ssmxidllem - Mathbox for Thierry Arnoux

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Theorem ssmxidllem 33379

Description: The set 𝑃 used in the proof of ssmxidl 33380 satisfies the condition of Zorn's Lemma. (Contributed by Thierry Arnoux, 10-Apr-2024.)

**Hypotheses**
Ref	Expression
ssmxidl.1	⊢ 𝐵 = (Base‘𝑅)
ssmxidllem.1	⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝)}
ssmxidllem.2	⊢ (𝜑 → 𝑅 ∈ Ring)
ssmxidllem.3	⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
ssmxidllem.4	⊢ (𝜑 → 𝐼 ≠ 𝐵)
ssmxidllem2.1	⊢ (𝜑 → 𝑍 ⊆ 𝑃)
ssmxidllem2.2	⊢ (𝜑 → 𝑍 ≠ ∅)
ssmxidllem2.3	⊢ (𝜑 → [⊊] Or 𝑍)

**Assertion**
Ref	Expression
ssmxidllem	⊢ (𝜑 → ∪ 𝑍 ∈ 𝑃)

Distinct variable groups: 𝐵,𝑝 𝐼,𝑝 𝑅,𝑝 𝑍,𝑝 Allowed substitution hints: 𝜑(𝑝) 𝑃(𝑝)

