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Theorem unichnlidl 21341

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 3925	. . . . 5 ⊢ (𝐶 ⊆ (LIdeal‘𝑅) ↔ ∀𝑖 ∈ 𝐶 𝑖 ∈ (LIdeal‘𝑅))
2		eqid 2761	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2761	. . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
4	2, 3	lidlss 21315	. . . . . . . 8 ⊢ (𝑖 ∈ (LIdeal‘𝑅) → 𝑖 ⊆ (Base‘𝑅))
5	4	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝑖 ∈ (LIdeal‘𝑅) → 𝑖 ⊆ (Base‘𝑅)))
6	5	ralimdv 3177	. . . . . 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 4905	. . . 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 21340	. . 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 4875	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐶 ↔ ∃𝑘 ∈ 𝐶 𝑥 ∈ 𝑘)
15		eluni2 4875	. . . . . . . 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 3961	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑘 → (𝑖 ⊆ 𝑗 ↔ 𝑘 ⊆ 𝑗))
22		sseq2 3962	. . . . . . . . . . . . . . . . . . 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 3962	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑙 → (𝑘 ⊆ 𝑗 ↔ 𝑘 ⊆ 𝑙))
25		sseq1 3961	. . . . . . . . . . . . . . . . . . 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 3591	. . . . . . . . . . . . . . . . 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 3930	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . 20 ⊢ (._r‘𝑅) = (._r‘𝑅)
45	3, 2, 44	lidlmcl 21329	. . . . . . . . . . . . . . . . . . 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 3930	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3930	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
65	3, 64	lidlacl 21325	. . . . . . . . . . . . . . . . . . . . . . 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 4876	. . . . . . . . . . . . . . . . . . . . . 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 3930	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 21325	. . . . . . . . . . . . . . . . . . . . . . 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 4876	. . . . . . . . . . . . . . . . . . . . . 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 3165	. . . . . . . 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 3154	. . . . . 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 3165	. . . . 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 3154	. . 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 3155	. 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 21319	. 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 2141 ≠ wne 2956 ∀wral 3077 ∃wrex 3087 ⊆ wss 3904 ∅c0 4285 ∪ cuni 4871 ‘cfv 6536 (class class class)co 7410 Basecbs 17268 +_gcplusg 17309 ._rcmulr 17310 Ringcrg 20314 LIdealclidl 21309

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-sca 17325 df-vsca 17326 df-ip 17327 df-0g 17493 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-grp 19002 df-minusg 19003 df-sbg 19004 df-subg 19188 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-subrg 20654 df-lmod 20962 df-lss 21032 df-sra 21273 df-rgmod 21274 df-lidl 21311

This theorem is referenced by: (None)

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