Theorem hbtlem5 36717

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	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ 𝑈)
3		eqid 2610	. . . . . . . . . 10 ⊢ (Base‘𝑃) = (Base‘𝑃)
4		hbtlem.u	. . . . . . . . . 10 ⊢ 𝑈 = (LIdeal‘𝑃)
5	3, 4	lidlss 19031	. . . . . . . . 9 ⊢ (𝐽 ∈ 𝑈 → 𝐽 ⊆ (Base‘𝑃))
6	2, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ⊆ (Base‘𝑃))
7	6	sselda 3568	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → 𝑎 ∈ (Base‘𝑃))
8		eqid 2610	. . . . . . . 8 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅)
9		hbtlem.p	. . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅)
10	8, 9, 3	deg1cl 23647	. . . . . . 7 ⊢ (𝑎 ∈ (Base‘𝑃) → (( deg1 ‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
11	7, 10	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (( deg1 ‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
12		elun 3715	. . . . . . 7 ⊢ ((( deg1 ‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}) ↔ ((( deg1 ‘𝑅)‘𝑎) ∈ ℕ0 ∨ (( deg1 ‘𝑅)‘𝑎) ∈ {-∞}))
13		nnssnn0 11172	. . . . . . . . 9 ⊢ ℕ ⊆ ℕ0
14		nn0re 11178	. . . . . . . . . 10 ⊢ ((( deg1 ‘𝑅)‘𝑎) ∈ ℕ0 → (( deg1 ‘𝑅)‘𝑎) ∈ ℝ)
15		arch 11166	. . . . . . . . . 10 ⊢ ((( deg1 ‘𝑅)‘𝑎) ∈ ℝ → ∃𝑏 ∈ ℕ (( deg1 ‘𝑅)‘𝑎) < 𝑏)
16	14, 15	syl 17	. . . . . . . . 9 ⊢ ((( deg1 ‘𝑅)‘𝑎) ∈ ℕ0 → ∃𝑏 ∈ ℕ (( deg1 ‘𝑅)‘𝑎) < 𝑏)
17		ssrexv 3630	. . . . . . . . 9 ⊢ (ℕ ⊆ ℕ0 → (∃𝑏 ∈ ℕ (( deg1 ‘𝑅)‘𝑎) < 𝑏 → ∃𝑏 ∈ ℕ0 (( deg1 ‘𝑅)‘𝑎) < 𝑏))
18	13, 16, 17	mpsyl 66	. . . . . . . 8 ⊢ ((( deg1 ‘𝑅)‘𝑎) ∈ ℕ0 → ∃𝑏 ∈ ℕ0 (( deg1 ‘𝑅)‘𝑎) < 𝑏)
19		elsni 4142	. . . . . . . . 9 ⊢ ((( deg1 ‘𝑅)‘𝑎) ∈ {-∞} → (( deg1 ‘𝑅)‘𝑎) = -∞)
20		0nn0 11184	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ0
21		mnflt0 11835	. . . . . . . . . . 11 ⊢ -∞ < 0
22		breq2 4587	. . . . . . . . . . . 12 ⊢ (𝑏 = 0 → (-∞ < 𝑏 ↔ -∞ < 0))
23	22	rspcev 3282	. . . . . . . . . . 11 ⊢ ((0 ∈ ℕ0 ∧ -∞ < 0) → ∃𝑏 ∈ ℕ0 -∞ < 𝑏)
24	20, 21, 23	mp2an 704	. . . . . . . . . 10 ⊢ ∃𝑏 ∈ ℕ0 -∞ < 𝑏
25		breq1 4586	. . . . . . . . . . 11 ⊢ ((( deg1 ‘𝑅)‘𝑎) = -∞ → ((( deg1 ‘𝑅)‘𝑎) < 𝑏 ↔ -∞ < 𝑏))
26	25	rexbidv 3034	. . . . . . . . . 10 ⊢ ((( deg1 ‘𝑅)‘𝑎) = -∞ → (∃𝑏 ∈ ℕ0 (( deg1 ‘𝑅)‘𝑎) < 𝑏 ↔ ∃𝑏 ∈ ℕ0 -∞ < 𝑏))
27	24, 26	mpbiri 247	. . . . . . . . 9 ⊢ ((( deg1 ‘𝑅)‘𝑎) = -∞ → ∃𝑏 ∈ ℕ0 (( deg1 ‘𝑅)‘𝑎) < 𝑏)
28	19, 27	syl 17	. . . . . . . 8 ⊢ ((( deg1 ‘𝑅)‘𝑎) ∈ {-∞} → ∃𝑏 ∈ ℕ0 (( deg1 ‘𝑅)‘𝑎) < 𝑏)
29	18, 28	jaoi 393	. . . . . . 7 ⊢ (((( deg1 ‘𝑅)‘𝑎) ∈ ℕ0 ∨ (( deg1 ‘𝑅)‘𝑎) ∈ {-∞}) → ∃𝑏 ∈ ℕ0 (( deg1 ‘𝑅)‘𝑎) < 𝑏)
30	12, 29	sylbi 206	. . . . . 6 ⊢ ((( deg1 ‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}) → ∃𝑏 ∈ ℕ0 (( deg1 ‘𝑅)‘𝑎) < 𝑏)
31	11, 30	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → ∃𝑏 ∈ ℕ0 (( deg1 ‘𝑅)‘𝑎) < 𝑏)
32		breq2 4587	. . . . . . . . . . . . 13 ⊢ (𝑐 = 0 → ((( deg1 ‘𝑅)‘𝑎) < 𝑐 ↔ (( deg1 ‘𝑅)‘𝑎) < 0))
33	32	imbi1d 330	. . . . . . . . . . . 12 ⊢ (𝑐 = 0 → (((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ((( deg1 ‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼)))
34	33	ralbidv 2969	. . . . . . . . . . 11 ⊢ (𝑐 = 0 → (∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼)))
35	34	imbi2d 329	. . . . . . . . . 10 ⊢ (𝑐 = 0 → ((𝜑 → ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼)) ↔ (𝜑 → ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼))))
36		breq2 4587	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → ((( deg1 ‘𝑅)‘𝑎) < 𝑐 ↔ (( deg1 ‘𝑅)‘𝑎) < 𝑏))
37	36	imbi1d 330	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑏 → (((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
38	37	ralbidv 2969	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑏 → (∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
39	38	imbi2d 329	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → ((𝜑 → ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼)) ↔ (𝜑 → ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))))
40		breq2 4587	. . . . . . . . . . . . . 14 ⊢ (𝑐 = (𝑏 + 1) → ((( deg1 ‘𝑅)‘𝑎) < 𝑐 ↔ (( deg1 ‘𝑅)‘𝑎) < (𝑏 + 1)))
