 9 ⊢ ℕ0 ⊆ ℝ*
32		simp3 1139	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
33	31, 32	sselid 3981	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈ ℝ*)
34		mnfle 13177	. . . . . . . 8 ⊢ (𝑋 ∈ ℝ* → -∞ ≤ 𝑋)
35	33, 34	syl 17	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → -∞ ≤ 𝑋)
36	28, 35	eqbrtrd 5165	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((deg1‘𝑅)‘(0g‘𝑃)) ≤ 𝑋)
37		eqid 2737	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
38	1, 24, 37	coe1z 22266	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
39	38	3ad2ant1 1134	. . . . . . . 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 7224	. . . . . . . 8 ⊢ (𝑋 ∈ ℕ0 → ((ℕ0 × {(0g‘𝑅)})‘𝑋) = (0g‘𝑅))
43	42	3ad2ant3 1136	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((ℕ0 × {(0g‘𝑅)})‘𝑋) = (0g‘𝑅))
44	40, 43	eqtr2d 2778	. . . . . 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 5153	. . . . . . . 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 2748	. . . . . . . 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 3622	. . . . . 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 2741	. . . . . . . 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 3179	. . . . . 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 3679	. . . . 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 4342	. . 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 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
62	1, 61, 11, 6	ply1sclf 22288	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑅 ∈ Ring → (algSc‘𝑃):(Base‘𝑅)⟶(Base‘𝑃))
63	62	3ad2ant1 1134	. . . . . . . . . . . . . . . . . . . . . . . . 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 7105	. . . . . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (.r‘𝑃) = (.r‘𝑃)
70	2, 6, 69	lidlmcl 21235	. . . . . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (+g‘𝑃) = (+g‘𝑃)
75	2, 74	lidlacl 21231	. . . . . . . . . . . . . . . . . . . . . 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 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 3984	. . . . . . . . . . . . . . . . . . . . . . 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 20247	. . . . . . . . . . . . . . . . . . . . . . 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 3984	. . . . . . . . . . . . . . . . . . . . . 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 3981	. . . . . . . . . . . . . . . . . . . . . 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 26121	. . . . . . . . . . . . . . . . . . . . . . . 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 26121	. . . . . . . . . . . . . . . . . . . . . . . 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 26156	. . . . . . . . . . . . . . . . . . . . . . . 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 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 13204	. . . . . . . . . . . . . . . . . . . . . 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 26141	. . . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (+g‘𝑅) = (+g‘𝑅)
98	1, 6, 74, 97	coe1addfv 22268	. . . . . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (.r‘𝑅) = (.r‘𝑅)
101	1, 6, 11, 61, 69, 100	coe1sclmulfv 22286	. . . . . . . . . . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . . . . . 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 5153	. . . . . . . . . . . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . . . . . . . . . . . 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 3622	. . . . . . . . . . . . . . . . . . . . 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 7464	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)((coe1‘𝑔)‘𝑋)) ∈ V
114		eqeq1 2741	. . . . . . . . . . . . . . . . . . . . . . 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 3179	. . . . . . . . . . . . . . . . . . . . 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 3679	. . . . . . . . . . . . . . . . . . . 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 7439	. . . . . . . . . . . . . . 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 2826	. . . . . . . . . . . . . 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 3155	. . . . . . . . . . 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 1927	. . . . . . . . . 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 2741	. . . . . . . . . . . . . 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 3179	. . . . . . . . . . . 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 5153	. . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . . 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 3238	. . . . . . . . . . . 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 3697	. . . . . . . . . 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 7439	. . . . . . . . . . . 12 ⊢ (𝑑 = ((coe1‘𝑓)‘𝑋) → (𝑐(.r‘𝑅)𝑑) = (𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋)))
144	143	oveq1d 7446	. . . . . . . . . . 11 ⊢ (𝑑 = ((coe1‘𝑓)‘𝑋) → ((𝑐(.r‘𝑅)𝑑)(+g‘𝑅)𝑒) = ((𝑐(.r‘𝑅)((coe1‘𝑓)‘𝑋))(+g‘𝑅)𝑒))
145	144	eleq1d 2826	. . . . . . . . . 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 3178	. . . . . . . . 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 3155	. . . . . 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 1927	. . . . 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 2741	. . . . . . . . 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 3179	. . . . . . 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 5153	. . . . . . . . 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 2748	. . . . . . . . 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 3238	. . . . . . 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 3697	. . . . 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 3146	. . 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 21225	. . 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 2841	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) ∈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wal 1538   = wceq 1540   ∈ wcel 2108  {cab 2714   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951  ∅c0 4333  {csn 4626   class class class wbr 5143   × cxp 5683  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  ℝcr 11154  -∞cmnf 11293  ℝ^*cxr 11294   ≤ cle 11296  ℕ₀cn0 12526  Basecbs 17247  +_gcplusg 17297  ._rcmulr 17298  0_gc0g 17484  Ringcrg 20230  LIdealclidl 21216  algSccascl 21872  Poly₁cpl1 22178  coe₁cco1 22179  deg₁cdg1 26093  ldgIdlSeqcldgis 43133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  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 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-cring 20233  df-subrng 20546  df-subrg 20570  df-lmod 20860  df-lss 20930  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-cnfld 21365  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184  df-mdeg 26094  df-deg1 26095  df-ldgis 43134

This theorem is referenced by:  hbtlem7  43137  hbtlem6  43141