ssmxidllem - Mathbox for Thierry Arnoux

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Theorem ssmxidllem 31637

Description: The set 𝑃 used in the proof of ssmxidl 31638 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 3008	. . . 4 ⊢ (𝑝 = ∪ 𝑍 → (𝑝 ≠ 𝐵 ↔ ∪ 𝑍 ≠ 𝐵))
2		sseq2 3952	. . . 4 ⊢ (𝑝 = ∪ 𝑍 → (𝐼 ⊆ 𝑝 ↔ 𝐼 ⊆ ∪ 𝑍))
3	1, 2	anbi12d 631	. . 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 4020	. . . . . . . . 9 ⊢ 𝑃 ⊆ (LIdeal‘𝑅)
7	4, 6	sstrdi 3938	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ⊆ (LIdeal‘𝑅))
8	7	sselda 3926	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (LIdeal‘𝑅))
9		ssmxidl.1	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
10		eqid 2740	. . . . . . . 8 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
11	9, 10	lidlss 20479	. . . . . . 7 ⊢ (𝑗 ∈ (LIdeal‘𝑅) → 𝑗 ⊆ 𝐵)
12	8, 11	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ⊆ 𝐵)
13	12	ralrimiva 3110	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 𝑗 ⊆ 𝐵)
14		unissb 4879	. . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑅 ∈ Ring)
19		eqid 2740	. . . . . . . . . . 11 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
20	10, 19	lidl0cl 20481	. . . . . . . . . 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 4273	. . . . . . . . 9 ⊢ ((0_g‘𝑅) ∈ 𝑗 → ¬ 𝑗 = ∅)
23	21, 22	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ¬ 𝑗 = ∅)
24	23	reximdva0 4291	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑍 ≠ ∅) → ∃𝑗 ∈ 𝑍 ¬ 𝑗 = ∅)
25	16, 24	mpdan 684	. . . . . 6 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ¬ 𝑗 = ∅)
26		rexnal 3168	. . . . . 6 ⊢ (∃𝑗 ∈ 𝑍 ¬ 𝑗 = ∅ ↔ ¬ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
27	25, 26	sylib 217	. . . . 5 ⊢ (𝜑 → ¬ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
28		uni0c 4874	. . . . . 6 ⊢ (∪ 𝑍 = ∅ ↔ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
29	28	necon3abii 2992	. . . . 5 ⊢ (∪ 𝑍 ≠ ∅ ↔ ¬ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
30	27, 29	sylibr 233	. . . 4 ⊢ (𝜑 → ∪ 𝑍 ≠ ∅)
31		eluni2 4849	. . . . . . . 8 ⊢ (𝑎 ∈ ∪ 𝑍 ↔ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖)
32		eluni2 4849	. . . . . . . 8 ⊢ (𝑏 ∈ ∪ 𝑍 ↔ ∃𝑗 ∈ 𝑍 𝑏 ∈ 𝑗)
33	31, 32	anbi12i 627	. . . . . . 7 ⊢ ((𝑎 ∈ ∪ 𝑍 ∧ 𝑏 ∈ ∪ 𝑍) ↔ (∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖 ∧ ∃𝑗 ∈ 𝑍 𝑏 ∈ 𝑗))
34		an32 643	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖) ∧ 𝑗 ∈ 𝑍) ↔ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖))
35	17	ad6antr 733	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑅 ∈ Ring)
36	7	ad5antr 731	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → 𝑍 ⊆ (LIdeal‘𝑅))
37		simp-4r 781	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → 𝑗 ∈ 𝑍)
38	36, 37	sseldd 3927	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → 𝑗 ∈ (LIdeal‘𝑅))
39	38	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑗 ∈ (LIdeal‘𝑅))
40		simp-6r 785	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑥 ∈ 𝐵)
41		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑖 ⊆ 𝑗)
42		simplr 766	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑎 ∈ 𝑖)
43	41, 42	sseldd 3927	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑎 ∈ 𝑗)
44		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑅) = (._r‘𝑅)
45	10, 9, 44	lidlmcl 20486	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝑗)) → (𝑥(._r‘𝑅)𝑎) ∈ 𝑗)
46	35, 39, 40, 43, 45	syl22anc 836	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → (𝑥(._r‘𝑅)𝑎) ∈ 𝑗)
47		simp-4r 781	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑏 ∈ 𝑗)
48		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
49	10, 48	lidlacl 20482	. . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑗 ∈ 𝑍)
52		elunii 4850	. . . . . . . . . . . . . 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 733	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑅 ∈ Ring)
55	36	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑍 ⊆ (LIdeal‘𝑅))
56		simplr 766	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → 𝑖 ∈ 𝑍)
57	56	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑖 ∈ 𝑍)
58	55, 57	sseldd 3927	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑖 ∈ (LIdeal‘𝑅))
59		simp-6r 785	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑥 ∈ 𝐵)
