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Theorem ig1pdvds 25701

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 20368	. . . . . . 7 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
2		ig1pval.p	. . . . . . . 8 ⊢ 𝑃 = (Poly₁‘𝑅)
3	2	ply1ring 21777	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
4	1, 3	syl 17	. . . . . 6 ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ Ring)
5	4	3ad2ant1 1133	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) → 𝑃 ∈ Ring)
6		eqid 2732	. . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘𝑃)
7		ig1pcl.u	. . . . . . . 8 ⊢ 𝑈 = (LIdeal‘𝑃)
8	6, 7	lidlss 20839	. . . . . . 7 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ⊆ (Base‘𝑃))
9	8	3ad2ant2 1134	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) → 𝐼 ⊆ (Base‘𝑃))
10		ig1pval.g	. . . . . . . 8 ⊢ 𝐺 = (idlGen_1p‘𝑅)
11	2, 10, 7	ig1pcl 25700	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈) → (𝐺‘𝐼) ∈ 𝐼)
12	11	3adant3 1132	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) → (𝐺‘𝐼) ∈ 𝐼)
13	9, 12	sseldd 3983	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) → (𝐺‘𝐼) ∈ (Base‘𝑃))
14		ig1pdvds.d	. . . . . 6 ⊢ ∥ = (∥_r‘𝑃)
15		eqid 2732	. . . . . 6 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
16	6, 14, 15	dvdsr01 20189	. . . . 5 ⊢ ((𝑃 ∈ Ring ∧ (𝐺‘𝐼) ∈ (Base‘𝑃)) → (𝐺‘𝐼) ∥ (0_g‘𝑃))
17	5, 13, 16	syl2anc 584	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) → (𝐺‘𝐼) ∥ (0_g‘𝑃))
18	17	adantr 481	. . 3 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 = {(0_g‘𝑃)}) → (𝐺‘𝐼) ∥ (0_g‘𝑃))
19		eleq2 2822	. . . . . 6 ⊢ (𝐼 = {(0_g‘𝑃)} → (𝑋 ∈ 𝐼 ↔ 𝑋 ∈ {(0_g‘𝑃)}))
20	19	biimpac 479	. . . . 5 ⊢ ((𝑋 ∈ 𝐼 ∧ 𝐼 = {(0_g‘𝑃)}) → 𝑋 ∈ {(0_g‘𝑃)})
21	20	3ad2antl3 1187	. . . 4 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 = {(0_g‘𝑃)}) → 𝑋 ∈ {(0_g‘𝑃)})
22		elsni 4645	. . . 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 5176	. 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 = {(0_g‘𝑃)}) → (𝐺‘𝐼) ∥ 𝑋)
25		simpl1 1191	. . . . . . . 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 1192	. . . . . . . . 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 1193	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → 𝑋 ∈ 𝐼)
30	28, 29	sseldd 3983	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → 𝑋 ∈ (Base‘𝑃))
31		simpr 485	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → 𝐼 ≠ {(0_g‘𝑃)})
32		eqid 2732	. . . . . . . . . . 11 ⊢ ( deg₁ ‘𝑅) = ( deg₁ ‘𝑅)
33		eqid 2732	. . . . . . . . . . 11 ⊢ (Monic_1p‘𝑅) = (Monic_1p‘𝑅)
34	2, 10, 15, 7, 32, 33	ig1pval3 25699	. . . . . . . . . 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 1371	. . . . . . . . 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 1143	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝐺‘𝐼) ∈ (Monic_1p‘𝑅))
37		eqid 2732	. . . . . . . . 9 ⊢ (Unic_1p‘𝑅) = (Unic_1p‘𝑅)
38	37, 33	mon1puc1p 25675	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘𝐼) ∈ (Monic_1p‘𝑅)) → (𝐺‘𝐼) ∈ (Unic_1p‘𝑅))
39	26, 36, 38	syl2anc 584	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝐺‘𝐼) ∈ (Unic_1p‘𝑅))
40		eqid 2732	. . . . . . . 8 ⊢ (rem_1p‘𝑅) = (rem_1p‘𝑅)
41	40, 2, 6, 37, 32	r1pdeglt 25683	. . . . . . 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 1371	. . . . . 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 1144	. . . . . 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 5174	. . . . 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 25606	. . . . . . 7 ⊢ ( deg₁ ‘𝑅):(Base‘𝑃)⟶ℝ^*
46	35	simp1d 1142	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝐺‘𝐼) ∈ 𝐼)
47	28, 46	sseldd 3983	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝐺‘𝐼) ∈ (Base‘𝑃))
48		eqid 2732	. . . . . . . . . . 11 ⊢ (quot_1p‘𝑅) = (quot_1p‘𝑅)
49		eqid 2732	. . . . . . . . . . 11 ⊢ (._r‘𝑃) = (._r‘𝑃)
50		eqid 2732	. . . . . . . . . . 11 ⊢ (-_g‘𝑃) = (-_g‘𝑃)
51	40, 2, 6, 48, 49, 50	r1pval 25681	. . . . . . . . . 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 584	. . . . . . . . 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 25680	. . . . . . . . . . . 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 1371	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝑋(quot_1p‘𝑅)(𝐺‘𝐼)) ∈ (Base‘𝑃))
56	7, 6, 49	lidlmcl 20846	. . . . . . . . . . 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 837	. . . . . . . . . 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 20845	. . . . . . . . . 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 837	. . . . . . . . 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 2833	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ∈ 𝐼)
61	28, 60	sseldd 3983	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝑋(rem_1p‘𝑅)(𝐺‘𝐼)) ∈ (Base‘𝑃))
62		ffvelcdm 7083	. . . . . . 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 587	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (( deg₁ ‘𝑅)‘(𝑋(rem_1p‘𝑅)(𝐺‘𝐼))) ∈ ℝ^*)
64	28	ssdifd 4140	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝐼 ∖ {(0_g‘𝑃)}) ⊆ ((Base‘𝑃) ∖ {(0_g‘𝑃)}))
