Theorem hbtlem5 43117

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 2735	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘𝑃)
4		hbtlem.u	. . . . . . . 8 ⊢ 𝑈 = (LIdeal‘𝑃)
5	3, 4	lidlss 21240	. . . . . . 7 ⊢ (𝐽 ∈ 𝑈 → 𝐽 ⊆ (Base‘𝑃))
6	2, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐽 ⊆ (Base‘𝑃))
7	6	sselda 3995	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → 𝑎 ∈ (Base‘𝑃))
8		eqid 2735	. . . . . 6 ⊢ (deg1‘𝑅) = (deg1‘𝑅)
9		hbtlem.p	. . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅)
10	8, 9, 3	deg1cl 26137	. . . . 5 ⊢ (𝑎 ∈ (Base‘𝑃) → ((deg1‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
11	7, 10	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → ((deg1‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
12		elun 4163	. . . . 5 ⊢ (((deg1‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}) ↔ (((deg1‘𝑅)‘𝑎) ∈ ℕ0 ∨ ((deg1‘𝑅)‘𝑎) ∈ {-∞}))
13		nnssnn0 12527	. . . . . . 7 ⊢ ℕ ⊆ ℕ0
14		nn0re 12533	. . . . . . . 8 ⊢ (((deg1‘𝑅)‘𝑎) ∈ ℕ0 → ((deg1‘𝑅)‘𝑎) ∈ ℝ)
15		arch 12521	. . . . . . . 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 4065	. . . . . . 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 4648	. . . . . . 7 ⊢ (((deg1‘𝑅)‘𝑎) ∈ {-∞} → ((deg1‘𝑅)‘𝑎) = -∞)
20		0nn0 12539	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
21		mnflt0 13165	. . . . . . . . 9 ⊢ -∞ < 0
22		breq2 5152	. . . . . . . . . 10 ⊢ (𝑏 = 0 → (-∞ < 𝑏 ↔ -∞ < 0))
23	22	rspcev 3622	. . . . . . . . 9 ⊢ ((0 ∈ ℕ0 ∧ -∞ < 0) → ∃𝑏 ∈ ℕ0 -∞ < 𝑏)
24	20, 21, 23	mp2an 692	. . . . . . . 8 ⊢ ∃𝑏 ∈ ℕ0 -∞ < 𝑏
25		breq1 5151	. . . . . . . . 9 ⊢ (((deg1‘𝑅)‘𝑎) = -∞ → (((deg1‘𝑅)‘𝑎) < 𝑏 ↔ -∞ < 𝑏))
26	25	rexbidv 3177	. . . . . . . 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 5152	. . . . . . . . . . 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 3176	. . . . . . . . 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 5152	. . . . . . . . . . 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 3176	. . . . . . . . 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 5152	. . . . . . . . . . . 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 3176	. . . . . . . . . 10 ⊢ (𝑐 = (𝑏 + 1) → (∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < (𝑏 + 1) → 𝑎 ∈ 𝐼)))
43		fveq2 6907	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑑 → ((deg1‘𝑅)‘𝑎) = ((deg1‘𝑅)‘𝑑))
44	43	breq1d 5158	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑑 → (((deg1‘𝑅)‘𝑎) < (𝑏 + 1) ↔ ((deg1‘𝑅)‘𝑑) < (𝑏 + 1)))
45		eleq1 2827	. . . . . . . . . . . 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 3235	. . . . . . . . . 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 2735	. . . . . . . . . . . 12 ⊢ (0g‘𝑃) = (0g‘𝑃)
53	8, 9, 52, 3	deg1lt0 26145	. . . . . . . . . . 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 22265	. . . . . . . . . . . . . 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 21248	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (0g‘𝑃) ∈ 𝐼)
59	56, 57, 58	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (0g‘𝑃) ∈ 𝐼)
60		eleq1a 2834	. . . . . . . . . . . 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 3144	. . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼))
65	6	3ad2ant2 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → 𝐽 ⊆ (Base‘𝑃))
66	65	sselda 3995	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → 𝑑 ∈ (Base‘𝑃))
67	8, 9, 3	deg1cl 26137	. . . . . . . . . . . . . 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 1190	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → 𝑏 ∈ ℕ0)
70	69	nn0zd 12637	. . . . . . . . . . . . 13 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → 𝑏 ∈ ℤ)
71		degltp1le 26127	. . . . . . . . . . . . 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 6907	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑏 → ((𝑆‘𝐽)‘𝑥) = ((𝑆‘𝐽)‘𝑏))
75		fveq2 6907	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑏 → ((𝑆‘𝐼)‘𝑥) = ((𝑆‘𝐼)‘𝑏))
76	74, 75	sseq12d 4029	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑏 → (((𝑆‘𝐽)‘𝑥) ⊆ ((𝑆‘𝐼)‘𝑥) ↔ ((𝑆‘𝐽)‘𝑏) ⊆ ((𝑆‘𝐼)‘𝑏)))
77	76	rspcva 3620	. . . . . . . . . . . . . . . . . . 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 43112	. . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . 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 43112	. . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → ((𝑆‘𝐼)‘𝑏) = {𝑐 ∣ ∃𝑒 ∈ 𝐼 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
88	78, 84, 87	3sstr3d 4042	. . . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏) → ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑑)‘𝑏))
94		fveq2 6907	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑒 = 𝑑 → ((deg1‘𝑅)‘𝑒) = ((deg1‘𝑅)‘𝑑))
95	94	breq1d 5158	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = 𝑑 → (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ↔ ((deg1‘𝑅)‘𝑑) ≤ 𝑏))
96		fveq2 6907	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑒 = 𝑑 → (coe1‘𝑒) = (coe1‘𝑑))
