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Theorem ig1pdvds 25694

Description: The monic generator of an ideal divides all elements of the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Proof shortened by AV, 25-Sep-2020.)

**Hypotheses**
Ref	Expression
ig1pval.p	⊢ 𝑃 = (Poly₁‘𝑅)
ig1pval.g	⊢ 𝐺 = (idlGen_1p‘𝑅)
ig1pcl.u	⊢ 𝑈 = (LIdeal‘𝑃)
ig1pdvds.d	⊢ ∥ = (∥_r‘𝑃)

**Assertion**
Ref	Expression
ig1pdvds	⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) → (𝐺‘𝐼) ∥ 𝑋)

**Proof of Theorem ig1pdvds**
Step	Hyp	Ref	Expression
1		drngring 20364	. . . . . . 7 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
2		ig1pval.p	. . . . . . . 8 ⊢ 𝑃 = (Poly₁‘𝑅)
3	2	ply1ring 21770	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
4	1, 3	syl 17	. . . . . 6 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ Ring)
5	4	3ad2ant1 1134	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) → 𝑃 ∈ Ring)
6		eqid 2733	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘𝑃)
7		ig1pcl.u	. . . . . . . 8 ⊢ 𝑈 = (LIdeal‘𝑃)
8	6, 7	lidlss 20833	. . . . . . 7 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ (Base‘𝑃))
9	8	3ad2ant2 1135	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) → 𝐼 ⊆ (Base‘𝑃))
10		ig1pval.g	. . . . . . . 8 ⊢ 𝐺 = (idlGen_1p‘𝑅)
11	2, 10, 7	ig1pcl 25693	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → (𝐺‘𝐼) ∈ 𝐼)
12	11	3adant3 1133	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) → (𝐺‘𝐼) ∈ 𝐼)
13	9, 12	sseldd 3984	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) → (𝐺‘𝐼) ∈ (Base‘𝑃))
14		ig1pdvds.d	. . . . . 6 ⊢ ∥ = (∥_r‘𝑃)
15		eqid 2733	. . . . . 6 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
16	6, 14, 15	dvdsr01 20185	. . . . 5 ⊢ ((𝑃 ∈ Ring ∧ (𝐺‘𝐼) ∈ (Base‘𝑃)) → (𝐺‘𝐼) ∥ (0_g‘𝑃))
17	5, 13, 16	syl2anc 585	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) → (𝐺‘𝐼) ∥ (0_g‘𝑃))
18	17	adantr 482	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 = {(0_g‘𝑃)}) → (𝐺‘𝐼) ∥ (0_g‘𝑃))
19		eleq2 2823	. . . . . 6 ⊢ (𝐼 = {(0_g‘𝑃)} → (𝑋 ∈ 𝐼 ↔ 𝑋 ∈ {(0_g‘𝑃)}))
20	19	biimpac 480	. . . . 5 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝐼 = {(0_g‘𝑃)}) → 𝑋 ∈ {(0_g‘𝑃)})
21	20	3ad2antl3 1188	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 = {(0_g‘𝑃)}) → 𝑋 ∈ {(0_g‘𝑃)})
22		elsni 4646	. . . 4 ⊢ (𝑋 ∈ {(0_g‘𝑃)} → 𝑋 = (0_g‘𝑃))
23	21, 22	syl 17	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 = {(0_g‘𝑃)}) → 𝑋 = (0_g‘𝑃))
24	18, 23	breqtrrd 5177	. 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 = {(0_g‘𝑃)}) → (𝐺‘𝐼) ∥ 𝑋)
25		simpl1 1192	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → 𝑅 ∈ DivRing)
26	25, 1	syl 17	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → 𝑅 ∈ Ring)
27		simpl2 1193	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → 𝐼 ∈ 𝑈)
28	27, 8	syl 17	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → 𝐼 ⊆ (Base‘𝑃))
29		simpl3 1194	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → 𝑋 ∈ 𝐼)
30	28, 29	sseldd 3984	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → 𝑋 ∈ (Base‘𝑃))
31		simpr 486	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → 𝐼 ≠ {(0_g‘𝑃)})
32		eqid 2733	. . . . . . . . . . 11 ⊢ ( deg₁ ‘𝑅) = ( deg₁ ‘𝑅)
33		eqid 2733	. . . . . . . . . . 11 ⊢ (Monic_1p‘𝑅) = (Monic_1p‘𝑅)
