Theorem hbtlem5 43140

Description: The leading ideal function is strictly monotone. (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)
hbtlem3.r	⊢ (𝜑 → 𝑅 ∈ Ring)
hbtlem3.i	⊢ (𝜑 → 𝐼 ∈ 𝑈)
hbtlem3.j	⊢ (𝜑 → 𝐽 ∈ 𝑈)
hbtlem3.ij	⊢ (𝜑 → 𝐼 ⊆ 𝐽)
hbtlem5.e	⊢ (𝜑 → ∀𝑥 ∈ ℕ0 ((𝑆‘𝐽)‘𝑥) ⊆ ((𝑆‘𝐼)‘𝑥))

Assertion

Ref	Expression
hbtlem5	⊢ (𝜑 → 𝐼 = 𝐽)

Distinct variable groups:   𝑥,𝐼   𝑥,𝐽   𝑥,𝑆

Allowed substitution hints:   𝜑(𝑥)   𝑃(𝑥)   𝑅(𝑥)   𝑈(𝑥)

Proof of Theorem hbtlem5

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

Step	Hyp	Ref	Expression
1		hbtlem3.ij	. 2 ⊢ (𝜑 → 𝐼 ⊆ 𝐽)
2		hbtlem3.j	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ 𝑈)
3		eqid 2730	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘𝑃)
4		hbtlem.u	. . . . . . . 8 ⊢ 𝑈 = (LIdeal‘𝑃)
5	3, 4	lidlss 21142	. . . . . . 7 ⊢ (𝐽 ∈ 𝑈 → 𝐽 ⊆ (Base‘𝑃))
6	2, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐽 ⊆ (Base‘𝑃))
7	6	sselda 3932	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → 𝑎 ∈ (Base‘𝑃))
8		eqid 2730	. . . . . 6 ⊢ (deg1‘𝑅) = (deg1‘𝑅)
9		hbtlem.p	. . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅)
10	8, 9, 3	deg1cl 26008	. . . . 5 ⊢ (𝑎 ∈ (Base‘𝑃) → ((deg1‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
11	7, 10	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → ((deg1‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
12		elun 4101	. . . . 5 ⊢ (((deg1‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}) ↔ (((deg1‘𝑅)‘𝑎) ∈ ℕ0 ∨ ((deg1‘𝑅)‘𝑎) ∈ {-∞}))
13		nnssnn0 12376	. . . . . . 7 ⊢ ℕ ⊆ ℕ0
14		nn0re 12382	. . . . . . . 8 ⊢ (((deg1‘𝑅)‘𝑎) ∈ ℕ0 → ((deg1‘𝑅)‘𝑎) ∈ ℝ)
15		arch 12370	. . . . . . . 8 ⊢ (((deg1‘𝑅)‘𝑎) ∈ ℝ → ∃𝑏 ∈ ℕ ((deg1‘𝑅)‘𝑎) < 𝑏)
16	14, 15	syl 17	. . . . . . 7 ⊢ (((deg1‘𝑅)‘𝑎) ∈ ℕ0 → ∃𝑏 ∈ ℕ ((deg1‘𝑅)‘𝑎) < 𝑏)
17		ssrexv 4002	. . . . . . 7 ⊢ (ℕ ⊆ ℕ0 → (∃𝑏 ∈ ℕ ((deg1‘𝑅)‘𝑎) < 𝑏 → ∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏))
18	13, 16, 17	mpsyl 68	. . . . . 6 ⊢ (((deg1‘𝑅)‘𝑎) ∈ ℕ0 → ∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏)
19		elsni 4591	. . . . . . 7 ⊢ (((deg1‘𝑅)‘𝑎) ∈ {-∞} → ((deg1‘𝑅)‘𝑎) = -∞)
20		0nn0 12388	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
21		mnflt0 13016	. . . . . . . . 9 ⊢ -∞ < 0
22		breq2 5093	. . . . . . . . . 10 ⊢ (𝑏 = 0 → (-∞ < 𝑏 ↔ -∞ < 0))
23	22	rspcev 3575	. . . . . . . . 9 ⊢ ((0 ∈ ℕ0 ∧ -∞ < 0) → ∃𝑏 ∈ ℕ0 -∞ < 𝑏)
24	20, 21, 23	mp2an 692	. . . . . . . 8 ⊢ ∃𝑏 ∈ ℕ0 -∞ < 𝑏
25		breq1 5092	. . . . . . . . 9 ⊢ (((deg1‘𝑅)‘𝑎) = -∞ → (((deg1‘𝑅)‘𝑎) < 𝑏 ↔ -∞ < 𝑏))
26	25	rexbidv 3154	. . . . . . . 8 ⊢ (((deg1‘𝑅)‘𝑎) = -∞ → (∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏 ↔ ∃𝑏 ∈ ℕ0 -∞ < 𝑏))
27	24, 26	mpbiri 258	. . . . . . 7 ⊢ (((deg1‘𝑅)‘𝑎) = -∞ → ∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏)
28	19, 27	syl 17	. . . . . 6 ⊢ (((deg1‘𝑅)‘𝑎) ∈ {-∞} → ∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏)
29	18, 28	jaoi 857	. . . . 5 ⊢ ((((deg1‘𝑅)‘𝑎) ∈ ℕ0 ∨ ((deg1‘𝑅)‘𝑎) ∈ {-∞}) → ∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏)
30	12, 29	sylbi 217	. . . 4 ⊢ (((deg1‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}) → ∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏)
