ssmxidllem - Mathbox for Thierry Arnoux

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Theorem ssmxidllem 33444

Description: The set 𝑃 used in the proof of ssmxidl 33445 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 2987	. . . 4 ⊢ (𝑝 = ∪ 𝑍 → (𝑝 ≠ 𝐵 ↔ ∪ 𝑍 ≠ 𝐵))
2		sseq2 3973	. . . 4 ⊢ (𝑝 = ∪ 𝑍 → (𝐼 ⊆ 𝑝 ↔ 𝐼 ⊆ ∪ 𝑍))
3	1, 2	anbi12d 632	. . 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 4045	. . . . . . . . 9 ⊢ 𝑃 ⊆ (LIdeal‘𝑅)
7	4, 6	sstrdi 3959	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ⊆ (LIdeal‘𝑅))
8	7	sselda 3946	. . . . . . 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 21122	. . . . . . 7 ⊢ (𝑗 ∈ (LIdeal‘𝑅) → 𝑗 ⊆ 𝐵)
12	8, 11	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ⊆ 𝐵)
13	12	ralrimiva 3125	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 𝑗 ⊆ 𝐵)
14		unissb 4903	. . . . 5 ⊢ (∪ 𝑍 ⊆ 𝐵 ↔ ∀𝑗 ∈ 𝑍 𝑗 ⊆ 𝐵)
15	13, 14	sylibr 234	. . . 4 ⊢ (𝜑 → ∪ 𝑍 ⊆ 𝐵)
16		ssmxidllem2.2	. . . . . . 7 ⊢ (𝜑 → 𝑍 ≠ ∅)
17		ssmxidllem.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ Ring)
18	17	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑅 ∈ Ring)
19		eqid 2729	. . . . . . . . . . 11 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
20	10, 19	lidl0cl 21130	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅)) → (0_g‘𝑅) ∈ 𝑗)
21	18, 8, 20	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (0_g‘𝑅) ∈ 𝑗)
22		n0i 4303	. . . . . . . . 9 ⊢ ((0_g‘𝑅) ∈ 𝑗 → ¬ 𝑗 = ∅)
23	21, 22	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ¬ 𝑗 = ∅)
24	23	reximdva0 4318	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑍 ≠ ∅) → ∃𝑗 ∈ 𝑍 ¬ 𝑗 = ∅)
25	16, 24	mpdan 687	. . . . . 6 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ¬ 𝑗 = ∅)
26		rexnal 3082	. . . . . 6 ⊢ (∃𝑗 ∈ 𝑍 ¬ 𝑗 = ∅ ↔ ¬ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
27	25, 26	sylib 218	. . . . 5 ⊢ (𝜑 → ¬ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
28		uni0c 4898	. . . . . 6 ⊢ (∪ 𝑍 = ∅ ↔ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
29	28	necon3abii 2971	. . . . 5 ⊢ (∪ 𝑍 ≠ ∅ ↔ ¬ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
30	27, 29	sylibr 234	. . . 4 ⊢ (𝜑 → ∪ 𝑍 ≠ ∅)
31		eluni2 4875	. . . . . . . 8 ⊢ (𝑎 ∈ ∪ 𝑍 ↔ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖)
32		eluni2 4875	. . . . . . . 8 ⊢ (𝑏 ∈ ∪ 𝑍 ↔ ∃𝑗 ∈ 𝑍 𝑏 ∈ 𝑗)
33	31, 32	anbi12i 628	. . . . . . 7 ⊢ ((𝑎 ∈ ∪ 𝑍 ∧ 𝑏 ∈ ∪ 𝑍) ↔ (∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖 ∧ ∃𝑗 ∈ 𝑍 𝑏 ∈ 𝑗))
34		an32 646	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖) ∧ 𝑗 ∈ 𝑍) ↔ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖))
35	17	ad6antr 736	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑅 ∈ Ring)
36	7	ad5antr 734	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → 𝑍 ⊆ (LIdeal‘𝑅))
37		simp-4r 783	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → 𝑗 ∈ 𝑍)
38	36, 37	sseldd 3947	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → 𝑗 ∈ (LIdeal‘𝑅))
39	38	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑗 ∈ (LIdeal‘𝑅))
40		simp-6r 787	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑥 ∈ 𝐵)
41		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑖 ⊆ 𝑗)
42		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑎 ∈ 𝑖)
