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Theorem unichnlidl 21331

Description: The union of a nonempty chain of ideals is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by AV, 28-Jun-2026.)

**Assertion**
Ref	Expression
unichnlidl	⊢ ((𝑅 ∈ Ring ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) → ∪ 𝐶 ∈ (LIdeal‘𝑅))

Distinct variable groups: 𝑅,𝑖 𝐶,𝑖,𝑗 Allowed substitution hint: 𝑅(𝑗)

Proof of Theorem unichnlidl Dummy variables 𝑏 𝑘 𝑙 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss3 3928	. . . . 5 ⊢ (𝐶 ⊆ (LIdeal‘𝑅) ↔ ∀𝑖 ∈ 𝐶 𝑖 ∈ (LIdeal‘𝑅))
2		eqid 2765	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2765	. . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
4	2, 3	lidlss 21305	. . . . . . . 8 ⊢ (𝑖 ∈ (LIdeal‘𝑅) → 𝑖 ⊆ (Base‘𝑅))
5	4	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝑖 ∈ (LIdeal‘𝑅) → 𝑖 ⊆ (Base‘𝑅)))
6	5	ralimdv 3179	. . . . . 6 ⊢ (𝑅 ∈ Ring → (∀𝑖 ∈ 𝐶 𝑖 ∈ (LIdeal‘𝑅) → ∀𝑖 ∈ 𝐶 𝑖 ⊆ (Base‘𝑅)))
7	6	imp 411	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ ∀𝑖 ∈ 𝐶 𝑖 ∈ (LIdeal‘𝑅)) → ∀𝑖 ∈ 𝐶 𝑖 ⊆ (Base‘𝑅))
8	1, 7	sylan2b 605	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → ∀𝑖 ∈ 𝐶 𝑖 ⊆ (Base‘𝑅))
9		unissb 4902	. . . 4 ⊢ (∪ 𝐶 ⊆ (Base‘𝑅) ↔ ∀𝑖 ∈ 𝐶 𝑖 ⊆ (Base‘𝑅))
10	8, 9	sylibr 237	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → ∪ 𝐶 ⊆ (Base‘𝑅))
11	10	3ad2antr2 1206	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) → ∪ 𝐶 ⊆ (Base‘𝑅))
12		lidlunin0 21330	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → ∪ 𝐶 ≠ ∅)
13	12	3adant3r3 1201	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) → ∪ 𝐶 ≠ ∅)
14		eluni2 4872	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐶 ↔ ∃𝑘 ∈ 𝐶 𝑥 ∈ 𝑘)
15		eluni2 4872	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ 𝐶 ↔ ∃𝑙 ∈ 𝐶 𝑦 ∈ 𝑙)
16		simpl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) → 𝑘 ∈ 𝐶)
17		simpl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙) → 𝑙 ∈ 𝐶)
18	16, 17	anim12i 624	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → (𝑘 ∈ 𝐶 ∧ 𝑙 ∈ 𝐶))
19	18	3adant1 1146	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → (𝑘 ∈ 𝐶 ∧ 𝑙 ∈ 𝐶))
20	19	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) → (𝑘 ∈ 𝐶 ∧ 𝑙 ∈ 𝐶))
21		sseq1 3964	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑘 → (𝑖 ⊆ 𝑗 ↔ 𝑘 ⊆ 𝑗))
22		sseq2 3965	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑘 → (𝑗 ⊆ 𝑖 ↔ 𝑗 ⊆ 𝑘))
23	21, 22	orbi12d 931	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑘 → ((𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖) ↔ (𝑘 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑘)))
24		sseq2 3965	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑙 → (𝑘 ⊆ 𝑗 ↔ 𝑘 ⊆ 𝑙))
25		sseq1 3964	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑙 → (𝑗 ⊆ 𝑘 ↔ 𝑙 ⊆ 𝑘))
26	24, 25	orbi12d 931	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑙 → ((𝑘 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑘) ↔ (𝑘 ⊆ 𝑙 ∨ 𝑙 ⊆ 𝑘)))
27	23, 26	rspc2v 3595	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ 𝐶 ∧ 𝑙 ∈ 𝐶) → (∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖) → (𝑘 ⊆ 𝑙 ∨ 𝑙 ⊆ 𝑘)))
28	20, 27	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) → (∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖) → (𝑘 ⊆ 𝑙 ∨ 𝑙 ⊆ 𝑘)))
29		simpl1 1208	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) → 𝑅 ∈ Ring)
30		ssel 3933	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐶 ⊆ (LIdeal‘𝑅) → (𝑘 ∈ 𝐶 → 𝑘 ∈ (LIdeal‘𝑅)))
31	30	com12 33	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ 𝐶 → (𝐶 ⊆ (LIdeal‘𝑅) → 𝑘 ∈ (LIdeal‘𝑅)))