31	11, 30	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → ∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏)
32		breq2 5093	. . . . . . . . . . 11 ⊢ (𝑐 = 0 → (((deg1‘𝑅)‘𝑎) < 𝑐 ↔ ((deg1‘𝑅)‘𝑎) < 0))
33	32	imbi1d 341	. . . . . . . . . 10 ⊢ (𝑐 = 0 → ((((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ (((deg1‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼)))
34	33	ralbidv 3153	. . . . . . . . 9 ⊢ (𝑐 = 0 → (∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼)))
35	34	imbi2d 340	. . . . . . . 8 ⊢ (𝑐 = 0 → ((𝜑 → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼)) ↔ (𝜑 → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼))))
36		breq2 5093	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑏 → (((deg1‘𝑅)‘𝑎) < 𝑐 ↔ ((deg1‘𝑅)‘𝑎) < 𝑏))
37	36	imbi1d 341	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → ((((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
38	37	ralbidv 3153	. . . . . . . . 9 ⊢ (𝑐 = 𝑏 → (∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
39	38	imbi2d 340	. . . . . . . 8 ⊢ (𝑐 = 𝑏 → ((𝜑 → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼)) ↔ (𝜑 → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))))
40		breq2 5093	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑏 + 1) → (((deg1‘𝑅)‘𝑎) < 𝑐 ↔ ((deg1‘𝑅)‘𝑎) < (𝑏 + 1)))
41	40	imbi1d 341	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑏 + 1) → ((((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ (((deg1‘𝑅)‘𝑎) < (𝑏 + 1) → 𝑎 ∈ 𝐼)))
42	41	ralbidv 3153	. . . . . . . . . 10 ⊢ (𝑐 = (𝑏 + 1) → (∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < (𝑏 + 1) → 𝑎 ∈ 𝐼)))
43		fveq2 6817	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑑 → ((deg1‘𝑅)‘𝑎) = ((deg1‘𝑅)‘𝑑))
44	43	breq1d 5099	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑑 → (((deg1‘𝑅)‘𝑎) < (𝑏 + 1) ↔ ((deg1‘𝑅)‘𝑑) < (𝑏 + 1)))
45		eleq1 2817	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑑 → (𝑎 ∈ 𝐼 ↔ 𝑑 ∈ 𝐼))
46	44, 45	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑑 → ((((deg1‘𝑅)‘𝑎) < (𝑏 + 1) → 𝑎 ∈ 𝐼) ↔ (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼)))
47	46	cbvralvw 3208	. . . . . . . . . 10 ⊢ (∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < (𝑏 + 1) → 𝑎 ∈ 𝐼) ↔ ∀𝑑 ∈ 𝐽 (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))
48	42, 47	bitrdi 287	. . . . . . . . 9 ⊢ (𝑐 = (𝑏 + 1) → (∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ∀𝑑 ∈ 𝐽 (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼)))
49	48	imbi2d 340	. . . . . . . 8 ⊢ (𝑐 = (𝑏 + 1) → ((𝜑 → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼)) ↔ (𝜑 → ∀𝑑 ∈ 𝐽 (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))))
50		hbtlem3.r	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Ring)
51	50	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → 𝑅 ∈ Ring)
52		eqid 2730	. . . . . . . . . . . 12 ⊢ (0g‘𝑃) = (0g‘𝑃)
53	8, 9, 52, 3	deg1lt0 26016	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑃)) → (((deg1‘𝑅)‘𝑎) < 0 ↔ 𝑎 = (0g‘𝑃)))
54	51, 7, 53	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (((deg1‘𝑅)‘𝑎) < 0 ↔ 𝑎 = (0g‘𝑃)))
55	9	ply1ring 22153	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
56	50, 55	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ Ring)