43	41, 42	sseldd 3947	. . . . . . . . . . . . . . . 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 21135	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝑗)) → (𝑥(._r‘𝑅)𝑎) ∈ 𝑗)
46	35, 39, 40, 43, 45	syl22anc 838	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → (𝑥(._r‘𝑅)𝑎) ∈ 𝑗)
47		simp-4r 783	. . . . . . . . . . . . . . 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 21131	. . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑗 ∈ 𝑍)
52		elunii 4876	. . . . . . . . . . . . . 14 ⊢ ((((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ 𝑗 ∧ 𝑗 ∈ 𝑍) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
53	50, 51, 52	syl2anc 584	. . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑅 ∈ Ring)
55	36	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑍 ⊆ (LIdeal‘𝑅))
56		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → 𝑖 ∈ 𝑍)
57	56	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑖 ∈ 𝑍)
58	55, 57	sseldd 3947	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑖 ∈ (LIdeal‘𝑅))
59		simp-6r 787	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑥 ∈ 𝐵)
60		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑎 ∈ 𝑖)
61	10, 9, 44	lidlmcl 21135	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝑖)) → (𝑥(._r‘𝑅)𝑎) ∈ 𝑖)
62	54, 58, 59, 60, 61	syl22anc 838	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → (𝑥(._r‘𝑅)𝑎) ∈ 𝑖)
63		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑗 ⊆ 𝑖)
64		simp-4r 783	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑏 ∈ 𝑗)
65	63, 64	sseldd 3947	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑏 ∈ 𝑖)
66	10, 48	lidlacl 21131	. . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ 𝑖)
68		elunii 4876	. . . . . . . . . . . . . 14 ⊢ ((((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ 𝑖 ∧ 𝑖 ∈ 𝑍) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
69	67, 57, 68	syl2anc 584	. . . . . . . . . . . . 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 734	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → [⊊] Or 𝑍)
72		sorpssi 7705	. . . . . . . . . . . . . 14 ⊢ (( [⊊] Or 𝑍 ∧ (𝑖 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍)) → (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))
73	71, 56, 37, 72	syl12anc 836	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))
74	53, 69, 73	mpjaodan 960	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
75	74	r19.29an 3137	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
76	75	an32s 652	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖) ∧ 𝑏 ∈ 𝑗) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
77	34, 76	sylanb 581	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
78	77	r19.29an 3137	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖) ∧ ∃𝑗 ∈ 𝑍 𝑏 ∈ 𝑗) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
79	78	anasss 466	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖 ∧ ∃𝑗 ∈ 𝑍 𝑏 ∈ 𝑗)) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
80	33, 79	sylan2b 594	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑎 ∈ ∪ 𝑍 ∧ 𝑏 ∈ ∪ 𝑍)) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
81	80	ralrimivva 3180	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑎 ∈ ∪ 𝑍∀𝑏 ∈ ∪ 𝑍((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
