ssmxidllem - Mathbox for Thierry Arnoux

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Theorem ssmxidllem 33501

Description: The set 𝑃 used in the proof of ssmxidl 33502 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 3003	. . . 4 ⊢ (𝑝 = ∪ 𝑍 → (𝑝 ≠ 𝐵 ↔ ∪ 𝑍 ≠ 𝐵))
2		sseq2 4010	. . . 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 4082	. . . . . . . . 9 ⊢ 𝑃 ⊆ (LIdeal‘𝑅)
7	4, 6	sstrdi 3996	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ⊆ (LIdeal‘𝑅))
8	7	sselda 3983	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (LIdeal‘𝑅))
9		ssmxidl.1	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
10		eqid 2737	. . . . . . . 8 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
11	9, 10	lidlss 21222	. . . . . . 7 ⊢ (𝑗 ∈ (LIdeal‘𝑅) → 𝑗 ⊆ 𝐵)
12	8, 11	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ⊆ 𝐵)
13	12	ralrimiva 3146	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 𝑗 ⊆ 𝐵)
14		unissb 4939	. . . . 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 2737	. . . . . . . . . . 11 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
20	10, 19	lidl0cl 21230	. . . . . . . . . 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 4340	. . . . . . . . 9 ⊢ ((0_g‘𝑅) ∈ 𝑗 → ¬ 𝑗 = ∅)
23	21, 22	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ¬ 𝑗 = ∅)
24	23	reximdva0 4355	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑍 ≠ ∅) → ∃𝑗 ∈ 𝑍 ¬ 𝑗 = ∅)
25	16, 24	mpdan 687	. . . . . 6 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ¬ 𝑗 = ∅)
26		rexnal 3100	. . . . . 6 ⊢ (∃𝑗 ∈ 𝑍 ¬ 𝑗 = ∅ ↔ ¬ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
27	25, 26	sylib 218	. . . . 5 ⊢ (𝜑 → ¬ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
28		uni0c 4934	. . . . . 6 ⊢ (∪ 𝑍 = ∅ ↔ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
29	28	necon3abii 2987	. . . . 5 ⊢ (∪ 𝑍 ≠ ∅ ↔ ¬ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
30	27, 29	sylibr 234	. . . 4 ⊢ (𝜑 → ∪ 𝑍 ≠ ∅)
31		eluni2 4911	. . . . . . . 8 ⊢ (𝑎 ∈ ∪ 𝑍 ↔ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖)
32		eluni2 4911	. . . . . . . 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 784	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → 𝑗 ∈ 𝑍)
38	36, 37	sseldd 3984	. . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑎 ∈ 𝑖)
43	41, 42	sseldd 3984	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑎 ∈ 𝑗)
44		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑅) = (._r‘𝑅)
45	10, 9, 44	lidlmcl 21235	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝑗)) → (𝑥(._r‘𝑅)𝑎) ∈ 𝑗)
46	35, 39, 40, 43, 45	syl22anc 839	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → (𝑥(._r‘𝑅)𝑎) ∈ 𝑗)
47		simp-4r 784	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑏 ∈ 𝑗)
48		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
49	10, 48	lidlacl 21231	. . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . 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 4912	. . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . 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 3984	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑖 ∈ (LIdeal‘𝑅))
59		simp-6r 788	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑥 ∈ 𝐵)
60		simplr 769	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑎 ∈ 𝑖)
61	10, 9, 44	lidlmcl 21235	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝑖)) → (𝑥(._r‘𝑅)𝑎) ∈ 𝑖)
62	54, 58, 59, 60, 61	syl22anc 839	. . . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑏 ∈ 𝑗)
65	63, 64	sseldd 3984	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑏 ∈ 𝑖)
66	10, 48	lidlacl 21231	. . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ 𝑖)
68		elunii 4912	. . . . . . . . . . . . . 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 7749	. . . . . . . . . . . . . 14 ⊢ (( [⊊] Or 𝑍 ∧ (𝑖 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍)) → (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))
73	71, 56, 37, 72	syl12anc 837	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))
74	53, 69, 73	mpjaodan 961	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
75	74	r19.29an 3158	. . . . . . . . . . 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 3158	. . . . . . . 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 3202	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑎 ∈ ∪ 𝑍∀𝑏 ∈ ∪ 𝑍((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
82	81	ralrimiva 3146	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ ∪ 𝑍∀𝑏 ∈ ∪ 𝑍((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
83	10, 9, 48, 44	islidl 21225	. . . 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 3983	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑃)
86		neeq1 3003	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑗 → (𝑝 ≠ 𝐵 ↔ 𝑗 ≠ 𝐵))
87		sseq2 4010	. . . . . . . . . . . 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 3695	. . . . . . . . . 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 772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ≠ 𝐵)
92		eqid 2737	. . . . . . . . 9 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
93	9, 92	pridln1 33471	. . . . . . . 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 3149	. . . . . 6 ⊢ (𝜑 → ¬ ∃𝑗 ∈ 𝑍 (1_r‘𝑅) ∈ 𝑗)
96		eluni2 4911	. . . . . 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 21236	. . . . . . 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 2978	. . . . 5 ⊢ (𝜑 → (¬ (1_r‘𝑅) ∈ ∪ 𝑍 ↔ ∪ 𝑍 ≠ 𝐵))
101	97, 100	mpbid 232	. . . 4 ⊢ (𝜑 → ∪ 𝑍 ≠ 𝐵)
102	90	simprrd 774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐼 ⊆ 𝑗)
103	102	ralrimiva 3146	. . . . . 6 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 𝐼 ⊆ 𝑗)
104		ssint 4964	. . . . . 6 ⊢ (𝐼 ⊆ ∩ 𝑍 ↔ ∀𝑗 ∈ 𝑍 𝐼 ⊆ 𝑗)
105	103, 104	sylibr 234	. . . . 5 ⊢ (𝜑 → 𝐼 ⊆ ∩ 𝑍)
106		intssuni 4970	. . . . . 6 ⊢ (𝑍 ≠ ∅ → ∩ 𝑍 ⊆ ∪ 𝑍)
107	16, 106	syl 17	. . . . 5 ⊢ (𝜑 → ∩ 𝑍 ⊆ ∪ 𝑍)
108	105, 107	sstrd 3994	. . . 4 ⊢ (𝜑 → 𝐼 ⊆ ∪ 𝑍)
109	101, 108	jca 511	. . 3 ⊢ (𝜑 → (∪ 𝑍 ≠ 𝐵 ∧ 𝐼 ⊆ ∪ 𝑍))
110	3, 84, 109	elrabd 3694	. 2 ⊢ (𝜑 → ∪ 𝑍 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝)})
111	110, 5	eleqtrrdi 2852	1 ⊢ (𝜑 → ∪ 𝑍 ∈ 𝑃)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3436 ⊆ wss 3951 ∅c0 4333 ∪ cuni 4907 ∩ cint 4946 Or wor 5591 ‘cfv 6561 (class class class)co 7431 [⊊] crpss 7742 Basecbs 17247 +_gcplusg 17297 ._rcmulr 17298 0_gc0g 17484 1_rcur 20178 Ringcrg 20230 LIdealclidl 21216

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-rpss 7743 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-subrg 20570 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-lidl 21218

This theorem is referenced by: ssmxidl 33502

W3C validator