32	31	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) → (𝐶 ⊆ (LIdeal‘𝑅) → 𝑘 ∈ (LIdeal‘𝑅)))
33	32	3ad2ant2 1150	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → (𝐶 ⊆ (LIdeal‘𝑅) → 𝑘 ∈ (LIdeal‘𝑅)))
34	33	com12 33	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐶 ⊆ (LIdeal‘𝑅) → ((𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → 𝑘 ∈ (LIdeal‘𝑅)))
35	34	3ad2ant3 1151	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → ((𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → 𝑘 ∈ (LIdeal‘𝑅)))
36	35	imp 411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) → 𝑘 ∈ (LIdeal‘𝑅))
37	29, 36	jca 520	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) → (𝑅 ∈ Ring ∧ 𝑘 ∈ (LIdeal‘𝑅)))
38		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) → 𝑥 ∈ 𝑘)
39	38	anim2i 628	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘)) → (𝑏 ∈ (Base‘𝑅) ∧ 𝑥 ∈ 𝑘))
40	39	3adant3 1148	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → (𝑏 ∈ (Base‘𝑅) ∧ 𝑥 ∈ 𝑘))
41	40	adantl 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) → (𝑏 ∈ (Base‘𝑅) ∧ 𝑥 ∈ 𝑘))
42	37, 41	jca 520	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) → ((𝑅 ∈ Ring ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ 𝑥 ∈ 𝑘)))
43	42	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑘 ⊆ 𝑙 ∨ 𝑙 ⊆ 𝑘)) → ((𝑅 ∈ Ring ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ 𝑥 ∈ 𝑘)))
44		eqid 2765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (._r‘𝑅) = (._r‘𝑅)
45	3, 2, 44	lidlmcl 21319	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ Ring ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ 𝑥 ∈ 𝑘)) → (𝑏(._r‘𝑅)𝑥) ∈ 𝑘)
46	43, 45	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑘 ⊆ 𝑙 ∨ 𝑙 ⊆ 𝑘)) → (𝑏(._r‘𝑅)𝑥) ∈ 𝑘)
47		ssel 3933	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐶 ⊆ (LIdeal‘𝑅) → (𝑙 ∈ 𝐶 → 𝑙 ∈ (LIdeal‘𝑅)))
48	47	com12 33	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑙 ∈ 𝐶 → (𝐶 ⊆ (LIdeal‘𝑅) → 𝑙 ∈ (LIdeal‘𝑅)))
49	48	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙) → (𝐶 ⊆ (LIdeal‘𝑅) → 𝑙 ∈ (LIdeal‘𝑅)))
50	49	3ad2ant3 1151	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → (𝐶 ⊆ (LIdeal‘𝑅) → 𝑙 ∈ (LIdeal‘𝑅)))
51	50	com12 33	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐶 ⊆ (LIdeal‘𝑅) → ((𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → 𝑙 ∈ (LIdeal‘𝑅)))
52	51	3ad2ant3 1151	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → ((𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → 𝑙 ∈ (LIdeal‘𝑅)))
53	52	imp 411	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) → 𝑙 ∈ (LIdeal‘𝑅))
54	29, 53	jca 520	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) → (𝑅 ∈ Ring ∧ 𝑙 ∈ (LIdeal‘𝑅)))
55	54	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) → (𝑅 ∈ Ring ∧ 𝑙 ∈ (LIdeal‘𝑅)))
56	55	adantr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) ∧ 𝑘 ⊆ 𝑙) → (𝑅 ∈ Ring ∧ 𝑙 ∈ (LIdeal‘𝑅)))
57		ssel 3933	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ⊆ 𝑙 → ((𝑏(._r‘𝑅)𝑥) ∈ 𝑘 → (𝑏(._r‘𝑅)𝑥) ∈ 𝑙))
58	57	com12 33	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑏(._r‘𝑅)𝑥) ∈ 𝑘 → (𝑘 ⊆ 𝑙 → (𝑏(._r‘𝑅)𝑥) ∈ 𝑙))
59	58	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) → (𝑘 ⊆ 𝑙 → (𝑏(._r‘𝑅)𝑥) ∈ 𝑙))
60	59	imp 411	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) ∧ 𝑘 ⊆ 𝑙) → (𝑏(._r‘𝑅)𝑥) ∈ 𝑙)
61		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙) → 𝑦 ∈ 𝑙)
62	61	3ad2ant3 1151	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → 𝑦 ∈ 𝑙)
63	62	ad3antlr 743	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) ∧ 𝑘 ⊆ 𝑙) → 𝑦 ∈ 𝑙)