82	81	ralrimiva 3125	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ ∪ 𝑍∀𝑏 ∈ ∪ 𝑍((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
83	10, 9, 48, 44	islidl 21125	. . . 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 1344	. . 3 ⊢ (𝜑 → ∪ 𝑍 ∈ (LIdeal‘𝑅))
85	4	sselda 3946	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑃)
86		neeq1 2987	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑗 → (𝑝 ≠ 𝐵 ↔ 𝑗 ≠ 𝐵))
87		sseq2 3973	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑗 → (𝐼 ⊆ 𝑝 ↔ 𝐼 ⊆ 𝑗))
88	86, 87	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑗 → ((𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝) ↔ (𝑗 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑗)))
89	88, 5	elrab2 3662	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑃 ↔ (𝑗 ∈ (LIdeal‘𝑅) ∧ (𝑗 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑗)))
90	85, 89	sylib 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 ∈ (LIdeal‘𝑅) ∧ (𝑗 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑗)))
91	90	simprld 771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ≠ 𝐵)
92		eqid 2729	. . . . . . . . 9 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
93	9, 92	pridln1 33414	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑗 ≠ 𝐵) → ¬ (1_r‘𝑅) ∈ 𝑗)
94	18, 8, 91, 93	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ¬ (1_r‘𝑅) ∈ 𝑗)
95	94	nrexdv 3128	. . . . . 6 ⊢ (𝜑 → ¬ ∃𝑗 ∈ 𝑍 (1_r‘𝑅) ∈ 𝑗)
96		eluni2 4875	. . . . . 6 ⊢ ((1_r‘𝑅) ∈ ∪ 𝑍 ↔ ∃𝑗 ∈ 𝑍 (1_r‘𝑅) ∈ 𝑗)
97	95, 96	sylnibr 329	. . . . 5 ⊢ (𝜑 → ¬ (1_r‘𝑅) ∈ ∪ 𝑍)
98	10, 9, 92	lidl1el 21136	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ∪ 𝑍 ∈ (LIdeal‘𝑅)) → ((1_r‘𝑅) ∈ ∪ 𝑍 ↔ ∪ 𝑍 = 𝐵))
99	17, 84, 98	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((1_r‘𝑅) ∈ ∪ 𝑍 ↔ ∪ 𝑍 = 𝐵))
100	99	necon3bbid 2962	. . . . 5 ⊢ (𝜑 → (¬ (1_r‘𝑅) ∈ ∪ 𝑍 ↔ ∪ 𝑍 ≠ 𝐵))
101	97, 100	mpbid 232	. . . 4 ⊢ (𝜑 → ∪ 𝑍 ≠ 𝐵)
102	90	simprrd 773	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐼 ⊆ 𝑗)
103	102	ralrimiva 3125	. . . . . 6 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 𝐼 ⊆ 𝑗)
104		ssint 4928	. . . . . 6 ⊢ (𝐼 ⊆ ∩ 𝑍 ↔ ∀𝑗 ∈ 𝑍 𝐼 ⊆ 𝑗)
105	103, 104	sylibr 234	. . . . 5 ⊢ (𝜑 → 𝐼 ⊆ ∩ 𝑍)
106		intssuni 4934	. . . . . 6 ⊢ (𝑍 ≠ ∅ → ∩ 𝑍 ⊆ ∪ 𝑍)
107	16, 106	syl 17	. . . . 5 ⊢ (𝜑 → ∩ 𝑍 ⊆ ∪ 𝑍)
108	105, 107	sstrd 3957	. . . 4 ⊢ (𝜑 → 𝐼 ⊆ ∪ 𝑍)
109	101, 108	jca 511	. . 3 ⊢ (𝜑 → (∪ 𝑍 ≠ 𝐵 ∧ 𝐼 ⊆ ∪ 𝑍))
110	3, 84, 109	elrabd 3661	. 2 ⊢ (𝜑 → ∪ 𝑍 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝)})
111	110, 5	eleqtrrdi 2839	1 ⊢ (𝜑 → ∪ 𝑍 ∈ 𝑃)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 {crab 3405 ⊆ wss 3914 ∅c0 4296 ∪ cuni 4871 ∩ cint 4910 Or wor 5545 ‘cfv 6511 (class class class)co 7387 [⊊] crpss 7698 Basecbs 17179 +_gcplusg 17220 ._rcmulr 17221 0_gc0g 17402 1_rcur 20090 Ringcrg 20142 LIdealclidl 21116

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-rpss 7699 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-ip 17238 df-0g 17404 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-subrg 20479 df-lmod 20768 df-lss 20838 df-sra 21080 df-rgmod 21081 df-lidl 21118

This theorem is referenced by: ssmxidl 33445

W3C validator