Proof of Theorem ssmxidllem Dummy variables 𝑎 𝑏 𝑖 𝑥 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		neeq1 2996	. . . 4 ⊢ (𝑝 = ∪ 𝑍 → (𝑝 ≠ 𝐵 ↔ ∪ 𝑍 ≠ 𝐵))
2		sseq2 4018	. . . 4 ⊢ (𝑝 = ∪ 𝑍 → (𝐼 ⊆ 𝑝 ↔ 𝐼 ⊆ ∪ 𝑍))
3	1, 2	anbi12d 630	. . 3 ⊢ (𝑝 = ∪ 𝑍 → ((𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝) ↔ (∪ 𝑍 ≠ 𝐵 ∧ 𝐼 ⊆ ∪ 𝑍)))
4		ssmxidllem2.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ⊆ 𝑃)
5		ssmxidllem.1	. . . . . . . . . 10 ⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝)}
6	5	ssrab3 4089	. . . . . . . . 9 ⊢ 𝑃 ⊆ (LIdeal‘𝑅)
7	4, 6	sstrdi 4004	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ⊆ (LIdeal‘𝑅))
8	7	sselda 3991	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (LIdeal‘𝑅))
9		ssmxidl.1	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
10		eqid 2729	. . . . . . . 8 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
11	9, 10	lidlss 21223	. . . . . . 7 ⊢ (𝑗 ∈ (LIdeal‘𝑅) → 𝑗 ⊆ 𝐵)
12	8, 11	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ⊆ 𝐵)
13	12	ralrimiva 3139	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 𝑗 ⊆ 𝐵)
14		unissb 4952	. . . . 5 ⊢ (∪ 𝑍 ⊆ 𝐵 ↔ ∀𝑗 ∈ 𝑍 𝑗 ⊆ 𝐵)
15	13, 14	sylibr 233	. . . 4 ⊢ (𝜑 → ∪ 𝑍 ⊆ 𝐵)
16		ssmxidllem2.2	. . . . . . 7 ⊢ (𝜑 → 𝑍 ≠ ∅)
17		ssmxidllem.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ Ring)
18	17	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑅 ∈ Ring)
19		eqid 2729	. . . . . . . . . . 11 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
20	10, 19	lidl0cl 21231	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅)) → (0_g‘𝑅) ∈ 𝑗)
21	18, 8, 20	syl2anc 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (0_g‘𝑅) ∈ 𝑗)
22		n0i 4346	. . . . . . . . 9 ⊢ ((0_g‘𝑅) ∈ 𝑗 → ¬ 𝑗 = ∅)
23	21, 22	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ¬ 𝑗 = ∅)
24	23	reximdva0 4361	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑍 ≠ ∅) → ∃𝑗 ∈ 𝑍 ¬ 𝑗 = ∅)
25	16, 24	mpdan 685	. . . . . 6 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ¬ 𝑗 = ∅)
26		rexnal 3093	. . . . . 6 ⊢ (∃𝑗 ∈ 𝑍 ¬ 𝑗 = ∅ ↔ ¬ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
27	25, 26	sylib 217	. . . . 5 ⊢ (𝜑 → ¬ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
28		uni0c 4947	. . . . . 6 ⊢ (∪ 𝑍 = ∅ ↔ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
29	28	necon3abii 2980	. . . . 5 ⊢ (∪ 𝑍 ≠ ∅ ↔ ¬ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
30	27, 29	sylibr 233	. . . 4 ⊢ (𝜑 → ∪ 𝑍 ≠ ∅)
31		eluni2 4922	. . . . . . . 8 ⊢ (𝑎 ∈ ∪ 𝑍 ↔ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖)
32		eluni2 4922	. . . . . . . 8 ⊢ (𝑏 ∈ ∪ 𝑍 ↔ ∃𝑗 ∈ 𝑍 𝑏 ∈ 𝑗)
33	31, 32	anbi12i 626	. . . . . . 7 ⊢ ((𝑎 ∈ ∪ 𝑍 ∧ 𝑏 ∈ ∪ 𝑍) ↔ (∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖 ∧ ∃𝑗 ∈ 𝑍 𝑏 ∈ 𝑗))
34		an32 644	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖) ∧ 𝑗 ∈ 𝑍) ↔ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖))
35	17	ad6antr 734	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑅 ∈ Ring)
36	7	ad5antr 732	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → 𝑍 ⊆ (LIdeal‘𝑅))
37		simp-4r 782	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → 𝑗 ∈ 𝑍)
38	36, 37	sseldd 3992	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → 𝑗 ∈ (LIdeal‘𝑅))
39	38	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑗 ∈ (LIdeal‘𝑅))
40		simp-6r 786	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑥 ∈ 𝐵)
41		simpr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑖 ⊆ 𝑗)
42		simplr 767	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑎 ∈ 𝑖)
43	41, 42	sseldd 3992	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑎 ∈ 𝑗)
44		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑅) = (._r‘𝑅)
45	10, 9, 44	lidlmcl 21236	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝑗)) → (𝑥(._r‘𝑅)𝑎) ∈ 𝑗)
46	35, 39, 40, 43, 45	syl22anc 837	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → (𝑥(._r‘𝑅)𝑎) ∈ 𝑗)
47		simp-4r 782	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑏 ∈ 𝑗)
48		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
49	10, 48	lidlacl 21232	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ ((𝑥(._r‘𝑅)𝑎) ∈ 𝑗 ∧ 𝑏 ∈ 𝑗)) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ 𝑗)
50	35, 39, 46, 47, 49	syl22anc 837	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ 𝑗)
51	37	adantr 479	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑗 ∈ 𝑍)
52		elunii 4923	. . . . . . . . . . . . . 14 ⊢ ((((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ 𝑗 ∧ 𝑗 ∈ 𝑍) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
53	50, 51, 52	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
54	17	ad6antr 734	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑅 ∈ Ring)
55	36	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑍 ⊆ (LIdeal‘𝑅))
56		simplr 767	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → 𝑖 ∈ 𝑍)
57	56	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑖 ∈ 𝑍)
58	55, 57	sseldd 3992	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑖 ∈ (LIdeal‘𝑅))
59		simp-6r 786	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑥 ∈ 𝐵)
60		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑎 ∈ 𝑖)
61	10, 9, 44	lidlmcl 21236	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝑖)) → (𝑥(._r‘𝑅)𝑎) ∈ 𝑖)