57		hbtlem3.i	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ 𝑈)
58	4, 52	lidl0cl 21150	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (0g‘𝑃) ∈ 𝐼)
59	56, 57, 58	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (0g‘𝑃) ∈ 𝐼)
60		eleq1a 2824	. . . . . . . . . . . 12 ⊢ ((0g‘𝑃) ∈ 𝐼 → (𝑎 = (0g‘𝑃) → 𝑎 ∈ 𝐼))
61	59, 60	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑎 = (0g‘𝑃) → 𝑎 ∈ 𝐼))
62	61	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (𝑎 = (0g‘𝑃) → 𝑎 ∈ 𝐼))
63	54, 62	sylbid 240	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (((deg1‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼))
64	63	ralrimiva 3122	. . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼))
65	6	3ad2ant2 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → 𝐽 ⊆ (Base‘𝑃))
66	65	sselda 3932	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → 𝑑 ∈ (Base‘𝑃))
67	8, 9, 3	deg1cl 26008	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ (Base‘𝑃) → ((deg1‘𝑅)‘𝑑) ∈ (ℕ0 ∪ {-∞}))
68	66, 67	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → ((deg1‘𝑅)‘𝑑) ∈ (ℕ0 ∪ {-∞}))
69		simpl1 1192	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → 𝑏 ∈ ℕ0)
70	69	nn0zd 12486	. . . . . . . . . . . . 13 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → 𝑏 ∈ ℤ)
71		degltp1le 25998	. . . . . . . . . . . . 13 ⊢ ((((deg1‘𝑅)‘𝑑) ∈ (ℕ0 ∪ {-∞}) ∧ 𝑏 ∈ ℤ) → (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) ↔ ((deg1‘𝑅)‘𝑑) ≤ 𝑏))
72	68, 70, 71	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) ↔ ((deg1‘𝑅)‘𝑑) ≤ 𝑏))
73		hbtlem5.e	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑥 ∈ ℕ0 ((𝑆‘𝐽)‘𝑥) ⊆ ((𝑆‘𝐼)‘𝑥))
74		fveq2 6817	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑏 → ((𝑆‘𝐽)‘𝑥) = ((𝑆‘𝐽)‘𝑏))
75		fveq2 6817	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑏 → ((𝑆‘𝐼)‘𝑥) = ((𝑆‘𝐼)‘𝑏))
76	74, 75	sseq12d 3966	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑏 → (((𝑆‘𝐽)‘𝑥) ⊆ ((𝑆‘𝐼)‘𝑥) ↔ ((𝑆‘𝐽)‘𝑏) ⊆ ((𝑆‘𝐼)‘𝑏)))
77	76	rspcva 3573	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 ((𝑆‘𝐽)‘𝑥) ⊆ ((𝑆‘𝐼)‘𝑥)) → ((𝑆‘𝐽)‘𝑏) ⊆ ((𝑆‘𝐼)‘𝑏))
78	73, 77	sylan2 593	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → ((𝑆‘𝐽)‘𝑏) ⊆ ((𝑆‘𝐼)‘𝑏))
79	50	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → 𝑅 ∈ Ring)
80	2	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → 𝐽 ∈ 𝑈)
81		simpl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → 𝑏 ∈ ℕ0)
82		hbtlem.s	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑆 = (ldgIdlSeq‘𝑅)
83	9, 4, 82, 8	hbtlem1 43135	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Ring ∧ 𝐽 ∈ 𝑈 ∧ 𝑏 ∈ ℕ0) → ((𝑆‘𝐽)‘𝑏) = {𝑐 ∣ ∃𝑒 ∈ 𝐽 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
84	79, 80, 81, 83	syl3anc 1373	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → ((𝑆‘𝐽)‘𝑏) = {𝑐 ∣ ∃𝑒 ∈ 𝐽 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
85	57	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → 𝐼 ∈ 𝑈)
86	9, 4, 82, 8	hbtlem1 43135	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝑏 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑏) = {𝑐 ∣ ∃𝑒 ∈ 𝐼 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