34	2, 10, 15, 7, 32, 33	ig1pval3 25692	. . . . . . . . . 10 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝐼 ≠ {(0_g‘𝑃)}) → ((𝐺‘𝐼) ∈ 𝐼 ∧ (𝐺‘𝐼) ∈ (Monic_1p‘𝑅) ∧ (( deg₁ ‘𝑅)‘(𝐺‘𝐼)) = inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < )))
35	25, 27, 31, 34	syl3anc 1372	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → ((𝐺‘𝐼) ∈ 𝐼 ∧ (𝐺‘𝐼) ∈ (Monic_1p‘𝑅) ∧ (( deg₁ ‘𝑅)‘(𝐺‘𝐼)) = inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < )))
36	35	simp2d 1144	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝐺‘𝐼) ∈ (Monic_1p‘𝑅))
37		eqid 2733	. . . . . . . . 9 ⊢ (Unic_1p‘𝑅) = (Unic_1p‘𝑅)
38	37, 33	mon1puc1p 25668	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘𝐼) ∈ (Monic_1p‘𝑅)) → (𝐺‘𝐼) ∈ (Unic_1p‘𝑅))
39	26, 36, 38	syl2anc 585	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝐺‘𝐼) ∈ (Unic_1p‘𝑅))
40		eqid 2733	. . . . . . . 8 ⊢ (rem_1p‘𝑅) = (rem_1p‘𝑅)
41	40, 2, 6, 37, 32	r1pdeglt 25676	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑃) ∧ (𝐺‘𝐼) ∈ (Unic_1p‘𝑅)) → (( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼))) < (( deg₁ ‘𝑅)‘(𝐺‘𝐼)))
42	26, 30, 39, 41	syl3anc 1372	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼))) < (( deg₁ ‘𝑅)‘(𝐺‘𝐼)))
43	35	simp3d 1145	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (( deg₁ ‘𝑅)‘(𝐺‘𝐼)) = inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < ))
44	42, 43	breqtrd 5175	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼))) < inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < ))
45	32, 2, 6	deg1xrf 25599	. . . . . . 7 ⊢ ( deg₁ ‘𝑅):(Base‘𝑃)⟶ℝ^*
46	35	simp1d 1143	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝐺‘𝐼) ∈ 𝐼)
47	28, 46	sseldd 3984	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝐺‘𝐼) ∈ (Base‘𝑃))
48		eqid 2733	. . . . . . . . . . 11 ⊢ (quot_1p‘𝑅) = (quot_1p‘𝑅)
49		eqid 2733	. . . . . . . . . . 11 ⊢ (._r‘𝑃) = (._r‘𝑃)
50		eqid 2733	. . . . . . . . . . 11 ⊢ (-_g‘𝑃) = (-_g‘𝑃)
51	40, 2, 6, 48, 49, 50	r1pval 25674	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (Base‘𝑃) ∧ (𝐺‘𝐼) ∈ (Base‘𝑃)) → (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) = (𝑋(-_g‘𝑃)((𝑋(quot_1p‘𝑅)(𝐺‘𝐼))(._r‘𝑃)(𝐺‘𝐼))))
52	30, 47, 51	syl2anc 585	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) = (𝑋(-_g‘𝑃)((𝑋(quot_1p‘𝑅)(𝐺‘𝐼))(._r‘𝑃)(𝐺‘𝐼))))
53	26, 3	syl 17	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → 𝑃 ∈ Ring)
54	48, 2, 6, 37	q1pcl 25673	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑃) ∧ (𝐺‘𝐼) ∈ (Unic_1p‘𝑅)) → (𝑋(quot_1p‘𝑅)(𝐺‘𝐼)) ∈ (Base‘𝑃))
55	26, 30, 39, 54	syl3anc 1372	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝑋(quot_1p‘𝑅)(𝐺‘𝐼)) ∈ (Base‘𝑃))
56	7, 6, 49	lidlmcl 20840	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ ((𝑋(quot_1p‘𝑅)(𝐺‘𝐼)) ∈ (Base‘𝑃) ∧ (𝐺‘𝐼) ∈ 𝐼)) → ((𝑋(quot_1p‘𝑅)(𝐺‘𝐼))(._r‘𝑃)(𝐺‘𝐼)) ∈ 𝐼)
57	53, 27, 55, 46, 56	syl22anc 838	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → ((𝑋(quot_1p‘𝑅)(𝐺‘𝐼))(._r‘𝑃)(𝐺‘𝐼)) ∈ 𝐼)
58	7, 50	lidlsubcl 20839	. . . . . . . . . 10 ⊢ (((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈) ∧ (𝑋 ∈ 𝐼 ∧ ((𝑋(quot_1p‘𝑅)(𝐺‘𝐼))(._r‘𝑃)(𝐺‘𝐼)) ∈ 𝐼)) → (𝑋(-_g‘𝑃)((𝑋(quot_1p‘𝑅)(𝐺‘𝐼))(._r‘𝑃)(𝐺‘𝐼))) ∈ 𝐼)
59	53, 27, 29, 57, 58	syl22anc 838	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝑋(-_g‘𝑃)((𝑋(quot_1p‘𝑅)(𝐺‘𝐼))(._r‘𝑃)(𝐺‘𝐼))) ∈ 𝐼)