41	40	imbi1d 330	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑏 + 1) → (((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ((( deg1 ‘𝑅)‘𝑎) < (𝑏 + 1) → 𝑎 ∈ 𝐼)))
42	41	ralbidv 2969	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑏 + 1) → (∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < (𝑏 + 1) → 𝑎 ∈ 𝐼)))
43		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑑 → (( deg1 ‘𝑅)‘𝑎) = (( deg1 ‘𝑅)‘𝑑))
44	43	breq1d 4593	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑑 → ((( deg1 ‘𝑅)‘𝑎) < (𝑏 + 1) ↔ (( deg1 ‘𝑅)‘𝑑) < (𝑏 + 1)))
45		eleq1 2676	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑑 → (𝑎 ∈ 𝐼 ↔ 𝑑 ∈ 𝐼))
46	44, 45	imbi12d 333	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑑 → (((( deg1 ‘𝑅)‘𝑎) < (𝑏 + 1) → 𝑎 ∈ 𝐼) ↔ ((( deg1 ‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼)))
47	46	cbvralv 3147	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < (𝑏 + 1) → 𝑎 ∈ 𝐼) ↔ ∀𝑑 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))
48	42, 47	syl6bb 275	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑏 + 1) → (∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ∀𝑑 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼)))
49	48	imbi2d 329	. . . . . . . . . 10 ⊢ (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	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ Ring)
51	50	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → 𝑅 ∈ Ring)
52		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑃) = (0g‘𝑃)
53	8, 9, 52, 3	deg1lt0 23655	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑃)) → ((( deg1 ‘𝑅)‘𝑎) < 0 ↔ 𝑎 = (0g‘𝑃)))
54	51, 7, 53	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → ((( deg1 ‘𝑅)‘𝑎) < 0 ↔ 𝑎 = (0g‘𝑃)))
55	9	ply1ring 19439	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
56	50, 55	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑃 ∈ Ring)
57		hbtlem3.i	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 ∈ 𝑈)
58	4, 52	lidl0cl 19033	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (0g‘𝑃) ∈ 𝐼)
59	56, 57, 58	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝑃) ∈ 𝐼)
60		eleq1a 2683	. . . . . . . . . . . . . 14 ⊢ ((0g‘𝑃) ∈ 𝐼 → (𝑎 = (0g‘𝑃) → 𝑎 ∈ 𝐼))
61	59, 60	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑎 = (0g‘𝑃) → 𝑎 ∈ 𝐼))
62	61	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (𝑎 = (0g‘𝑃) → 𝑎 ∈ 𝐼))
63	54, 62	sylbid 229	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → ((( deg1 ‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼))
64	63	ralrimiva 2949	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼))
65	6	3ad2ant2 1076	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → 𝐽 ⊆ (Base‘𝑃))
66	65	sselda 3568	. . . . . . . . . . . . . . . 16 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → 𝑑 ∈ (Base‘𝑃))
67	8, 9, 3	deg1cl 23647	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ (Base‘𝑃) → (( deg1 ‘𝑅)‘𝑑) ∈ (ℕ0 ∪ {-∞}))
68	66, 67	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → (( deg1 ‘𝑅)‘𝑑) ∈ (ℕ0 ∪ {-∞}))
69		simpl1 1057	. . . . . . . . . . . . . . . 16 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → 𝑏 ∈ ℕ0)
70	69	nn0zd 11356	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → 𝑏 ∈ ℤ)
71		degltp1le 23637	. . . . . . . . . . . . . . 15 ⊢ (((( deg1 ‘𝑅)‘𝑑) ∈ (ℕ0 ∪ {-∞}) ∧ 𝑏 ∈ ℤ) → ((( deg1 ‘𝑅)‘𝑑) < (𝑏 + 1) ↔ (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏))
72	68, 70, 71	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((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	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ∀𝑥 ∈ ℕ0 ((𝑆‘𝐽)‘𝑥) ⊆ ((𝑆‘𝐼)‘𝑥))
74		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑏 → ((𝑆‘𝐽)‘𝑥) = ((𝑆‘𝐽)‘𝑏))
75		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑏 → ((𝑆‘𝐼)‘𝑥) = ((𝑆‘𝐼)‘𝑏))
76	74, 75	sseq12d 3597	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑏 → (((𝑆‘𝐽)‘𝑥) ⊆ ((𝑆‘𝐼)‘𝑥) ↔ ((𝑆‘𝐽)‘𝑏) ⊆ ((𝑆‘𝐼)‘𝑏)))
77	76	rspcva 3280	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 ((𝑆‘𝐽)‘𝑥) ⊆ ((𝑆‘𝐼)‘𝑥)) → ((𝑆‘𝐽)‘𝑏) ⊆ ((𝑆‘𝐼)‘𝑏))
