Theorem hbtlem5 37698

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 2622	. . . . . . . . . 10 ⊢ (Base‘𝑃) = (Base‘𝑃)
4		hbtlem.u	. . . . . . . . . 10 ⊢ 𝑈 = (LIdeal‘𝑃)
5	3, 4	lidlss 19210	. . . . . . . . 9 ⊢ (𝐽 ∈ 𝑈 → 𝐽 ⊆ (Base‘𝑃))
6	2, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ⊆ (Base‘𝑃))
7	6	sselda 3603	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → 𝑎 ∈ (Base‘𝑃))
8		eqid 2622	. . . . . . . 8 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅)
9		hbtlem.p	. . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅)
10	8, 9, 3	deg1cl 23843	. . . . . . 7 ⊢ (𝑎 ∈ (Base‘𝑃) → (( deg1 ‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
11	7, 10	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (( deg1 ‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
12		elun 3753	. . . . . . 7 ⊢ ((( deg1 ‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}) ↔ ((( deg1 ‘𝑅)‘𝑎) ∈ ℕ0 ∨ (( deg1 ‘𝑅)‘𝑎) ∈ {-∞}))
13		nnssnn0 11295	. . . . . . . . 9 ⊢ ℕ ⊆ ℕ0
14		nn0re 11301	. . . . . . . . . 10 ⊢ ((( deg1 ‘𝑅)‘𝑎) ∈ ℕ0 → (( deg1 ‘𝑅)‘𝑎) ∈ ℝ)
15		arch 11289	. . . . . . . . . 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 3667	. . . . . . . . 9 ⊢ (ℕ ⊆ ℕ0 → (∃𝑏 ∈ ℕ (( deg1 ‘𝑅)‘𝑎) < 𝑏 → ∃𝑏 ∈ ℕ0 (( deg1 ‘𝑅)‘𝑎) < 𝑏))
18	13, 16, 17	mpsyl 68	. . . . . . . 8 ⊢ ((( deg1 ‘𝑅)‘𝑎) ∈ ℕ0 → ∃𝑏 ∈ ℕ0 (( deg1 ‘𝑅)‘𝑎) < 𝑏)
19		elsni 4194	. . . . . . . . 9 ⊢ ((( deg1 ‘𝑅)‘𝑎) ∈ {-∞} → (( deg1 ‘𝑅)‘𝑎) = -∞)
20		0nn0 11307	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ0
21		mnflt0 11959	. . . . . . . . . . 11 ⊢ -∞ < 0
22		breq2 4657	. . . . . . . . . . . 12 ⊢ (𝑏 = 0 → (-∞ < 𝑏 ↔ -∞ < 0))
23	22	rspcev 3309	. . . . . . . . . . 11 ⊢ ((0 ∈ ℕ0 ∧ -∞ < 0) → ∃𝑏 ∈ ℕ0 -∞ < 𝑏)
24	20, 21, 23	mp2an 708	. . . . . . . . . 10 ⊢ ∃𝑏 ∈ ℕ0 -∞ < 𝑏
25		breq1 4656	. . . . . . . . . . 11 ⊢ ((( deg1 ‘𝑅)‘𝑎) = -∞ → ((( deg1 ‘𝑅)‘𝑎) < 𝑏 ↔ -∞ < 𝑏))
26	25	rexbidv 3052	. . . . . . . . . 10 ⊢ ((( deg1 ‘𝑅)‘𝑎) = -∞ → (∃𝑏 ∈ ℕ0 (( deg1 ‘𝑅)‘𝑎) < 𝑏 ↔ ∃𝑏 ∈ ℕ0 -∞ < 𝑏))
27	24, 26	mpbiri 248	. . . . . . . . 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 394	. . . . . . 7 ⊢ (((( deg1 ‘𝑅)‘𝑎) ∈ ℕ0 ∨ (( deg1 ‘𝑅)‘𝑎) ∈ {-∞}) → ∃𝑏 ∈ ℕ0 (( deg1 ‘𝑅)‘𝑎) < 𝑏)
30	12, 29	sylbi 207	. . . . . 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 4657	. . . . . . . . . . . . 13 ⊢ (𝑐 = 0 → ((( deg1 ‘𝑅)‘𝑎) < 𝑐 ↔ (( deg1 ‘𝑅)‘𝑎) < 0))
33	32	imbi1d 331	. . . . . . . . . . . 12 ⊢ (𝑐 = 0 → (((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ((( deg1 ‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼)))
34	33	ralbidv 2986	. . . . . . . . . . 11 ⊢ (𝑐 = 0 → (∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼)))
35	34	imbi2d 330	. . . . . . . . . 10 ⊢ (𝑐 = 0 → ((𝜑 → ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼)) ↔ (𝜑 → ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼))))
36		breq2 4657	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → ((( deg1 ‘𝑅)‘𝑎) < 𝑐 ↔ (( deg1 ‘𝑅)‘𝑎) < 𝑏))
37	36	imbi1d 331	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑏 → (((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
38	37	ralbidv 2986	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑏 → (∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
39	38	imbi2d 330	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → ((𝜑 → ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼)) ↔ (𝜑 → ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))))
40		breq2 4657	. . . . . . . . . . . . . 14 ⊢ (𝑐 = (𝑏 + 1) → ((( deg1 ‘𝑅)‘𝑎) < 𝑐 ↔ (( deg1 ‘𝑅)‘𝑎) < (𝑏 + 1)))
41	40	imbi1d 331	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑏 + 1) → (((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ((( deg1 ‘𝑅)‘𝑎) < (𝑏 + 1) → 𝑎 ∈ 𝐼)))
42	41	ralbidv 2986	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑏 + 1) → (∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < (𝑏 + 1) → 𝑎 ∈ 𝐼)))
43		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑑 → (( deg1 ‘𝑅)‘𝑎) = (( deg1 ‘𝑅)‘𝑑))
44	43	breq1d 4663	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑑 → ((( deg1 ‘𝑅)‘𝑎) < (𝑏 + 1) ↔ (( deg1 ‘𝑅)‘𝑑) < (𝑏 + 1)))