64		eqid 2765	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
65	3, 64	lidlacl 21315	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑅 ∈ Ring ∧ 𝑙 ∈ (LIdeal‘𝑅)) ∧ ((𝑏(._r‘𝑅)𝑥) ∈ 𝑙 ∧ 𝑦 ∈ 𝑙)) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ 𝑙)
66	56, 60, 63, 65	syl12anc 849	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) ∧ 𝑘 ⊆ 𝑙) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ 𝑙)
67	17	3ad2ant3 1151	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → 𝑙 ∈ 𝐶)
68	67	ad3antlr 743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) ∧ 𝑘 ⊆ 𝑙) → 𝑙 ∈ 𝐶)
69		elunii 4873	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ 𝑙 ∧ 𝑙 ∈ 𝐶) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶)
70	66, 68, 69	syl2anc 595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) ∧ 𝑘 ⊆ 𝑙) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶)
71	70	ex 417	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) → (𝑘 ⊆ 𝑙 → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶))
72	37	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) → (𝑅 ∈ Ring ∧ 𝑘 ∈ (LIdeal‘𝑅)))
73	72	adantr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) ∧ 𝑙 ⊆ 𝑘) → (𝑅 ∈ Ring ∧ 𝑘 ∈ (LIdeal‘𝑅)))
74		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) → (𝑏(._r‘𝑅)𝑥) ∈ 𝑘)
75	74	adantr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) ∧ 𝑙 ⊆ 𝑘) → (𝑏(._r‘𝑅)𝑥) ∈ 𝑘)
76		ssel 3933	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑙 ⊆ 𝑘 → (𝑦 ∈ 𝑙 → 𝑦 ∈ 𝑘))
77	76	com12 33	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ 𝑙 → (𝑙 ⊆ 𝑘 → 𝑦 ∈ 𝑘))
78	77	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙) → (𝑙 ⊆ 𝑘 → 𝑦 ∈ 𝑘))
79	78	3ad2ant3 1151	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → (𝑙 ⊆ 𝑘 → 𝑦 ∈ 𝑘))
80	79	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) → (𝑙 ⊆ 𝑘 → 𝑦 ∈ 𝑘))
81	80	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) → (𝑙 ⊆ 𝑘 → 𝑦 ∈ 𝑘))
82	81	imp 411	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) ∧ 𝑙 ⊆ 𝑘) → 𝑦 ∈ 𝑘)
83	3, 64	lidlacl 21315	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑅 ∈ Ring ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ ((𝑏(._r‘𝑅)𝑥) ∈ 𝑘 ∧ 𝑦 ∈ 𝑘)) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ 𝑘)
84	73, 75, 82, 83	syl12anc 849	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) ∧ 𝑙 ⊆ 𝑘) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ 𝑘)
85	16	3ad2ant2 1150	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → 𝑘 ∈ 𝐶)
86	85	ad3antlr 743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) ∧ 𝑙 ⊆ 𝑘) → 𝑘 ∈ 𝐶)
87		elunii 4873	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ 𝑘 ∧ 𝑘 ∈ 𝐶) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶)
88	84, 86, 87	syl2anc 595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) ∧ 𝑙 ⊆ 𝑘) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶)
89	88	ex 417	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) → (𝑙 ⊆ 𝑘 → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶))
90	71, 89	jaod 872	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑏(._r‘𝑅)𝑥) ∈ 𝑘) → ((𝑘 ⊆ 𝑙 ∨ 𝑙 ⊆ 𝑘) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶))
91	90	impancom 456	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑘 ⊆ 𝑙 ∨ 𝑙 ⊆ 𝑘)) → ((𝑏(._r‘𝑅)𝑥) ∈ 𝑘 → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶))
92	46, 91	mpd 16	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) ∧ (𝑘 ⊆ 𝑙 ∨ 𝑙 ⊆ 𝑘)) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶)
93	92	ex 417	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) → ((𝑘 ⊆ 𝑙 ∨ 𝑙 ⊆ 𝑘) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶))
