ssmxidllem - Mathbox for Thierry Arnoux

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Theorem ssmxidllem 33553

Description: The set 𝑃 used in the proof of ssmxidl 33554 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 2995	. . . 4 ⊢ (𝑝 = ∪ 𝑍 → (𝑝 ≠ 𝐵 ↔ ∪ 𝑍 ≠ 𝐵))
2		sseq2 3949	. . . 4 ⊢ (𝑝 = ∪ 𝑍 → (𝐼 ⊆ 𝑝 ↔ 𝐼 ⊆ ∪ 𝑍))
3	1, 2	anbi12d 633	. . 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 4023	. . . . . . . . 9 ⊢ 𝑃 ⊆ (LIdeal‘𝑅)
7	4, 6	sstrdi 3935	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ⊆ (LIdeal‘𝑅))
8	7	sselda 3922	. . . . . . 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 21200	. . . . . . 7 ⊢ (𝑗 ∈ (LIdeal‘𝑅) → 𝑗 ⊆ 𝐵)
12	8, 11	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ⊆ 𝐵)
13	12	ralrimiva 3130	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 𝑗 ⊆ 𝐵)
14		unissb 4884	. . . . 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 21208	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅)) → (0_g‘𝑅) ∈ 𝑗)
21	18, 8, 20	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (0_g‘𝑅) ∈ 𝑗)
22		n0i 4281	. . . . . . . . 9 ⊢ ((0_g‘𝑅) ∈ 𝑗 → ¬ 𝑗 = ∅)
23	21, 22	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ¬ 𝑗 = ∅)
24	23	reximdva0 4296	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑍 ≠ ∅) → ∃𝑗 ∈ 𝑍 ¬ 𝑗 = ∅)
25	16, 24	mpdan 688	. . . . . 6 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ¬ 𝑗 = ∅)
26		rexnal 3090	. . . . . 6 ⊢ (∃𝑗 ∈ 𝑍 ¬ 𝑗 = ∅ ↔ ¬ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
27	25, 26	sylib 218	. . . . 5 ⊢ (𝜑 → ¬ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
28		uni0c 4878	. . . . . 6 ⊢ (∪ 𝑍 = ∅ ↔ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
29	28	necon3abii 2979	. . . . 5 ⊢ (∪ 𝑍 ≠ ∅ ↔ ¬ ∀𝑗 ∈ 𝑍 𝑗 = ∅)
30	27, 29	sylibr 234	. . . 4 ⊢ (𝜑 → ∪ 𝑍 ≠ ∅)
31		eluni2 4855	. . . . . . . 8 ⊢ (𝑎 ∈ ∪ 𝑍 ↔ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖)
32		eluni2 4855	. . . . . . . 8 ⊢ (𝑏 ∈ ∪ 𝑍 ↔ ∃𝑗 ∈ 𝑍 𝑏 ∈ 𝑗)
33	31, 32	anbi12i 629	. . . . . . 7 ⊢ ((𝑎 ∈ ∪ 𝑍 ∧ 𝑏 ∈ ∪ 𝑍) ↔ (∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖 ∧ ∃𝑗 ∈ 𝑍 𝑏 ∈ 𝑗))
34		an32 647	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖) ∧ 𝑗 ∈ 𝑍) ↔ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖))
35	17	ad6antr 737	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑖 ⊆ 𝑗) → 𝑅 ∈ Ring)
36	7	ad5antr 735	. . . . . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . . . . 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 21213	. . . . . . . . . . . . . . . 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 21209	. . . . . . . . . . . . . . 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 4856	. . . . . . . . . . . . . 14 ⊢ ((((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ 𝑗 ∧ 𝑗 ∈ 𝑍) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
53	50, 51, 52	syl2anc 585	. . . . . . . . . . . . 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 737	. . . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . . . 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 21213	. . . . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) ∧ 𝑗 ⊆ 𝑖) → 𝑏 ∈ 𝑖)