87	79, 85, 81, 86	syl3anc 1373	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → ((𝑆‘𝐼)‘𝑏) = {𝑐 ∣ ∃𝑒 ∈ 𝐼 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
88	78, 84, 87	3sstr3d 3987	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → {𝑐 ∣ ∃𝑒 ∈ 𝐽 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))} ⊆ {𝑐 ∣ ∃𝑒 ∈ 𝐼 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
89	88	3adant3 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → {𝑐 ∣ ∃𝑒 ∈ 𝐽 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))} ⊆ {𝑐 ∣ ∃𝑒 ∈ 𝐼 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
90	89	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) → {𝑐 ∣ ∃𝑒 ∈ 𝐽 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))} ⊆ {𝑐 ∣ ∃𝑒 ∈ 𝐼 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
91		simpl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏) → 𝑑 ∈ 𝐽)
92		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏) → ((deg1‘𝑅)‘𝑑) ≤ 𝑏)
93		eqidd 2731	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏) → ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑑)‘𝑏))
94		fveq2 6817	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑒 = 𝑑 → ((deg1‘𝑅)‘𝑒) = ((deg1‘𝑅)‘𝑑))
95	94	breq1d 5099	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = 𝑑 → (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ↔ ((deg1‘𝑅)‘𝑑) ≤ 𝑏))
96		fveq2 6817	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑒 = 𝑑 → (coe1‘𝑒) = (coe1‘𝑑))
97	96	fveq1d 6819	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑒 = 𝑑 → ((coe1‘𝑒)‘𝑏) = ((coe1‘𝑑)‘𝑏))
98	97	eqeq2d 2741	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = 𝑑 → (((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏) ↔ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑑)‘𝑏)))
99	95, 98	anbi12d 632	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝑑 → ((((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)) ↔ (((deg1‘𝑅)‘𝑑) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑑)‘𝑏))))
100	99	rspcev 3575	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ 𝐽 ∧ (((deg1‘𝑅)‘𝑑) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑑)‘𝑏))) → ∃𝑒 ∈ 𝐽 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))
101	91, 92, 93, 100	syl12anc 836	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏) → ∃𝑒 ∈ 𝐽 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))
102		fvex 6830	. . . . . . . . . . . . . . . . . 18 ⊢ ((coe1‘𝑑)‘𝑏) ∈ V
103		eqeq1 2734	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = ((coe1‘𝑑)‘𝑏) → (𝑐 = ((coe1‘𝑒)‘𝑏) ↔ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))
104	103	anbi2d 630	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = ((coe1‘𝑑)‘𝑏) → ((((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏)) ↔ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏))))
105	104	rexbidv 3154	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = ((coe1‘𝑑)‘𝑏) → (∃𝑒 ∈ 𝐽 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏)) ↔ ∃𝑒 ∈ 𝐽 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏))))
106	102, 105	elab 3633	. . . . . . . . . . . . . . . . 17 ⊢ (((coe1‘𝑑)‘𝑏) ∈ {𝑐 ∣ ∃𝑒 ∈ 𝐽 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))} ↔ ∃𝑒 ∈ 𝐽 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))