94	28, 93	syld 48	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) ∧ (𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙))) → (∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶))
95	94	ex 417	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → ((𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → (∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶)))
96	95	com23 87	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → (∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖) → ((𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶)))
97	96	3exp 1135	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (𝐶 ≠ ∅ → (𝐶 ⊆ (LIdeal‘𝑅) → (∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖) → ((𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶)))))
98	97	3imp2 1366	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) → ((𝑏 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶))
99	98	3expd 1370	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) → (𝑏 ∈ (Base‘𝑅) → ((𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘) → ((𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶))))
100	99	imp41 430	. . . . . . . . 9 ⊢ (((((𝑅 ∈ Ring ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘)) ∧ (𝑙 ∈ 𝐶 ∧ 𝑦 ∈ 𝑙)) → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶)
101	100	rexlimdvaa 3167	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘)) → (∃𝑙 ∈ 𝐶 𝑦 ∈ 𝑙 → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶))
102	15, 101	biimtrid 245	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘)) → (𝑦 ∈ ∪ 𝐶 → ((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶))
103	102	ralrimiv 3156	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) ∧ 𝑏 ∈ (Base‘𝑅)) ∧ (𝑘 ∈ 𝐶 ∧ 𝑥 ∈ 𝑘)) → ∀𝑦 ∈ ∪ 𝐶((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶)
104	103	rexlimdvaa 3167	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) ∧ 𝑏 ∈ (Base‘𝑅)) → (∃𝑘 ∈ 𝐶 𝑥 ∈ 𝑘 → ∀𝑦 ∈ ∪ 𝐶((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶))
105	14, 104	biimtrid 245	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑥 ∈ ∪ 𝐶 → ∀𝑦 ∈ ∪ 𝐶((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶))
106	105	ralrimiv 3156	. . 3 ⊢ (((𝑅 ∈ Ring ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) ∧ 𝑏 ∈ (Base‘𝑅)) → ∀𝑥 ∈ ∪ 𝐶∀𝑦 ∈ ∪ 𝐶((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶)
107	106	ralrimiva 3157	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) → ∀𝑏 ∈ (Base‘𝑅)∀𝑥 ∈ ∪ 𝐶∀𝑦 ∈ ∪ 𝐶((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶)
108	3, 2, 64, 44	islidl 21309	. 2 ⊢ (∪ 𝐶 ∈ (LIdeal‘𝑅) ↔ (∪ 𝐶 ⊆ (Base‘𝑅) ∧ ∪ 𝐶 ≠ ∅ ∧ ∀𝑏 ∈ (Base‘𝑅)∀𝑥 ∈ ∪ 𝐶∀𝑦 ∈ ∪ 𝐶((𝑏(._r‘𝑅)𝑥)(+_g‘𝑅)𝑦) ∈ ∪ 𝐶))
109	11, 13, 107, 108	syl3anbrc 1360	1 ⊢ ((𝑅 ∈ Ring ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) → ∪ 𝐶 ∈ (LIdeal‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 ∈ wcel 2145 ≠ wne 2960 ∀wral 3079 ∃wrex 3089 ⊆ wss 3907 ∅c0 4288 ∪ cuni 4868 ‘cfv 6525 (class class class)co 7400 Basecbs 17259 +_gcplusg 17300 ._rcmulr 17301 Ringcrg 20306 LIdealclidl 21299

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-sca 17316 df-vsca 17317 df-ip 17318 df-0g 17484 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-grp 18993 df-minusg 18994 df-sbg 18995 df-subg 19180 df-cmn 19843 df-abl 19844 df-mgp 20208 df-rng 20222 df-ur 20255 df-ring 20308 df-subrg 20646 df-lmod 20952 df-lss 21022 df-sra 21263 df-rgmod 21264 df-lidl 21301

This theorem is referenced by: (None)

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