Theorem hbtlem5 44073

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 2760	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘𝑃)
4		hbtlem.u	. . . . . . . 8 ⊢ 𝑈 = (LIdeal‘𝑃)
5	3, 4	lidlss 21451	. . . . . . 7 ⊢ (𝐽 ∈ 𝑈 → 𝐽 ⊆ (Base‘𝑃))
6	2, 5	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐽 ⊆ (Base‘𝑃))
7	6	sselda 3930	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → 𝑎 ∈ (Base‘𝑃))
8		eqid 2760	. . . . . 6 ⊢ (deg1‘𝑅) = (deg1‘𝑅)
9		hbtlem.p	. . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅)
10	8, 9, 3	deg1cl 26362	. . . . 5 ⊢ (𝑎 ∈ (Base‘𝑃) → ((deg1‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
11	7, 10	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → ((deg1‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}))
12		elun 4099	. . . . 5 ⊢ (((deg1‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}) ↔ (((deg1‘𝑅)‘𝑎) ∈ ℕ0 ∨ ((deg1‘𝑅)‘𝑎) ∈ {-∞}))
13		nnssnn0 12578	. . . . . . 7 ⊢ ℕ ⊆ ℕ0
14		nn0re 12584	. . . . . . . 8 ⊢ (((deg1‘𝑅)‘𝑎) ∈ ℕ0 → ((deg1‘𝑅)‘𝑎) ∈ ℝ)
15		arch 12572	. . . . . . . 8 ⊢ (((deg1‘𝑅)‘𝑎) ∈ ℝ → ∃𝑏 ∈ ℕ ((deg1‘𝑅)‘𝑎) < 𝑏)
16	14, 15	syl 18	. . . . . . 7 ⊢ (((deg1‘𝑅)‘𝑎) ∈ ℕ0 → ∃𝑏 ∈ ℕ ((deg1‘𝑅)‘𝑎) < 𝑏)
17		ssrexv 4000	. . . . . . 7 ⊢ (ℕ ⊆ ℕ0 → (∃𝑏 ∈ ℕ ((deg1‘𝑅)‘𝑎) < 𝑏 → ∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏))
18	13, 16, 17	mpsyl 69	. . . . . 6 ⊢ (((deg1‘𝑅)‘𝑎) ∈ ℕ0 → ∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏)
19		elsni 4600	. . . . . . 7 ⊢ (((deg1‘𝑅)‘𝑎) ∈ {-∞} → ((deg1‘𝑅)‘𝑎) = -∞)
20		0nn0 12590	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
21		mnflt0 13223	. . . . . . . . 9 ⊢ -∞ < 0
22		breq2 5106	. . . . . . . . . 10 ⊢ (𝑏 = 0 → (-∞ < 𝑏 ↔ -∞ < 0))
23	22	rspcev 3576	. . . . . . . . 9 ⊢ ((0 ∈ ℕ0 ∧ -∞ < 0) → ∃𝑏 ∈ ℕ0 -∞ < 𝑏)
24	20, 21, 23	mp2an 705	. . . . . . . 8 ⊢ ∃𝑏 ∈ ℕ0 -∞ < 𝑏
25		breq1 5105	. . . . . . . . 9 ⊢ (((deg1‘𝑅)‘𝑎) = -∞ → (((deg1‘𝑅)‘𝑎) < 𝑏 ↔ -∞ < 𝑏))
26	25	rexbidv 3186	. . . . . . . 8 ⊢ (((deg1‘𝑅)‘𝑎) = -∞ → (∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏 ↔ ∃𝑏 ∈ ℕ0 -∞ < 𝑏))
27	24, 26	mpbiri 261	. . . . . . 7 ⊢ (((deg1‘𝑅)‘𝑎) = -∞ → ∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏)
28	19, 27	syl 18	. . . . . 6 ⊢ (((deg1‘𝑅)‘𝑎) ∈ {-∞} → ∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏)
29	18, 28	jaoi 871	. . . . 5 ⊢ ((((deg1‘𝑅)‘𝑎) ∈ ℕ0 ∨ ((deg1‘𝑅)‘𝑎) ∈ {-∞}) → ∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏)
30	12, 29	sylbi 220	. . . 4 ⊢ (((deg1‘𝑅)‘𝑎) ∈ (ℕ0 ∪ {-∞}) → ∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏)
31	11, 30	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → ∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏)
32		breq2 5106	. . . . . . . . . . 11 ⊢ (𝑐 = 0 → (((deg1‘𝑅)‘𝑎) < 𝑐 ↔ ((deg1‘𝑅)‘𝑎) < 0))
33	32	imbi1d 344	. . . . . . . . . 10 ⊢ (𝑐 = 0 → ((((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ (((deg1‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼)))
34	33	ralbidv 3185	. . . . . . . . 9 ⊢ (𝑐 = 0 → (∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼)))
35	34	imbi2d 343	. . . . . . . 8 ⊢ (𝑐 = 0 → ((𝜑 → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼)) ↔ (𝜑 → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼))))
36		breq2 5106	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑏 → (((deg1‘𝑅)‘𝑎) < 𝑐 ↔ ((deg1‘𝑅)‘𝑎) < 𝑏))
37	36	imbi1d 344	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → ((((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
38	37	ralbidv 3185	. . . . . . . . 9 ⊢ (𝑐 = 𝑏 → (∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