62	54, 58, 59, 60, 61	syl22anc 837	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → (𝑥(._r‘𝑅)𝑎) ∈ 𝑖)
63		simpr 483	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑗 ⊆ 𝑖)
64		simp-4r 782	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑏 ∈ 𝑗)
65	63, 64	sseldd 3992	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑏 ∈ 𝑖)
66	10, 48	lidlacl 21232	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ ((𝑥(._r‘𝑅)𝑎) ∈ 𝑖 ∧ 𝑏 ∈ 𝑖)) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ 𝑖)
67	54, 58, 62, 65, 66	syl22anc 837	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ 𝑖)
68		elunii 4923	. . . . . . . . . . . . . 14 ⊢ ((((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ 𝑖 ∧ 𝑖 ∈ 𝑍) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
69	67, 57, 68	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
70		ssmxidllem2.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → [⊊] Or 𝑍)
71	70	ad5antr 732	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → [⊊] Or 𝑍)
72		sorpssi 7745	. . . . . . . . . . . . . 14 ⊢ (( [⊊] Or 𝑍 ∧ (𝑖 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍)) → (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))
73	71, 56, 37, 72	syl12anc 835	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))
74	53, 69, 73	mpjaodan 956	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
75	74	r19.29an 3151	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
76	75	an32s 650	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖) ∧ 𝑏 ∈ 𝑗) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
77	34, 76	sylanb 579	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
78	77	r19.29an 3151	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖) ∧ ∃𝑗 ∈ 𝑍 𝑏 ∈ 𝑗) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
79	78	anasss 465	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖 ∧ ∃𝑗 ∈ 𝑍 𝑏 ∈ 𝑗)) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
80	33, 79	sylan2b 592	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑎 ∈ ∪ 𝑍 ∧ 𝑏 ∈ ∪ 𝑍)) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
81	80	ralrimivva 3195	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑎 ∈ ∪ 𝑍∀𝑏 ∈ ∪ 𝑍((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
82	81	ralrimiva 3139	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ ∪ 𝑍∀𝑏 ∈ ∪ 𝑍((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
83	10, 9, 48, 44	islidl 21226	. . . 4 ⊢ (∪ 𝑍 ∈ (LIdeal‘𝑅) ↔ (∪ 𝑍 ⊆ 𝐵 ∧ ∪ 𝑍 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ ∪ 𝑍∀𝑏 ∈ ∪ 𝑍((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍))
84	15, 30, 82, 83	syl3anbrc 1340	. . 3 ⊢ (𝜑 → ∪ 𝑍 ∈ (LIdeal‘𝑅))
85	4	sselda 3991	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑃)
86		neeq1 2996	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑗 → (𝑝 ≠ 𝐵 ↔ 𝑗 ≠ 𝐵))
87		sseq2 4018	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑗 → (𝐼 ⊆ 𝑝 ↔ 𝐼 ⊆ 𝑗))
88	86, 87	anbi12d 630	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑗 → ((𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝) ↔ (𝑗 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑗)))
89	88, 5	elrab2 3696	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑃 ↔ (𝑗 ∈ (LIdeal‘𝑅) ∧ (𝑗 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑗)))
90	85, 89	sylib 217	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 ∈ (LIdeal‘𝑅) ∧ (𝑗 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑗)))
91	90	simprld 770	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ≠ 𝐵)
92		eqid 2729	. . . . . . . . 9 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
93	9, 92	pridln1 33349	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑗 ≠ 𝐵) → ¬ (1_r‘𝑅) ∈ 𝑗)
94	18, 8, 91, 93	syl3anc 1368	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ¬ (1_r‘𝑅) ∈ 𝑗)
95	94	nrexdv 3142	. . . . . 6 ⊢ (𝜑 → ¬ ∃𝑗 ∈ 𝑍 (1_r‘𝑅) ∈ 𝑗)
96		eluni2 4922	. . . . . 6 ⊢ ((1_r‘𝑅) ∈ ∪ 𝑍 ↔ ∃𝑗 ∈ 𝑍 (1_r‘𝑅) ∈ 𝑗)
97	95, 96	sylnibr 328	. . . . 5 ⊢ (𝜑 → ¬ (1_r‘𝑅) ∈ ∪ 𝑍)
98	10, 9, 92	lidl1el 21237	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ∪ 𝑍 ∈ (LIdeal‘𝑅)) → ((1_r‘𝑅) ∈ ∪ 𝑍 ↔ ∪ 𝑍 = 𝐵))
99	17, 84, 98	syl2anc 582	. . . . . 6 ⊢ (𝜑 → ((1_r‘𝑅) ∈ ∪ 𝑍 ↔ ∪ 𝑍 = 𝐵))
100	99	necon3bbid 2971	. . . . 5 ⊢ (𝜑 → (¬ (1_r‘𝑅) ∈ ∪ 𝑍 ↔ ∪ 𝑍 ≠ 𝐵))
101	97, 100	mpbid 231	. . . 4 ⊢ (𝜑 → ∪ 𝑍 ≠ 𝐵)
102	90	simprrd 772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐼 ⊆ 𝑗)
103	102	ralrimiva 3139	. . . . . 6 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 𝐼 ⊆ 𝑗)
104		ssint 4977	. . . . . 6 ⊢ (𝐼 ⊆ ∩ 𝑍 ↔ ∀𝑗 ∈ 𝑍 𝐼 ⊆ 𝑗)
105	103, 104	sylibr 233	. . . . 5 ⊢ (𝜑 → 𝐼 ⊆ ∩ 𝑍)
106		intssuni 4983	. . . . . 6 ⊢ (𝑍 ≠ ∅ → ∩ 𝑍 ⊆ ∪ 𝑍)
107	16, 106	syl 17	. . . . 5 ⊢ (𝜑 → ∩ 𝑍 ⊆ ∪ 𝑍)
108	105, 107	sstrd 4002	. . . 4 ⊢ (𝜑 → 𝐼 ⊆ ∪ 𝑍)
109	101, 108	jca 510	. . 3 ⊢ (𝜑 → (∪ 𝑍 ≠ 𝐵 ∧ 𝐼 ⊆ ∪ 𝑍))
110	3, 84, 109	elrabd 3695	. 2 ⊢ (𝜑 → ∪ 𝑍 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝)})
111	110, 5	eleqtrrdi 2840	1 ⊢ (𝜑 → ∪ 𝑍 ∈ 𝑃)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1534 ∈ wcel 2100 ≠ wne 2933 ∀wral 3054 ∃wrex 3063 {crab 3428 ⊆ wss 3959 ∅c0 4335 ∪ cuni 4918 ∩ cint 4959 Or wor 5597 ‘cfv 6558 (class class class)co 7430 [⊊] crpss 7738 Basecbs 17234 +_gcplusg 17287 ._rcmulr 17288 0_gc0g 17475 1_rcur 20186 Ringcrg 20238 LIdealclidl 21217