107	101, 106	sylibr 234	. . . . . . . . . . . . . . . 16 ⊢ ((𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏) → ((coe1‘𝑑)‘𝑏) ∈ {𝑐 ∣ ∃𝑒 ∈ 𝐽 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
108	107	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) → ((coe1‘𝑑)‘𝑏) ∈ {𝑐 ∣ ∃𝑒 ∈ 𝐽 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
109	90, 108	sseldd 3933	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) → ((coe1‘𝑑)‘𝑏) ∈ {𝑐 ∣ ∃𝑒 ∈ 𝐼 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
110	104	rexbidv 3154	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = ((coe1‘𝑑)‘𝑏) → (∃𝑒 ∈ 𝐼 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏)) ↔ ∃𝑒 ∈ 𝐼 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏))))
111	102, 110	elab 3633	. . . . . . . . . . . . . . 15 ⊢ (((coe1‘𝑑)‘𝑏) ∈ {𝑐 ∣ ∃𝑒 ∈ 𝐼 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))} ↔ ∃𝑒 ∈ 𝐼 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))
112		simpll2 1214	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → 𝜑)
113	112, 56	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → 𝑃 ∈ Ring)
114		ringgrp 20149	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp)
115	113, 114	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → 𝑃 ∈ Grp)
116	112, 6	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → 𝐽 ⊆ (Base‘𝑃))
117		simplrl 776	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → 𝑑 ∈ 𝐽)
118	116, 117	sseldd 3933	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → 𝑑 ∈ (Base‘𝑃))
119	3, 4	lidlss 21142	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ (Base‘𝑃))
120	57, 119	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐼 ⊆ (Base‘𝑃))
121	112, 120	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → 𝐼 ⊆ (Base‘𝑃))
122		simprl 770	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → 𝑒 ∈ 𝐼)
123	121, 122	sseldd 3933	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → 𝑒 ∈ (Base‘𝑃))
124		eqid 2730	. . . . . . . . . . . . . . . . . . 19 ⊢ (+g‘𝑃) = (+g‘𝑃)
125		eqid 2730	. . . . . . . . . . . . . . . . . . 19 ⊢ (-g‘𝑃) = (-g‘𝑃)
126	3, 124, 125	grpnpcan 18937	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ Grp ∧ 𝑑 ∈ (Base‘𝑃) ∧ 𝑒 ∈ (Base‘𝑃)) → ((𝑑(-g‘𝑃)𝑒)(+g‘𝑃)𝑒) = 𝑑)
127	115, 118, 123, 126	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → ((𝑑(-g‘𝑃)𝑒)(+g‘𝑃)𝑒) = 𝑑)
128	57	3ad2ant2 1134	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → 𝐼 ∈ 𝑈)
129	128	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → 𝐼 ∈ 𝑈)
130		simpll1 1213	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → 𝑏 ∈ ℕ0)
131	112, 50	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → 𝑅 ∈ Ring)
132		simplrr 777	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → ((deg1‘𝑅)‘𝑑) ≤ 𝑏)
133		simprrl 780	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → ((deg1‘𝑅)‘𝑒) ≤ 𝑏)
134		eqid 2730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (coe1‘𝑑) = (coe1‘𝑑)
135		eqid 2730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (coe1‘𝑒) = (coe1‘𝑒)
136		simprrr 781	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏))