39	38	imbi2d 343	. . . . . . . 8 ⊢ (𝑐 = 𝑏 → ((𝜑 → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼)) ↔ (𝜑 → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))))
40		breq2 5106	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑏 + 1) → (((deg1‘𝑅)‘𝑎) < 𝑐 ↔ ((deg1‘𝑅)‘𝑎) < (𝑏 + 1)))
41	40	imbi1d 344	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑏 + 1) → ((((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ (((deg1‘𝑅)‘𝑎) < (𝑏 + 1) → 𝑎 ∈ 𝐼)))
42	41	ralbidv 3185	. . . . . . . . . 10 ⊢ (𝑐 = (𝑏 + 1) → (∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < (𝑏 + 1) → 𝑎 ∈ 𝐼)))
43		fveq2 6873	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑑 → ((deg1‘𝑅)‘𝑎) = ((deg1‘𝑅)‘𝑑))
44	43	breq1d 5112	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑑 → (((deg1‘𝑅)‘𝑎) < (𝑏 + 1) ↔ ((deg1‘𝑅)‘𝑑) < (𝑏 + 1)))
45		eleq1 2848	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑑 → (𝑎 ∈ 𝐼 ↔ 𝑑 ∈ 𝐼))
46	44, 45	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑑 → ((((deg1‘𝑅)‘𝑎) < (𝑏 + 1) → 𝑎 ∈ 𝐼) ↔ (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼)))
47	46	cbvralvw 3240	. . . . . . . . . 10 ⊢ (∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < (𝑏 + 1) → 𝑎 ∈ 𝐼) ↔ ∀𝑑 ∈ 𝐽 (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))
48	42, 47	bitrdi 290	. . . . . . . . 9 ⊢ (𝑐 = (𝑏 + 1) → (∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑐 → 𝑎 ∈ 𝐼) ↔ ∀𝑑 ∈ 𝐽 (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼)))
49	48	imbi2d 343	. . . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → 𝑅 ∈ Ring)
52		eqid 2760	. . . . . . . . . . . 12 ⊢ (0g‘𝑃) = (0g‘𝑃)
53	8, 9, 52, 3	deg1lt0 26370	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑃)) → (((deg1‘𝑅)‘𝑎) < 0 ↔ 𝑎 = (0g‘𝑃)))
54	51, 7, 53	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (((deg1‘𝑅)‘𝑎) < 0 ↔ 𝑎 = (0g‘𝑃)))
55	9	ply1ring 22526	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
56	50, 55	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ Ring)
57		hbtlem3.i	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ 𝑈)
58	4, 52	lidl0cl 21460	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (0g‘𝑃) ∈ 𝐼)
59	56, 57, 58	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (0g‘𝑃) ∈ 𝐼)
60		eleq1a 2855	. . . . . . . . . . . 12 ⊢ ((0g‘𝑃) ∈ 𝐼 → (𝑎 = (0g‘𝑃) → 𝑎 ∈ 𝐼))
61	59, 60	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑎 = (0g‘𝑃) → 𝑎 ∈ 𝐼))
62	61	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (𝑎 = (0g‘𝑃) → 𝑎 ∈ 𝐼))
63	54, 62	sylbid 243	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (((deg1‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼))
64	63	ralrimiva 3154	. . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 0 → 𝑎 ∈ 𝐼))
65	6	3ad2ant2 1152	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → 𝐽 ⊆ (Base‘𝑃))
66	65	sselda 3930	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → 𝑑 ∈ (Base‘𝑃))
67	8, 9, 3	deg1cl 26362	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ (Base‘𝑃) → ((deg1‘𝑅)‘𝑑) ∈ (ℕ0 ∪ {-∞}))
68	66, 67	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → ((deg1‘𝑅)‘𝑑) ∈ (ℕ0 ∪ {-∞}))
69		simpl1 1210	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → 𝑏 ∈ ℕ0)
70	69	nn0zd 12687	. . . . . . . . . . . . 13 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → 𝑏 ∈ ℤ)
71		degltp1le 26352	. . . . . . . . . . . . 13 ⊢ ((((deg1‘𝑅)‘𝑑) ∈ (ℕ0 ∪ {-∞}) ∧ 𝑏 ∈ ℤ) → (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) ↔ ((deg1‘𝑅)‘𝑑) ≤ 𝑏))