60	52, 59	eqeltrd 2834	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ∈ 𝐼)
61	28, 60	sseldd 3984	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ∈ (Base‘𝑃))
62		ffvelcdm 7084	. . . . . . 7 ⊢ ((( deg₁ ‘𝑅):(Base‘𝑃)⟶ℝ^* ∧ (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ∈ (Base‘𝑃)) → (( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼))) ∈ ℝ^*)
63	45, 61, 62	sylancr 588	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼))) ∈ ℝ^*)
64	28	ssdifd 4141	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝐼 ∖ {(0_g‘𝑃)}) ⊆ ((Base‘𝑃) ∖ {(0_g‘𝑃)}))
65		imass2 6102	. . . . . . . . . 10 ⊢ ((𝐼 ∖ {(0_g‘𝑃)}) ⊆ ((Base‘𝑃) ∖ {(0_g‘𝑃)}) → (( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})) ⊆ (( deg₁ ‘𝑅) “ ((Base‘𝑃) ∖ {(0_g‘𝑃)})))
66	64, 65	syl 17	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})) ⊆ (( deg₁ ‘𝑅) “ ((Base‘𝑃) ∖ {(0_g‘𝑃)})))
67	32, 2, 15, 6	deg1n0ima 25607	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (( deg₁ ‘𝑅) “ ((Base‘𝑃) ∖ {(0_g‘𝑃)})) ⊆ ℕ₀)
68	26, 67	syl 17	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (( deg₁ ‘𝑅) “ ((Base‘𝑃) ∖ {(0_g‘𝑃)})) ⊆ ℕ₀)
69		nn0uz 12864	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
70	68, 69	sseqtrdi 4033	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (( deg₁ ‘𝑅) “ ((Base‘𝑃) ∖ {(0_g‘𝑃)})) ⊆ (ℤ_≥‘0))
71	66, 70	sstrd 3993	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})) ⊆ (ℤ_≥‘0))
72		uzssz 12843	. . . . . . . . 9 ⊢ (ℤ_≥‘0) ⊆ ℤ
73		zssre 12565	. . . . . . . . . 10 ⊢ ℤ ⊆ ℝ
74		ressxr 11258	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ^*
75	73, 74	sstri 3992	. . . . . . . . 9 ⊢ ℤ ⊆ ℝ^*
76	72, 75	sstri 3992	. . . . . . . 8 ⊢ (ℤ_≥‘0) ⊆ ℝ^*
77	71, 76	sstrdi 3995	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})) ⊆ ℝ^*)
78	7, 15	lidl0cl 20835	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (0_g‘𝑃) ∈ 𝐼)
79	53, 27, 78	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (0_g‘𝑃) ∈ 𝐼)
80	79	snssd 4813	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → {(0_g‘𝑃)} ⊆ 𝐼)
81	31	necomd 2997	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → {(0_g‘𝑃)} ≠ 𝐼)
82		pssdifn0 4366	. . . . . . . . . 10 ⊢ (({(0_g‘𝑃)} ⊆ 𝐼 ∧ {(0_g‘𝑃)} ≠ 𝐼) → (𝐼 ∖ {(0_g‘𝑃)}) ≠ ∅)
83	80, 81, 82	syl2anc 585	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝐼 ∖ {(0_g‘𝑃)}) ≠ ∅)
84		ffn 6718	. . . . . . . . . . . 12 ⊢ (( deg₁ ‘𝑅):(Base‘𝑃)⟶ℝ^* → ( deg₁ ‘𝑅) Fn (Base‘𝑃))
85	45, 84	ax-mp 5	. . . . . . . . . . 11 ⊢ ( deg₁ ‘𝑅) Fn (Base‘𝑃)
86	28	ssdifssd 4143	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝐼 ∖ {(0_g‘𝑃)}) ⊆ (Base‘𝑃))
87		fnimaeq0 6684	. . . . . . . . . . 11 ⊢ ((( deg₁ ‘𝑅) Fn (Base‘𝑃) ∧ (𝐼 ∖ {(0_g‘𝑃)}) ⊆ (Base‘𝑃)) → ((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})) = ∅ ↔ (𝐼 ∖ {(0_g‘𝑃)}) = ∅))
88	85, 86, 87	sylancr 588	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → ((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})) = ∅ ↔ (𝐼 ∖ {(0_g‘𝑃)}) = ∅))
89	88	necon3bid 2986	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → ((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})) ≠ ∅ ↔ (𝐼 ∖ {(0_g‘𝑃)}) ≠ ∅))
90	83, 89	mpbird 257	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})) ≠ ∅)