60		simplr 766	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑎 ∈ 𝑖)
61	10, 9, 44	lidlmcl 20486	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑖 ∈ (LIdeal‘𝑅)) ∧ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝑖)) → (𝑥(._r‘𝑅)𝑎) ∈ 𝑖)
62	54, 58, 59, 60, 61	syl22anc 836	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → (𝑥(._r‘𝑅)𝑎) ∈ 𝑖)
63		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑗 ⊆ 𝑖)
64		simp-4r 781	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑏 ∈ 𝑗)
65	63, 64	sseldd 3927	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑏 ∈ 𝑖)
66	10, 48	lidlacl 20482	. . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ 𝑖)
68		elunii 4850	. . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → [⊊] Or 𝑍)
72		sorpssi 7576	. . . . . . . . . . . . . 14 ⊢ (( [⊊] Or 𝑍 ∧ (𝑖 ∈ 𝑍 ∧ 𝑗 ∈ 𝑍)) → (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))
73	71, 56, 37, 72	syl12anc 834	. . . . . . . . . . . . 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 3219	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
76	75	an32s 649	. . . . . . . . . 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 3219	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖) ∧ ∃𝑗 ∈ 𝑍 𝑏 ∈ 𝑗) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
79	78	anasss 467	. . . . . . 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 3117	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑎 ∈ ∪ 𝑍∀𝑏 ∈ ∪ 𝑍((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
82	81	ralrimiva 3110	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ ∪ 𝑍∀𝑏 ∈ ∪ 𝑍((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
83	10, 9, 48, 44	islidl 20480	. . . 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 1342	. . 3 ⊢ (𝜑 → ∪ 𝑍 ∈ (LIdeal‘𝑅))
85	4	sselda 3926	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑃)
86		neeq1 3008	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑗 → (𝑝 ≠ 𝐵 ↔ 𝑗 ≠ 𝐵))
87		sseq2 3952	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑗 → (𝐼 ⊆ 𝑝 ↔ 𝐼 ⊆ 𝑗))
88	86, 87	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑗 → ((𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝) ↔ (𝑗 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑗)))
89	88, 5	elrab2 3629	. . . . . . . . . 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 769	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ≠ 𝐵)
92		eqid 2740	. . . . . . . . 9 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
93	9, 92	pridln1 31614	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑗 ≠ 𝐵) → ¬ (1_r‘𝑅) ∈ 𝑗)
94	18, 8, 91, 93	syl3anc 1370	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ¬ (1_r‘𝑅) ∈ 𝑗)
95	94	nrexdv 3200	. . . . . 6 ⊢ (𝜑 → ¬ ∃𝑗 ∈ 𝑍 (1_r‘𝑅) ∈ 𝑗)
96		eluni2 4849	. . . . . 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 20487	. . . . . . 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 2983	. . . . 5 ⊢ (𝜑 → (¬ (1_r‘𝑅) ∈ ∪ 𝑍 ↔ ∪ 𝑍 ≠ 𝐵))
101	97, 100	mpbid 231	. . . 4 ⊢ (𝜑 → ∪ 𝑍 ≠ 𝐵)
102	90	simprrd 771	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐼 ⊆ 𝑗)
103	102	ralrimiva 3110	. . . . . 6 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 𝐼 ⊆ 𝑗)
104		ssint 4901	. . . . . 6 ⊢ (𝐼 ⊆ ∩ 𝑍 ↔ ∀𝑗 ∈ 𝑍 𝐼 ⊆ 𝑗)
105	103, 104	sylibr 233	. . . . 5 ⊢ (𝜑 → 𝐼 ⊆ ∩ 𝑍)
106		intssuni 4907	. . . . . 6 ⊢ (𝑍 ≠ ∅ → ∩ 𝑍 ⊆ ∪ 𝑍)
107	16, 106	syl 17	. . . . 5 ⊢ (𝜑 → ∩ 𝑍 ⊆ ∪ 𝑍)
108	105, 107	sstrd 3936	. . . 4 ⊢ (𝜑 → 𝐼 ⊆ ∪ 𝑍)
109	101, 108	jca 512	. . 3 ⊢ (𝜑 → (∪ 𝑍 ≠ 𝐵 ∧ 𝐼 ⊆ ∪ 𝑍))
110	3, 84, 109	elrabd 3628	. 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 205 ∧ wa 396 ∨ wo 844 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ∀wral 3066 ∃wrex 3067 {crab 3070 ⊆ wss 3892 ∅c0 4262 ∪ cuni 4845 ∩ cint 4885 Or wor 5503 ‘cfv 6432 (class class class)co 7271 [⊊] crpss 7569 Basecbs 16910 +_gcplusg 16960 ._rcmulr 16961 0_gc0g 17148 1_rcur 19735 Ringcrg 19781 LIdealclidl 20430

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-rpss 7570 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-ip 16978 df-0g 17150 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-mgp 19719 df-ur 19736 df-ring 19783 df-subrg 20020 df-lmod 20123 df-lss 20192 df-sra 20432 df-rgmod 20433 df-lidl 20434

This theorem is referenced by: ssmxidl 31638

W3C validator