72	68, 70, 71	syl2anc 596	. . . . . . . . . . . 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 6873	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑏 → ((𝑆‘𝐽)‘𝑥) = ((𝑆‘𝐽)‘𝑏))
75		fveq2 6873	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑏 → ((𝑆‘𝐼)‘𝑥) = ((𝑆‘𝐼)‘𝑏))
76	74, 75	sseq12d 3963	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑏 → (((𝑆‘𝐽)‘𝑥) ⊆ ((𝑆‘𝐼)‘𝑥) ↔ ((𝑆‘𝐽)‘𝑏) ⊆ ((𝑆‘𝐼)‘𝑏)))
77	76	rspcva 3574	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 ((𝑆‘𝐽)‘𝑥) ⊆ ((𝑆‘𝐼)‘𝑥)) → ((𝑆‘𝐽)‘𝑏) ⊆ ((𝑆‘𝐼)‘𝑏))
78	73, 77	sylan2 605	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → ((𝑆‘𝐽)‘𝑏) ⊆ ((𝑆‘𝐼)‘𝑏))
79	50	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → 𝑅 ∈ Ring)
80	2	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → 𝐽 ∈ 𝑈)
81		simpl 488	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → 𝑏 ∈ ℕ0)
82		hbtlem.s	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑆 = (ldgIdlSeq‘𝑅)
83	9, 4, 82, 8	hbtlem1 44068	. . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → ((𝑆‘𝐽)‘𝑏) = {𝑐 ∣ ∃𝑒 ∈ 𝐽 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
85	57	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → 𝐼 ∈ 𝑈)
86	9, 4, 82, 8	hbtlem1 44068	. . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑) → ((𝑆‘𝐼)‘𝑏) = {𝑐 ∣ ∃𝑒 ∈ 𝐼 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
88	78, 84, 87	3sstr3d 3984	. . . . . . . . . . . . . . . . 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 1150	. . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏) → 𝑑 ∈ 𝐽)
92		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏) → ((deg1‘𝑅)‘𝑑) ≤ 𝑏)
93		eqidd 2761	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏) → ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑑)‘𝑏))
94		fveq2 6873	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑒 = 𝑑 → ((deg1‘𝑅)‘𝑒) = ((deg1‘𝑅)‘𝑑))
95	94	breq1d 5112	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = 𝑑 → (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ↔ ((deg1‘𝑅)‘𝑑) ≤ 𝑏))
96		fveq2 6873	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑒 = 𝑑 → (coe1‘𝑒) = (coe1‘𝑑))
97	96	fveq1d 6875	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑒 = 𝑑 → ((coe1‘𝑒)‘𝑏) = ((coe1‘𝑑)‘𝑏))
98	97	eqeq2d 2771	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = 𝑑 → (((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏) ↔ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑑)‘𝑏)))
99	95, 98	anbi12d 644	. . . . . . . . . . . . . . . . . . 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 3576	. . . . . . . . . . . . . . . . . 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 850	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏) → ∃𝑒 ∈ 𝐽 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))
102		fvex 6886	. . . . . . . . . . . . . . . . . 18 ⊢ ((coe1‘𝑑)‘𝑏) ∈ V
103		eqeq1 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = ((coe1‘𝑑)‘𝑏) → (𝑐 = ((coe1‘𝑒)‘𝑏) ↔ ((coe1‘𝑑)‘𝑏) = ((coe1‘𝑒)‘𝑏)))
104	103	anbi2d 642	. . . . . . . . . . . . . . . . . . 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 3186	. . . . . . . . . . . . . . . . . 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 3632	. . . . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . . . 16 ⊢ ((𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏) → ((coe1‘𝑑)‘𝑏) ∈ {𝑐 ∣ ∃𝑒 ∈ 𝐽 (((deg1‘𝑅)‘𝑒) ≤ 𝑏 ∧ 𝑐 = ((coe1‘𝑒)‘𝑏))})