97	96	fveq1d 6909	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑒 = 𝑑 → ((coe1‘𝑒)‘𝑏) = ((coe1‘𝑑)‘𝑏))
98	97	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . 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 3622	. . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏) → ∃𝑒 ∈ 𝐽 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))
102		fvex 6920	. . . . . . . . . . . . . . . . . 18 ⊢ ((coe1‘𝑑)‘𝑏) ∈ V
103		eqeq1 2739	. . . . . . . . . . . . . . . . . . . 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 3177	. . . . . . . . . . . . . . . . . 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 3681	. . . . . . . . . . . . . . . . 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 3996	. . . . . . . . . . . . . 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 3177	. . . . . . . . . . . . . . . 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 3681	. . . . . . . . . . . . . . 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 1212	. . . . . . . . . . . . . . . . . . . 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 20256	. . . . . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . . . 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 3996	. . . . . . . . . . . . . . . . . 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 21240	. . . . . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . . 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 3996	. . . . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ (+g‘𝑃) = (+g‘𝑃)
125		eqid 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ (-g‘𝑃) = (-g‘𝑃)
126	3, 124, 125	grpnpcan 19063	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ Grp ∧ 𝑑 ∈ (Base‘𝑃) ∧ 𝑒 ∈ (Base‘𝑃)) → ((𝑑(-g‘𝑃)𝑒)(+g‘𝑃)𝑒) = 𝑑)
127	115, 118, 123, 126	syl3anc 1370	. . . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . . . . 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 1211	. . . . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . . . 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 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)))) → ((deg1‘𝑅)‘𝑒) ≤ 𝑏)
134		eqid 2735	. . . . . . . . . . . . . . . . . . . 20 ⊢ (coe1‘𝑑) = (coe1‘𝑑)
135		eqid 2735	. . . . . . . . . . . . . . . . . . . 20 ⊢ (coe1‘𝑒) = (coe1‘𝑒)
136		simprrr 782	. . . . . . . . . . . . . . . . . . . 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 26164	. . . . . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . . . . . . . . . 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 3996	. . . . . . . . . . . . . . . . . . . . 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 21252	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑃 ∈ Ring ∧ 𝐽 ∈ 𝑈) ∧ (𝑑 ∈ 𝐽 ∧ 𝑒 ∈ 𝐽)) → (𝑑(-g‘𝑃)𝑒) ∈ 𝐽)
143	113, 138, 117, 141, 142	syl22anc 839	. . . . . . . . . . . . . . . . . . . 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 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)))) → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))
145		fveq2 6907	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = (𝑑(-g‘𝑃)𝑒) → ((deg1‘𝑅)‘𝑎) = ((deg1‘𝑅)‘(𝑑(-g‘𝑃)𝑒)))
146	145	breq1d 5158	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = (𝑑(-g‘𝑃)𝑒) → (((deg1‘𝑅)‘𝑎) < 𝑏 ↔ ((deg1‘𝑅)‘(𝑑(-g‘𝑃)𝑒)) < 𝑏))
147		eleq1 2827	. . . . . . . . . . . . . . . . . . . . . 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 3620	. . . . . . . . . . . . . . . . . . . 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 21249	. . . . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . 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 3154	. . . . . . . . . . . . . . 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 3144	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → ∀𝑑 ∈ 𝐽 (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))
161	160	3exp 1118	. . . . . . . . 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 12711	. . . . . . 7 ⊢ (𝑏 ∈ ℕ0 → (𝜑 → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
164		rsp 3245	. . . . . . 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 3151	. . 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 4017	1 ⊢ (𝜑 → 𝐼 = 𝐽)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  {cab 2712  ∀wral 3059  ∃wrex 3068   ∪ cun 3961   ⊆ wss 3963  {csn 4631   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  ℝcr 11152  0cc0 11153  1c1 11154   + caddc 11156  -∞cmnf 11291   < clt 11293   ≤ cle 11294  ℕcn 12264  ℕ₀cn0 12524  ℤcz 12611  Basecbs 17245  +_gcplusg 17298  0_gc0g 17486  Grpcgrp 18964  -_gcsg 18966  Ringcrg 20251  LIdealclidl 21234  Poly₁cpl1 22194  coe₁cco1 22195  deg₁cdg1 26108  ldgIdlSeqcldgis 43110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-subrng 20563  df-subrg 20587  df-rlreg 20711  df-lmod 20877  df-lss 20948  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-cnfld 21383  df-psr 21947  df-mpl 21949  df-opsr 21951  df-psr1 22197  df-ply1 22199  df-coe1 22200  df-mdeg 26109  df-deg1 26110  df-ldgis 43111

This theorem is referenced by:  hbt  43119