91		infssuzcl 12916	. . . . . . . 8 ⊢ (((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})) ⊆ (ℤ_≥‘0) ∧ (( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})) ≠ ∅) → inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < ) ∈ (( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})))
92	71, 90, 91	syl2anc 585	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < ) ∈ (( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})))
93	77, 92	sseldd 3984	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < ) ∈ ℝ^*)
94		xrltnle 11281	. . . . . 6 ⊢ (((( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼))) ∈ ℝ^* ∧ inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < ) ∈ ℝ^*) → ((( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼))) < inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < ) ↔ ¬ inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < ) ≤ (( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼)))))
95	63, 93, 94	syl2anc 585	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → ((( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼))) < inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < ) ↔ ¬ inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < ) ≤ (( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼)))))
96	44, 95	mpbid 231	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → ¬ inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < ) ≤ (( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼))))
97	71	adantr 482	. . . . . . 7 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) ∧ (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ≠ (0_g‘𝑃)) → (( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})) ⊆ (ℤ_≥‘0))
98	60	adantr 482	. . . . . . . . 9 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) ∧ (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ≠ (0_g‘𝑃)) → (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ∈ 𝐼)
99		simpr 486	. . . . . . . . 9 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) ∧ (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ≠ (0_g‘𝑃)) → (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ≠ (0_g‘𝑃))
100		eldifsn 4791	. . . . . . . . 9 ⊢ ((𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ∈ (𝐼 ∖ {(0_g‘𝑃)}) ↔ ((𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ∈ 𝐼 ∧ (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ≠ (0_g‘𝑃)))
101	98, 99, 100	sylanbrc 584	. . . . . . . 8 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) ∧ (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ≠ (0_g‘𝑃)) → (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ∈ (𝐼 ∖ {(0_g‘𝑃)}))
102		fnfvima 7235	. . . . . . . 8 ⊢ ((( deg₁ ‘𝑅) Fn (Base‘𝑃) ∧ (𝐼 ∖ {(0_g‘𝑃)}) ⊆ (Base‘𝑃) ∧ (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ∈ (𝐼 ∖ {(0_g‘𝑃)})) → (( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼))) ∈ (( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})))
103	85, 86, 101, 102	mp3an2ani 1469	. . . . . . 7 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) ∧ (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ≠ (0_g‘𝑃)) → (( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼))) ∈ (( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})))