108	107	adantl 487	. . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . 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 3186	. . . . . . . . . . . . . . . 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 3632	. . . . . . . . . . . . . . 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 1232	. . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . 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 20425	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Grp)
115	113, 114	syl 18	. . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . . . . . 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 21451	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ (Base‘𝑃))
120	57, 119	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐼 ⊆ (Base‘𝑃))
121	112, 120	syl 18	. . . . . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . . . 19 ⊢ (+g‘𝑃) = (+g‘𝑃)
125		eqid 2760	. . . . . . . . . . . . . . . . . . 19 ⊢ (-g‘𝑃) = (-g‘𝑃)
126	3, 124, 125	grpnpcan 19203	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ Grp ∧ 𝑑 ∈ (Base‘𝑃) ∧ 𝑒 ∈ (Base‘𝑃)) → ((𝑑(-g‘𝑃)𝑒)(+g‘𝑃)𝑒) = 𝑑)
127	115, 118, 123, 126	syl3anc 1398	. . . . . . . . . . . . . . . . 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 1152	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → 𝐼 ∈ 𝑈)
129	128	ad2antrr 739	. . . . . . . . . . . . . . . . . 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 1231	. . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (coe1‘𝑑) = (coe1‘𝑑)
135		eqid 2760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (coe1‘𝑒) = (coe1‘𝑒)
136		simprrr 794	. . . . . . . . . . . . . . . . . . . 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 26389	. . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . 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 1152	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → 𝐼 ⊆ 𝐽)
140	139	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . . . . . . . . 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 21464	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑃 ∈ Ring ∧ 𝐽 ∈ 𝑈) ∧ (𝑑 ∈ 𝐽 ∧ 𝑒 ∈ 𝐽)) → (𝑑(-g‘𝑃)𝑒) ∈ 𝐽)
143	113, 138, 117, 141, 142	syl22anc 852	. . . . . . . . . . . . . . . . . . . 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 1233	. . . . . . . . . . . . . . . . . . . 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 6873	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = (𝑑(-g‘𝑃)𝑒) → ((deg1‘𝑅)‘𝑎) = ((deg1‘𝑅)‘(𝑑(-g‘𝑃)𝑒)))
146	145	breq1d 5112	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = (𝑑(-g‘𝑃)𝑒) → (((deg1‘𝑅)‘𝑎) < 𝑏 ↔ ((deg1‘𝑅)‘(𝑑(-g‘𝑃)𝑒)) < 𝑏))
147		eleq1 2848	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = (𝑑(-g‘𝑃)𝑒) → (𝑎 ∈ 𝐼 ↔ (𝑑(-g‘𝑃)𝑒) ∈ 𝐼))
148	146, 147	imbi12d 347	. . . . . . . . . . . . . . . . . . . . 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 3574	. . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . . . 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 21461	. . . . . . . . . . . . . . . . . 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 852	. . . . . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . . . 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 3164	. . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . . 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 16	. . . . . . . . . . . . 13 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ (𝑑 ∈ 𝐽 ∧ ((deg1‘𝑅)‘𝑑) ≤ 𝑏)) → 𝑑 ∈ 𝐼)
158	157	expr 462	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → (((deg1‘𝑅)‘𝑑) ≤ 𝑏 → 𝑑 ∈ 𝐼))
159	72, 158	sylbid 243	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) ∧ 𝑑 ∈ 𝐽) → (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))