137	8, 9, 3, 125, 130, 131, 118, 132, 123, 133, 134, 135, 136	deg1sublt 26035	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → ((deg1‘𝑅)‘(𝑑(-g‘𝑃)𝑒)) < 𝑏)
138	112, 2	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → 𝐽 ∈ 𝑈)
139	1	3ad2ant2 1134	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → 𝐼 ⊆ 𝐽)
140	139	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → 𝐼 ⊆ 𝐽)
141	140, 122	sseldd 3933	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → 𝑒 ∈ 𝐽)
142	4, 125	lidlsubcl 21154	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑃 ∈ Ring ∧ 𝐽 ∈ 𝑈) ∧ (𝑑 ∈ 𝐽 ∧ 𝑒 ∈ 𝐽)) → (𝑑(-g‘𝑃)𝑒) ∈ 𝐽)
143	113, 138, 117, 141, 142	syl22anc 838	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → (𝑑(-g‘𝑃)𝑒) ∈ 𝐽)
144		simpll3 1215	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))
145		fveq2 6817	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = (𝑑(-g‘𝑃)𝑒) → ((deg1‘𝑅)‘𝑎) = ((deg1‘𝑅)‘(𝑑(-g‘𝑃)𝑒)))
146	145	breq1d 5099	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = (𝑑(-g‘𝑃)𝑒) → (((deg1‘𝑅)‘𝑎) < 𝑏 ↔ ((deg1‘𝑅)‘(𝑑(-g‘𝑃)𝑒)) < 𝑏))
147		eleq1 2817	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = (𝑑(-g‘𝑃)𝑒) → (𝑎 ∈ 𝐼 ↔ (𝑑(-g‘𝑃)𝑒) ∈ 𝐼))
148	146, 147	imbi12d 344	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (𝑑(-g‘𝑃)𝑒) → ((((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼) ↔ (((deg1‘𝑅)‘(𝑑(-g‘𝑃)𝑒)) < 𝑏 → (𝑑(-g‘𝑃)𝑒) ∈ 𝐼)))
149	148	rspcva 3573	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑑(-g‘𝑃)𝑒) ∈ 𝐽 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → (((deg1‘𝑅)‘(𝑑(-g‘𝑃)𝑒)) < 𝑏 → (𝑑(-g‘𝑃)𝑒) ∈ 𝐼))
150	143, 144, 149	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → (((deg1‘𝑅)‘(𝑑(-g‘𝑃)𝑒)) < 𝑏 → (𝑑(-g‘𝑃)𝑒) ∈ 𝐼))
151	137, 150	mpd 15	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → (𝑑(-g‘𝑃)𝑒) ∈ 𝐼)
152	4, 124	lidlacl 21151	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ ((𝑑(-g‘𝑃)𝑒) ∈ 𝐼 ∧ 𝑒 ∈ 𝐼)) → ((𝑑(-g‘𝑃)𝑒)(+g‘𝑃)𝑒) ∈ 𝐼)
153	113, 129, 151, 122, 152	syl22anc 838	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → ((𝑑(-g‘𝑃)𝑒)(+g‘𝑃)𝑒) ∈ 𝐼)
154	127, 153	eqeltrrd 2830	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) ∧ (𝑒 ∈ 𝐼 ∧ (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))) → 𝑑 ∈ 𝐼)
155	154	rexlimdvaa 3132	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) → (∃𝑒 ∈ 𝐼 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)) → 𝑑 ∈ 𝐼))
156	111, 155	biimtrid 242	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) → (((coe1‘𝑑)‘𝑏) ∈ {𝑐 ∣ ∃𝑒 ∈ 𝐼 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))} → 𝑑 ∈ 𝐼))
157	109, 156	mpd 15	. . . . . . . . . . . . 13 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) → 𝑑 ∈ 𝐼)
158	157	expr 456	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → (((deg1‘𝑅)‘𝑑) ≤ 𝑏 → 𝑑 ∈ 𝐼))
159	72, 158	sylbid 240	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))
160	159	ralrimiva 3122	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → ∀𝑑 ∈ 𝐽 (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))
161	160	3exp 1119	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ0 → (𝜑 → (∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼) → ∀𝑑 ∈ 𝐽 (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))))