45		eleq1 2689	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑑 → (𝑎 ∈ 𝐼 ↔ 𝑑 ∈ 𝐼))
46	44, 45	imbi12d 334	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑑 → (((( deg1 ‘𝑅)‘𝑎) < (𝑏 + 1) → 𝑎 ∈ 𝐼) ↔ ((( deg1 ‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼)))
47	46	cbvralv 3171	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < (𝑏 + 1) → 𝑎 ∈ 𝐼) ↔ ∀𝑑 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))
48	42, 47	syl6bb 276	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑏 + 1) → (∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ∀𝑑 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼)))
49	48	imbi2d 330	. . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → 𝑅 ∈ Ring)
52		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑃) = (0g‘𝑃)
53	8, 9, 52, 3	deg1lt0 23851	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑃)) → ((( deg1 ‘𝑅)‘𝑎) < 0 ↔ 𝑎 = (0g‘𝑃)))
54	51, 7, 53	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → ((( deg1 ‘𝑅)‘𝑎) < 0 ↔ 𝑎 = (0g‘𝑃)))
55	9	ply1ring 19618	. . . . . . . . . . . . . . . 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 19212	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (0g‘𝑃) ∈ 𝐼)
59	56, 57, 58	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0g‘𝑃) ∈ 𝐼)
60		eleq1a 2696	. . . . . . . . . . . . . 14 ⊢ ((0g‘𝑃) ∈ 𝐼 → (𝑎 = (0g‘𝑃) → 𝑎 ∈ 𝐼))
61	59, 60	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑎 = (0g‘𝑃) → 𝑎 ∈ 𝐼))
62	61	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (𝑎 = (0g‘𝑃) → 𝑎 ∈ 𝐼))
63	54, 62	sylbid 230	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → ((( deg1 ‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼))
64	63	ralrimiva 2966	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼))
65	6	3ad2ant2 1083	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → 𝐽 ⊆ (Base‘𝑃))
66	65	sselda 3603	. . . . . . . . . . . . . . . 16 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → 𝑑 ∈ (Base‘𝑃))
67	8, 9, 3	deg1cl 23843	. . . . . . . . . . . . . . . 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 1064	. . . . . . . . . . . . . . . 16 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → 𝑏 ∈ ℕ0)
70	69	nn0zd 11480	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → 𝑏 ∈ ℤ)
71		degltp1le 23833	. . . . . . . . . . . . . . 15 ⊢ (((( deg1 ‘𝑅)‘𝑑) ∈ (ℕ0 ∪ {-∞}) ∧ 𝑏 ∈ ℤ) → ((( deg1 ‘𝑅)‘𝑑) < (𝑏 + 1) ↔ (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏))
72	68, 70, 71	syl2anc 693	. . . . . . . . . . . . . 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 6191	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑏 → ((𝑆‘𝐽)‘𝑥) = ((𝑆‘𝐽)‘𝑏))
75		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑏 → ((𝑆‘𝐼)‘𝑥) = ((𝑆‘𝐼)‘𝑏))
76	74, 75	sseq12d 3634	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑏 → (((𝑆‘𝐽)‘𝑥) ⊆ ((𝑆‘𝐼)‘𝑥) ↔ ((𝑆‘𝐽)‘𝑏) ⊆ ((𝑆‘𝐼)‘𝑏)))
77	76	rspcva 3307	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 ((𝑆‘𝐽)‘𝑥) ⊆ ((𝑆‘𝐼)‘𝑥)) → ((𝑆‘𝐽)‘𝑏) ⊆ ((𝑆‘𝐼)‘𝑏))
78	73, 77	sylan2 491	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → ((𝑆‘𝐽)‘𝑏) ⊆ ((𝑆‘𝐼)‘𝑏))
79	50	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → 𝑅 ∈ Ring)
80	2	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → 𝐽 ∈ 𝑈)
81		simpl 473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → 𝑏 ∈ ℕ0)
82		hbtlem.s	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑆 = (ldgIdlSeq‘𝑅)
83	9, 4, 82, 8	hbtlem1 37693	. . . . . . . . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → ((𝑆‘𝐽)‘𝑏) = {𝑐 ∣ ∃𝑒 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
85	57	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → 𝐼 ∈ 𝑈)
86	9, 4, 82, 8	hbtlem1 37693	. . . . . . . . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → ((𝑆‘𝐼)‘𝑏) = {𝑐 ∣ ∃𝑒 ∈ 𝐼 ((( deg1 ‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
88	78, 84, 87	3sstr3d 3647	. . . . . . . . . . . . . . . . . . 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 1081	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ 𝐽 ∧ (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏) → 𝑑 ∈ 𝐽)