78	73, 77	sylan2 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → ((𝑆‘𝐽)‘𝑏) ⊆ ((𝑆‘𝐼)‘𝑏))
79	50	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → 𝑅 ∈ Ring)
80	2	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → 𝐽 ∈ 𝑈)
81		simpl 472	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → 𝑏 ∈ ℕ0)
82		hbtlem.s	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑆 = (ldgIdlSeq‘𝑅)
83	9, 4, 82, 8	hbtlem1 36712	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((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 1318	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → ((𝑆‘𝐽)‘𝑏) = {𝑐 ∣ ∃𝑒 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
85	57	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → 𝐼 ∈ 𝑈)
86	9, 4, 82, 8	hbtlem1 36712	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((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 1318	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → ((𝑆‘𝐼)‘𝑏) = {𝑐 ∣ ∃𝑒 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
88	78, 84, 87	3sstr3d 3610	. . . . . . . . . . . . . . . . . . 19 ⊢ ((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 1074	. . . . . . . . . . . . . . . . . 18 ⊢ ((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	. . . . . . . . . . . . . . . . 17 ⊢ (((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 472	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ 𝐽 ∧ (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏) → 𝑑 ∈ 𝐽)
92		simpr 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ 𝐽 ∧ (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏) → (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏)
93		eqidd 2611	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ 𝐽 ∧ (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏) → ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑑)‘𝑏))
94		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 = 𝑑 → (( deg1 ‘𝑅)‘𝑒) = (( deg1 ‘𝑅)‘𝑑))
95	94	breq1d 4593	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑒 = 𝑑 → ((( deg1 ‘𝑅)‘𝑒) ≤ 𝑏 ↔ (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏))
96		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = 𝑑 → (coe1‘𝑒) = (coe1‘𝑑))
97	96	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 = 𝑑 → ((coe1‘𝑒)‘𝑏) = ((coe1‘𝑑)‘𝑏))
98	97	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑒 = 𝑑 → (((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏) ↔ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑑)‘𝑏)))
99	95, 98	anbi12d 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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 3282	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((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 1316	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑑 ∈ 𝐽 ∧ (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏) → ∃𝑒 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))
102		fvex 6113	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((coe1‘𝑑)‘𝑏) ∈ V
103		eqeq1 2614	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = ((coe1‘𝑑)‘𝑏) → (𝑐 = ((coe1‘𝑒)‘𝑏) ↔ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))
104	103	anbi2d 736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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 3034	. . . . . . . . . . . . . . . . . . . 20 ⊢ (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 3319	. . . . . . . . . . . . . . . . . . 19 ⊢ (((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 223	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ 𝐽 ∧ (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏) → ((coe1‘𝑑)‘𝑏) ∈ {𝑐 ∣ ∃𝑒 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
108	107	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((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 3569	. . . . . . . . . . . . . . . 16 ⊢ (((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 3034	. . . . . . . . . . . . . . . . . 18 ⊢ (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 3319	. . . . . . . . . . . . . . . . 17 ⊢ (((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 1094	. . . . . . . . . . . . . . . . . . . . . 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)))) → 𝜑)