162	161	a2d 29	. . . . . . . 8 ⊢ (𝑏 ∈ ℕ0 → ((𝜑 → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → (𝜑 → ∀𝑑 ∈ 𝐽 (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))))
163	35, 39, 49, 39, 64, 162	nn0ind 12560	. . . . . . 7 ⊢ (𝑏 ∈ ℕ0 → (𝜑 → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
164		rsp 3218	. . . . . . 7 ⊢ (∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼) → (𝑎 ∈ 𝐽 → (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
165	163, 164	syl6com 37	. . . . . 6 ⊢ (𝜑 → (𝑏 ∈ ℕ0 → (𝑎 ∈ 𝐽 → (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))))
166	165	com23 86	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐽 → (𝑏 ∈ ℕ0 → (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))))
167	166	imp 406	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (𝑏 ∈ ℕ0 → (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
168	167	rexlimdv 3129	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))
169	31, 168	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → 𝑎 ∈ 𝐼)
170	1, 169	eqelssd 3954	1 ⊢ (𝜑 → 𝐼 = 𝐽)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110  {cab 2708  ∀wral 3045  ∃wrex 3054   ∪ cun 3898   ⊆ wss 3900  {csn 4574   class class class wbr 5089  ‘cfv 6477  (class class class)co 7341  ℝcr 10997  0cc0 10998  1c1 10999   + caddc 11001  -∞cmnf 11136   < clt 11138   ≤ cle 11139  ℕcn 12117  ℕ₀cn0 12373  ℤcz 12460  Basecbs 17112  +_gcplusg 17153  0_gc0g 17335  Grpcgrp 18838  -_gcsg 18840  Ringcrg 20144  LIdealclidl 21136  Poly₁cpl1 22082  coe₁cco1 22083  deg₁cdg1 25979  ldgIdlSeqcldgis 43133

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-isom 6486  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-ofr 7606  df-om 7792  df-1st 7916  df-2nd 7917  df-supp 8086  df-tpos 8151  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-er 8617  df-map 8747  df-pm 8748  df-ixp 8817  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-fsupp 9241  df-sup 9321  df-oi 9391  df-card 9824  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-2 12180  df-3 12181  df-4 12182  df-5 12183  df-6 12184  df-7 12185  df-8 12186  df-9 12187  df-n0 12374  df-z 12461  df-dec 12581  df-uz 12725  df-fz 13400  df-fzo 13547  df-seq 13901  df-hash 14230  df-struct 17050  df-sets 17067  df-slot 17085  df-ndx 17097  df-base 17113  df-ress 17134  df-plusg 17166  df-mulr 17167  df-starv 17168  df-sca 17169  df-vsca 17170  df-ip 17171  df-tset 17172  df-ple 17173  df-ds 17175  df-unif 17176  df-hom 17177  df-cco 17178  df-0g 17337  df-gsum 17338  df-prds 17343  df-pws 17345  df-mre 17480  df-mrc 17481  df-acs 17483  df-mgm 18540  df-sgrp 18619  df-mnd 18635  df-mhm 18683  df-submnd 18684  df-grp 18841  df-minusg 18842  df-sbg 18843  df-mulg 18973  df-subg 19028  df-ghm 19118  df-cntz 19222  df-cmn 19687  df-abl 19688  df-mgp 20052  df-rng 20064  df-ur 20093  df-ring 20146  df-cring 20147  df-oppr 20248  df-dvdsr 20268  df-unit 20269  df-invr 20299  df-subrng 20454  df-subrg 20478  df-rlreg 20602  df-lmod 20788  df-lss 20858  df-sra 21100  df-rgmod 21101  df-lidl 21138  df-cnfld 21285  df-psr 21839  df-mpl 21841  df-opsr 21843  df-psr1 22085  df-ply1 22087  df-coe1 22088  df-mdeg 25980  df-deg1 25981  df-ldgis 43134

This theorem is referenced by:  hbt  43142