92		simpr 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ 𝐽 ∧ (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏) → (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏)
93		eqidd 2623	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ 𝐽 ∧ (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏) → ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑑)‘𝑏))
94		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 = 𝑑 → (( deg1 ‘𝑅)‘𝑒) = (( deg1 ‘𝑅)‘𝑑))
95	94	breq1d 4663	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑒 = 𝑑 → ((( deg1 ‘𝑅)‘𝑒) ≤ 𝑏 ↔ (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏))
96		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑒 = 𝑑 → (coe1‘𝑒) = (coe1‘𝑑))
97	96	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑒 = 𝑑 → ((coe1‘𝑒)‘𝑏) = ((coe1‘𝑑)‘𝑏))
98	97	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑒 = 𝑑 → (((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏) ↔ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑑)‘𝑏)))
99	95, 98	anbi12d 747	. . . . . . . . . . . . . . . . . . . . 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 3309	. . . . . . . . . . . . . . . . . . . 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 1324	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑑 ∈ 𝐽 ∧ (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏) → ∃𝑒 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))
102		fvex 6201	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((coe1‘𝑑)‘𝑏) ∈ V
103		eqeq1 2626	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = ((coe1‘𝑑)‘𝑏) → (𝑐 = ((coe1‘𝑒)‘𝑏) ↔ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))
104	103	anbi2d 740	. . . . . . . . . . . . . . . . . . . . 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 3052	. . . . . . . . . . . . . . . . . . . 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 3350	. . . . . . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ 𝐽 ∧ (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏) → ((coe1‘𝑑)‘𝑏) ∈ {𝑐 ∣ ∃𝑒 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
108	107	adantl 482	. . . . . . . . . . . . . . . . 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 3604	. . . . . . . . . . . . . . . 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 3052	. . . . . . . . . . . . . . . . . 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 3350	. . . . . . . . . . . . . . . . 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 1101	. . . . . . . . . . . . . . . . . . . . . 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 18552	. . . . . . . . . . . . . . . . . . . . 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 800	. . . . . . . . . . . . . . . . . . . . 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 3604	. . . . . . . . . . . . . . . . . . . 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 19210	. . . . . . . . . . . . . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . . . . . . 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 3604	. . . . . . . . . . . . . . . . . . . 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 2622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (+g‘𝑃) = (+g‘𝑃)
125		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (-g‘𝑃) = (-g‘𝑃)
126	3, 124, 125	grpnpcan 17507	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ∈ Grp ∧ 𝑑 ∈ (Base‘𝑃) ∧ 𝑒 ∈ (Base‘𝑃)) → ((𝑑(-g‘𝑃)𝑒)(+g‘𝑃)𝑒) = 𝑑)
127	115, 118, 123, 126	syl3anc 1326	. . . . . . . . . . . . . . . . . . 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 1083	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → 𝐼 ∈ 𝑈)
129	128	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 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 1100	. . . . . . . . . . . . . . . . . . . . . 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 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)))) → (( deg1 ‘𝑅)‘𝑑) ≤ 𝑏)
133		simprrl 804	. . . . . . . . . . . . . . . . . . . . . 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 2622	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (coe1‘𝑑) = (coe1‘𝑑)