113	112, 56	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)))) → 𝑃 ∈ Ring)
114		ringgrp 18375	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp)
115	113, 114	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)))) → 𝑃 ∈ Grp)
116	112, 6	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 ⊆ (Base‘𝑃))
117		simplrl 796	. . . . . . . . . . . . . . . . . . . . 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)))) → 𝑑 ∈ 𝐽)
118	116, 117	sseldd 3569	. . . . . . . . . . . . . . . . . . . 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 ∈ (Base‘𝑃))
119	3, 4	lidlss 19031	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ (Base‘𝑃))
120	57, 119	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐼 ⊆ (Base‘𝑃))
121	112, 120	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)))) → 𝐼 ⊆ (Base‘𝑃))
122		simprl 790	. . . . . . . . . . . . . . . . . . . . 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 ∈ 𝐼)
123	121, 122	sseldd 3569	. . . . . . . . . . . . . . . . . . . 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)))) → 𝑒 ∈ (Base‘𝑃))
124		eqid 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (+g‘𝑃) = (+g‘𝑃)
125		eqid 2610	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (-g‘𝑃) = (-g‘𝑃)
126	3, 124, 125	grpnpcan 17330	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ∈ Grp ∧ 𝑑 ∈ (Base‘𝑃) ∧ 𝑒 ∈ (Base‘𝑃)) → ((𝑑(-g‘𝑃)𝑒)(+g‘𝑃)𝑒) = 𝑑)
127	115, 118, 123, 126	syl3anc 1318	. . . . . . . . . . . . . . . . . . 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(-g‘𝑃)𝑒)(+g‘𝑃)𝑒) = 𝑑)
128	57	3ad2ant2 1076	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → 𝐼 ∈ 𝑈)
129	128	ad2antrr 758	. . . . . . . . . . . . . . . . . . . 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)))) → 𝐼 ∈ 𝑈)
130		simpll1 1093	. . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑏 ∈ ℕ0)
131	112, 50	syl 17	. . . . . . . . . . . . . . . . . . . . . 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)))) → 𝑅 ∈ Ring)
132		simplrr 797	. . . . . . . . . . . . . . . . . . . . . 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)))) → (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏)
133		simprrl 800	. . . . . . . . . . . . . . . . . . . . . 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)))) → (( deg1 ‘𝑅)‘𝑒) ≤ 𝑏)
134		eqid 2610	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (coe1‘𝑑) = (coe1‘𝑑)
135		eqid 2610	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (coe1‘𝑒) = (coe1‘𝑒)
136		simprrr 801	. . . . . . . . . . . . . . . . . . . . . 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)))) → ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏))
137	8, 9, 3, 125, 130, 131, 118, 132, 123, 133, 134, 135, 136	deg1sublt 23674	. . . . . . . . . . . . . . . . . . . . 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)))) → (( deg1 ‘𝑅)‘(𝑑(-g‘𝑃)𝑒)) < 𝑏)
138	112, 2	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((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 1076	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → 𝐼 ⊆ 𝐽)
140	139	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((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 3569	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((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 19037	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑃 ∈ Ring ∧ 𝐽 ∈ 𝑈) ∧ (𝑑 ∈ 𝐽 ∧ 𝑒 ∈ 𝐽)) → (𝑑(-g‘𝑃)𝑒) ∈ 𝐽)
143	113, 138, 117, 141, 142	syl22anc 1319	. . . . . . . . . . . . . . . . . . . . . 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)))) → (𝑑(-g‘𝑃)𝑒) ∈ 𝐽)
144		simpll3 1095	. . . . . . . . . . . . . . . . . . . . . 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)))) → ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))
145		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (𝑑(-g‘𝑃)𝑒) → (( deg1 ‘𝑅)‘𝑎) = (( deg1 ‘𝑅)‘(𝑑(-g‘𝑃)𝑒)))