65		imass2 6101	. . . . . . . . . 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 25614	. . . . . . . . . . 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 12866	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
70	68, 69	sseqtrdi 4032	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (( deg₁ ‘𝑅) “ ((Base‘𝑃) ∖ {(0_g‘𝑃)})) ⊆ (ℤ_≥‘0))
71	66, 70	sstrd 3992	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})) ⊆ (ℤ_≥‘0))
72		uzssz 12845	. . . . . . . . 9 ⊢ (ℤ_≥‘0) ⊆ ℤ
73		zssre 12567	. . . . . . . . . 10 ⊢ ℤ ⊆ ℝ
74		ressxr 11260	. . . . . . . . . 10 ⊢ ℝ ⊆ ℝ^*
75	73, 74	sstri 3991	. . . . . . . . 9 ⊢ ℤ ⊆ ℝ^*
76	72, 75	sstri 3991	. . . . . . . 8 ⊢ (ℤ_≥‘0) ⊆ ℝ^*
77	71, 76	sstrdi 3994	. . . . . . 7 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})) ⊆ ℝ^*)
78	7, 15	lidl0cl 20841	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (0_g‘𝑃) ∈ 𝐼)
79	53, 27, 78	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (0_g‘𝑃) ∈ 𝐼)
80	79	snssd 4812	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → {(0_g‘𝑃)} ⊆ 𝐼)
81	31	necomd 2996	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → {(0_g‘𝑃)} ≠ 𝐼)
82		pssdifn0 4365	. . . . . . . . . 10 ⊢ (({(0_g‘𝑃)} ⊆ 𝐼 ∧ {(0_g‘𝑃)} ≠ 𝐼) → (𝐼 ∖ {(0_g‘𝑃)}) ≠ ∅)
83	80, 81, 82	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝐼 ∖ {(0_g‘𝑃)}) ≠ ∅)
84		ffn 6717	. . . . . . . . . . . 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 4142	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝐼 ∖ {(0_g‘𝑃)}) ⊆ (Base‘𝑃))
87		fnimaeq0 6683	. . . . . . . . . . 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 587	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → ((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})) = ∅ ↔ (𝐼 ∖ {(0_g‘𝑃)}) = ∅))
89	88	necon3bid 2985	. . . . . . . . 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 256	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})) ≠ ∅)
91		infssuzcl 12918	. . . . . . . 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 584	. . . . . . 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 3983	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → inf((( deg₁ ‘𝑅) “ (𝐼 ∖ {(0_g‘𝑃)})), ℝ, < ) ∈ ℝ^*)
94		xrltnle 11283	. . . . . 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 584	. . . . 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 481	. . . . . . 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 481	. . . . . . . . 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 485	. . . . . . . . 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 4790	. . . . . . . . 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 583	. . . . . . . 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 7237	. . . . . . . 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 1468	. . . . . . 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 12917	. . . . . . 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 584	. . . . . 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 413	. . . . 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 2958	. . . 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 25686	. . . 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 1371	. . 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 256	. 2 ⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ 𝐼) ∧ 𝐼 ≠ {(0_g‘𝑃)}) → (𝐺‘𝐼) ∥ 𝑋)
112	24, 111	pm2.61dane 3029	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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∖ cdif 3945 ⊆ wss 3948 ∅c0 4322 {csn 4628 class class class wbr 5148 “ cima 5679 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 infcinf 9438 ℝcr 11111 0cc0 11112 ℝ^*cxr 11249 < clt 11250 ≤ cle 11251 ℕ₀cn0 12474 ℤcz 12560 ℤ_≥cuz 12824 Basecbs 17146 ._rcmulr 17200 0_gc0g 17387 -_gcsg 18823 Ringcrg 20058 ∥_rcdsr 20172 DivRingcdr 20361 LIdealclidl 20789 Poly₁cpl1 21707 deg₁ cdg1 25576 Monic_1pcmn1 25650 Unic_1pcuc1p 25651 quot_1pcq1p 25652 rem_1pcr1p 25653 idlGen_1pcig1p 25654

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-fz 13487 df-fzo 13630 df-seq 13969 df-hash 14293 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-hom 17223 df-cco 17224 df-0g 17389 df-gsum 17390 df-prds 17395 df-pws 17397 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-mhm 18673 df-submnd 18674 df-grp 18824 df-minusg 18825 df-sbg 18826 df-mulg 18953 df-subg 19005 df-ghm 19092 df-cntz 19183 df-cmn 19652 df-abl 19653 df-mgp 19990 df-ur 20007 df-ring 20060 df-cring 20061 df-oppr 20154 df-dvdsr 20175 df-unit 20176 df-invr 20206 df-subrg 20321 df-drng 20363 df-lmod 20477 df-lss 20548 df-sra 20791 df-rgmod 20792 df-lidl 20793 df-rlreg 20905 df-cnfld 20951 df-ascl 21416 df-psr 21468 df-mvr 21469 df-mpl 21470 df-opsr 21472 df-psr1 21710 df-vr1 21711 df-ply1 21712 df-coe1 21713 df-mdeg 25577 df-deg1 25578 df-mon1 25655 df-uc1p 25656 df-q1p 25657 df-r1p 25658 df-ig1p 25659

This theorem is referenced by: ig1prsp 25702

W3C validator