160	159	ralrimiva 3154	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)) → ∀𝑑 ∈ 𝐽 (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))
161	160	3exp 1137	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ0 → (𝜑 → (∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼) → ∀𝑑 ∈ 𝐽 (((deg1‘𝑅)‘𝑑) < (𝑏 + 1) → 𝑑 ∈ 𝐼))))
162	161	a2d 30	. . . . . . . 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 12763	. . . . . . 7 ⊢ (𝑏 ∈ ℕ0 → (𝜑 → ∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
164		rsp 3250	. . . . . . 7 ⊢ (∀𝑎 ∈ 𝐽 (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼) → (𝑎 ∈ 𝐽 → (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
165	163, 164	syl6com 38	. . . . . 6 ⊢ (𝜑 → (𝑏 ∈ ℕ0 → (𝑎 ∈ 𝐽 → (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))))
166	165	com23 87	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐽 → (𝑏 ∈ ℕ0 → (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))))
167	166	imp 412	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (𝑏 ∈ ℕ0 → (((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼)))
168	167	rexlimdv 3161	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → (∃𝑏 ∈ ℕ0 ((deg1‘𝑅)‘𝑎) < 𝑏 → 𝑎 ∈ 𝐼))
169	31, 168	mpd 16	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐽) → 𝑎 ∈ 𝐼)
170	1, 169	eqelssd 3951	1 ⊢ (𝜑 → 𝐼 = 𝐽)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086   ∪ cun 3896   ⊆ wss 3898  {csn 4583   class class class wbr 5102  ‘cfv 6527  (class class class)co 7408  ℝcr 11170  0cc0 11171  1c1 11172   + caddc 11174  -∞cmnf 11312   < clt 11314   ≤ cle 11315  ℕcn 12304  ℕ₀cn0 12575  ℤcz 12662  Basecbs 17348  +_gcplusg 17389  0_gc0g 17571  Grpcgrp 19105  -_gcsg 19107  Ringcrg 20420  LIdealclidl 21445  Poly₁cpl1 22456  coe₁cco1 22457  deg₁cdg1 26333  ldgIdlSeqcldgis 44066

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248  ax-pre-sup 11249  ax-addf 11250

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7676  df-ofr 7677  df-om 7861  df-1st 7984  df-2nd 7985  df-supp 8156  df-tpos 8221  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-er 8695  df-map 8827  df-pm 8828  df-ixp 8904  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-fsupp 9332  df-sup 9412  df-oi 9482  df-card 9991  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-nn 12305  df-2 12374  df-3 12375  df-4 12376  df-5 12377  df-6 12378  df-7 12379  df-8 12380  df-9 12381  df-n0 12576  df-z 12663  df-dec 12784  df-uz 12935  df-fz 13609  df-fzo 13757  df-seq 14113  df-hash 14442  df-struct 17286  df-sets 17303  df-slot 17321  df-ndx 17333  df-base 17349  df-ress 17370  df-plusg 17402  df-mulr 17403  df-starv 17404  df-sca 17405  df-vsca 17406  df-ip 17407  df-tset 17408  df-ple 17409  df-ds 17411  df-unif 17412  df-hom 17413  df-cco 17414  df-0g 17573  df-gsum 17574  df-prds 17579  df-pws 17581  df-mre 17717  df-mrc 17718  df-acs 17720  df-mgm 18777  df-sgrp 18869  df-mnd 18885  df-mhm 18939  df-submnd 18940  df-grp 19108  df-minusg 19109  df-sbg 19110  df-mulg 19239  df-subg 19294  df-ghm 19389  df-cntz 19492  df-cmn 19957  df-abl 19958  df-mgp 20322  df-rng 20336  df-ur 20369  df-ring 20422  df-cring 20423  df-oppr 20528  df-dvdsr 20548  df-unit 20549  df-invr 20579  df-subrng 20759  df-subrg 20783  df-rlreg 20907  df-lmod 21098  df-lss 21168  df-sra 21409  df-rgmod 21410  df-lidl 21447  df-cnfld 21640  df-psr 22178  df-mpl 22180  df-opsr 22182  df-psr1 22459  df-ply1 22461  df-coe1 22462  df-mdeg 26334  df-deg1 26335  df-ldgis 44067

This theorem is used by:  hbt  44075