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2102 ax-9 2110 ax-10 2133 ax-11 2150 ax-12 2170 ax-ext 2700 ax-rep 5293 ax-sep 5307 ax-nul 5314 ax-pow 5373 ax-pr 5437 ax-un 7751 ax-cnex 11221 ax-resscn 11222 ax-1cn 11223 ax-icn 11224 ax-addcl 11225 ax-addrcl 11226 ax-mulcl 11227 ax-mulrcl 11228 ax-mulcom 11229 ax-addass 11230 ax-mulass 11231 ax-distr 11232 ax-i2m1 11233 ax-1ne0 11234 ax-1rid 11235 ax-rnegex 11236 ax-rrecex 11237 ax-cnre 11238 ax-pre-lttri 11239 ax-pre-lttrn 11240 ax-pre-ltadd 11241 ax-pre-mulgt0 11242

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2062 df-mo 2532 df-eu 2561 df-clab 2707 df-cleq 2721 df-clel 2806 df-nfc 2881 df-ne 2934 df-nel 3040 df-ral 3055 df-rex 3064 df-rmo 3373 df-reu 3374 df-rab 3429 df-v 3474 df-sbc 3789 df-csb 3905 df-dif 3962 df-un 3964 df-in 3966 df-ss 3976 df-pss 3979 df-nul 4336 df-if 4537 df-pw 4612 df-sn 4637 df-pr 4639 df-op 4643 df-uni 4919 df-int 4960 df-iun 5008 df-br 5157 df-opab 5219 df-mpt 5240 df-tr 5274 df-id 5584 df-eprel 5590 df-po 5598 df-so 5599 df-fr 5641 df-we 5643 df-xp 5692 df-rel 5693 df-cnv 5694 df-co 5695 df-dm 5696 df-rn 5697 df-res 5698 df-ima 5699 df-pred 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7386 df-ov 7433 df-oprab 7434 df-mpo 7435 df-rpss 7739 df-om 7885 df-1st 8011 df-2nd 8012 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8742 df-en 8983 df-dom 8984 df-sdom 8985 df-pnf 11307 df-mnf 11308 df-xr 11309 df-ltxr 11310 df-le 11311 df-sub 11503 df-neg 11504 df-nn 12275 df-2 12337 df-3 12338 df-4 12339 df-5 12340 df-6 12341 df-7 12342 df-8 12343 df-sets 17187 df-slot 17205 df-ndx 17217 df-base 17235 df-ress 17264 df-plusg 17300 df-mulr 17301 df-sca 17303 df-vsca 17304 df-ip 17305 df-0g 17477 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18952 df-minusg 18953 df-sbg 18954 df-subg 19139 df-cmn 19802 df-abl 19803 df-mgp 20140 df-rng 20158 df-ur 20187 df-ring 20240 df-subrg 20575 df-lmod 20860 df-lss 20931 df-sra 21173 df-rgmod 21174 df-lidl 21219

This theorem is referenced by: ssmxidl 33380

W3C validator