146	145	breq1d 4593	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (𝑑(-g‘𝑃)𝑒) → ((( deg1 ‘𝑅)‘𝑎) < 𝑏 ↔ (( deg1 ‘𝑅)‘(𝑑(-g‘𝑃)𝑒)) < 𝑏))
147		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (𝑑(-g‘𝑃)𝑒) → (𝑎 ∈ 𝐼 ↔ (𝑑(-g‘𝑃)𝑒) ∈ 𝐼))
148	146, 147	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (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 3280	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((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 691	. . . . . . . . . . . . . . . . . . . . 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)))) → ((( deg1 ‘𝑅)‘(𝑑(-g‘𝑃)𝑒)) < 𝑏 → (𝑑(-g‘𝑃)𝑒) ∈ 𝐼))
151	137, 150	mpd 15	. . . . . . . . . . . . . . . . . . . 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) ∈ 𝐼)
152	4, 124	lidlacl 19034	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((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 1319	. . . . . . . . . . . . . . . . . . 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(-g‘𝑃)𝑒)(+g‘𝑃)𝑒) ∈ 𝐼)
154	127, 153	eqeltrrd 2689	. . . . . . . . . . . . . . . . . 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 ∈ 𝐼)
155	154	rexlimdvaa 3014	. . . . . . . . . . . . . . . . 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 ∈ 𝐼))
156	111, 155	syl5bi 231	. . . . . . . . . . . . . . . 16 ⊢ (((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	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏)) → 𝑑 ∈ 𝐼)
158	157	expr 641	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → ((( deg1 ‘𝑅)‘𝑑) ≤ 𝑏 → 𝑑 ∈ 𝐼))
159	72, 158	sylbid 229	. . . . . . . . . . . . 13 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → ((( deg1 ‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))
160	159	ralrimiva 2949	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → ∀𝑑 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))
161	160	3exp 1256	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ℕ0 → (𝜑 → (∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼) → ∀𝑑 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))))
162	161	a2d 29	. . . . . . . . . 10 ⊢ (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 11348	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ0 → (𝜑 → ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
164		rsp 2913	. . . . . . . . 9 ⊢ (∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼) → (𝑎 ∈ 𝐽 → ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
165	163, 164	syl6com 36	. . . . . . . 8 ⊢ (𝜑 → (𝑏 ∈ ℕ0 → (𝑎 ∈ 𝐽 → ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))))
166	165	com23 84	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ 𝐽 → (𝑏 ∈ ℕ0 → ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))))
167	166	imp 444	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (𝑏 ∈ ℕ0 → ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
168	167	rexlimdv 3012	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (∃𝑏 ∈ ℕ0 (( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))
169	31, 168	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → 𝑎 ∈ 𝐼)
170	169	ex 449	. . 3 ⊢ (𝜑 → (𝑎 ∈ 𝐽 → 𝑎 ∈ 𝐼))
171	170	ssrdv 3574	. 2 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
172	1, 171	eqssd 3585	1 ⊢ (𝜑 → 𝐼 = 𝐽)

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897   ∪ cun 3538   ⊆ wss 3540  {csn 4125   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818  -∞cmnf 9951   < clt 9953   ≤ cle 9954  ℕcn 10897  ℕ₀cn0 11169  ℤcz 11254  Basecbs 15695  +_gcplusg 15768  0_gc0g 15923  Grpcgrp 17245  -_gcsg 17247  Ringcrg 18370  LIdealclidl 18991  Poly₁cpl1 19368  coe₁cco1 19369   deg₁ cdg1 23618  ldgIdlSeqcldgis 36710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-subrg 18601  df-lmod 18688  df-lss 18754  df-sra 18993  df-rgmod 18994  df-lidl 18995  df-rlreg 19104  df-psr 19177  df-mpl 19179  df-opsr 19181  df-psr1 19371  df-ply1 19373  df-coe1 19374  df-cnfld 19568  df-mdeg 23619  df-deg1 23620  df-ldgis 36711

This theorem is referenced by:  hbt  36719