135		eqid 2622	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (coe1‘𝑒) = (coe1‘𝑒)
136		simprrr 805	. . . . . . . . . . . . . . . . . . . . . 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 23870	. . . . . . . . . . . . . . . . . . . . 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 1083	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → 𝐼 ⊆ 𝐽)
140	139	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . 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 3604	. . . . . . . . . . . . . . . . . . . . . . 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 19216	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑃 ∈ Ring ∧ 𝐽 ∈ 𝑈) ∧ (𝑑 ∈ 𝐽 ∧ 𝑒 ∈ 𝐽)) → (𝑑(-g‘𝑃)𝑒) ∈ 𝐽)
143	113, 138, 117, 141, 142	syl22anc 1327	. . . . . . . . . . . . . . . . . . . . . 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 1102	. . . . . . . . . . . . . . . . . . . . . 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 6191	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = (𝑑(-g‘𝑃)𝑒) → (( deg1 ‘𝑅)‘𝑎) = (( deg1 ‘𝑅)‘(𝑑(-g‘𝑃)𝑒)))
146	145	breq1d 4663	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (𝑑(-g‘𝑃)𝑒) → ((( deg1 ‘𝑅)‘𝑎) < 𝑏 ↔ (( deg1 ‘𝑅)‘(𝑑(-g‘𝑃)𝑒)) < 𝑏))
147		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (𝑑(-g‘𝑃)𝑒) → (𝑎 ∈ 𝐼 ↔ (𝑑(-g‘𝑃)𝑒) ∈ 𝐼))
148	146, 147	imbi12d 334	. . . . . . . . . . . . . . . . . . . . . . 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 3307	. . . . . . . . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . . . . . . . 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 19213	. . . . . . . . . . . . . . . . . . . 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 1327	. . . . . . . . . . . . . . . . . . 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 2702	. . . . . . . . . . . . . . . . . 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 3032	. . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → ((( deg1 ‘𝑅)‘𝑑) ≤ 𝑏 → 𝑑 ∈ 𝐼))
159	72, 158	sylbid 230	. . . . . . . . . . . . 13 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → ((( deg1 ‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))
160	159	ralrimiva 2966	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → ∀𝑑 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))
161	160	3exp 1264	. . . . . . . . . . 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 11472	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ0 → (𝜑 → ∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
164		rsp 2929	. . . . . . . . 9 ⊢ (∀𝑎 ∈ 𝐽 ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼) → (𝑎 ∈ 𝐽 → ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
165	163, 164	syl6com 37	. . . . . . . 8 ⊢ (𝜑 → (𝑏 ∈ ℕ0 → (𝑎 ∈ 𝐽 → ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))))
166	165	com23 86	. . . . . . 7 ⊢ (𝜑 → (𝑎 ∈ 𝐽 → (𝑏 ∈ ℕ0 → ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))))
167	166	imp 445	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (𝑏 ∈ ℕ0 → ((( deg1 ‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
168	167	rexlimdv 3030	. . . . 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 450	. . 3 ⊢ (𝜑 → (𝑎 ∈ 𝐽 → 𝑎 ∈ 𝐼))
171	170	ssrdv 3609	. 2 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
172	1, 171	eqssd 3620	1 ⊢ (𝜑 → 𝐼 = 𝐽)

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913   ∪ cun 3572   ⊆ wss 3574  {csn 4177   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939  -∞cmnf 10072   < clt 10074   ≤ cle 10075  ℕcn 11020  ℕ₀cn0 11292  ℤcz 11377  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100  Grpcgrp 17422  -_gcsg 17424  Ringcrg 18547  LIdealclidl 19170  Poly₁cpl1 19547  coe₁cco1 19548   deg₁ cdg1 23814  ldgIdlSeqcldgis 37691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-subrg 18778  df-lmod 18865  df-lss 18933  df-sra 19172  df-rgmod 19173  df-lidl 19174  df-rlreg 19283  df-psr 19356  df-mpl 19358  df-opsr 19360  df-psr1 19550  df-ply1 19552  df-coe1 19553  df-cnfld 19747  df-mdeg 23815  df-deg1 23816  df-ldgis 37692

This theorem is referenced by:  hbt  37700