104		infssuzle 12915	. . . . . . 7 ⊢ (((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})) ⊆ (ℤ_≥‘0) ∧ (( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼))) ∈ (( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)}))) → inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < ) ≤ (( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼))))
105	97, 103, 104	syl2anc 585	. . . . . 6 ⊢ ((((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) ∧ (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ≠ (0_g‘𝑃)) → inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < ) ≤ (( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼))))
106	105	ex 414	. . . . 5 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → ((𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ≠ (0_g‘𝑃) → inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < ) ≤ (( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼)))))
107	106	necon1bd 2959	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (¬ inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < ) ≤ (( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼))) → (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) = (0_g‘𝑃)))
108	96, 107	mpd 15	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) = (0_g‘𝑃))
109	2, 14, 6, 37, 15, 40	dvdsr1p 25679	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑃) ∧ (𝐺‘𝐼) ∈ (Unic_1p‘𝑅)) → ((𝐺‘𝐼) ∥ 𝑋 ↔ (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) = (0_g‘𝑃)))
110	26, 30, 39, 109	syl3anc 1372	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → ((𝐺‘𝐼) ∥ 𝑋 ↔ (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) = (0_g‘𝑃)))
111	108, 110	mpbird 257	. 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝐺‘𝐼) ∥ 𝑋)
112	24, 111	pm2.61dane 3030	1 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) → (𝐺‘𝐼) ∥ 𝑋)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3946 ⊆ wss 3949 ∅c0 4323 {csn 4629 class class class wbr 5149 “ cima 5680 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 infcinf 9436 ℝcr 11109 0cc0 11110 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 Basecbs 17144 ._rcmulr 17198 0_gc0g 17385 -_gcsg 18821 Ringcrg 20056 ∥_rcdsr 20168 DivRingcdr 20357 LIdealclidl 20783 Poly₁cpl1 21701 deg₁ cdg1 25569 Monic_1pcmn1 25643 Unic_1pcuc1p 25644 quot_1pcq1p 25645 rem_1pcr1p 25646 idlGen_1pcig1p 25647

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-ofr 7671 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-fzo 13628 df-seq 13967 df-hash 14291 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-0g 17387 df-gsum 17388 df-prds 17393 df-pws 17395 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-submnd 18672 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mulg 18951 df-subg 19003 df-ghm 19090 df-cntz 19181 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-oppr 20150 df-dvdsr 20171 df-unit 20172 df-invr 20202 df-subrg 20317 df-drng 20359 df-lmod 20473 df-lss 20543 df-sra 20785 df-rgmod 20786 df-lidl 20787 df-rlreg 20899 df-cnfld 20945 df-ascl 21410 df-psr 21462 df-mvr 21463 df-mpl 21464 df-opsr 21466 df-psr1 21704 df-vr1 21705 df-ply1 21706 df-coe1 21707 df-mdeg 25570 df-deg1 25571 df-mon1 25648 df-uc1p 25649 df-q1p 25650 df-r1p 25651 df-ig1p 25652

This theorem is referenced by: ig1prsp 25695

W3C validator