66	10, 48	lidlacl 21209	. . . . . . . . . . . . . . 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 4856	. . . . . . . . . . . . . 14 ⊢ ((((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ 𝑖 ∧ 𝑖 ∈ 𝑍) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
69	67, 57, 68	syl2anc 585	. . . . . . . . . . . . 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 735	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ 𝑖 ∈ 𝑍) ∧ 𝑎 ∈ 𝑖) → [⊊] Or 𝑍)
72		sorpssi 7674	. . . . . . . . . . . . . 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 3142	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
76	75	an32s 653	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑗 ∈ 𝑍) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖) ∧ 𝑏 ∈ 𝑗) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
77	34, 76	sylanb 582	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ ∃𝑖 ∈ 𝑍 𝑎 ∈ 𝑖) ∧ 𝑗 ∈ 𝑍) ∧ 𝑏 ∈ 𝑗) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
78	77	r19.29an 3142	. . . . . . . 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 595	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (𝑎 ∈ ∪ 𝑍 ∧ 𝑏 ∈ ∪ 𝑍)) → ((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
81	80	ralrimivva 3181	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑎 ∈ ∪ 𝑍∀𝑏 ∈ ∪ 𝑍((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
82	81	ralrimiva 3130	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ ∪ 𝑍∀𝑏 ∈ ∪ 𝑍((𝑥(._r‘𝑅)𝑎)(+_g‘𝑅)𝑏) ∈ ∪ 𝑍)
83	10, 9, 48, 44	islidl 21203	. . . 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 1345	. . 3 ⊢ (𝜑 → ∪ 𝑍 ∈ (LIdeal‘𝑅))
85	4	sselda 3922	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑃)
86		neeq1 2995	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑗 → (𝑝 ≠ 𝐵 ↔ 𝑗 ≠ 𝐵))
87		sseq2 3949	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑗 → (𝐼 ⊆ 𝑝 ↔ 𝐼 ⊆ 𝑗))
88	86, 87	anbi12d 633	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑗 → ((𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝) ↔ (𝑗 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑗)))
89	88, 5	elrab2 3638	. . . . . . . . . 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 33523	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑗 ≠ 𝐵) → ¬ (1_r‘𝑅) ∈ 𝑗)
94	18, 8, 91, 93	syl3anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ¬ (1_r‘𝑅) ∈ 𝑗)
95	94	nrexdv 3133	. . . . . 6 ⊢ (𝜑 → ¬ ∃𝑗 ∈ 𝑍 (1_r‘𝑅) ∈ 𝑗)
96		eluni2 4855	. . . . . 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 21214	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ∪ 𝑍 ∈ (LIdeal‘𝑅)) → ((1_r‘𝑅) ∈ ∪ 𝑍 ↔ ∪ 𝑍 = 𝐵))
99	17, 84, 98	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ((1_r‘𝑅) ∈ ∪ 𝑍 ↔ ∪ 𝑍 = 𝐵))
100	99	necon3bbid 2970	. . . . 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 3130	. . . . . 6 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 𝐼 ⊆ 𝑗)
104		ssint 4907	. . . . . 6 ⊢ (𝐼 ⊆ ∩ 𝑍 ↔ ∀𝑗 ∈ 𝑍 𝐼 ⊆ 𝑗)
105	103, 104	sylibr 234	. . . . 5 ⊢ (𝜑 → 𝐼 ⊆ ∩ 𝑍)
106		intssuni 4913	. . . . . 6 ⊢ (𝑍 ≠ ∅ → ∩ 𝑍 ⊆ ∪ 𝑍)
107	16, 106	syl 17	. . . . 5 ⊢ (𝜑 → ∩ 𝑍 ⊆ ∪ 𝑍)
108	105, 107	sstrd 3933	. . . 4 ⊢ (𝜑 → 𝐼 ⊆ ∪ 𝑍)
109	101, 108	jca 511	. . 3 ⊢ (𝜑 → (∪ 𝑍 ≠ 𝐵 ∧ 𝐼 ⊆ ∪ 𝑍))
110	3, 84, 109	elrabd 3637	. 2 ⊢ (𝜑 → ∪ 𝑍 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝)})
111	110, 5	eleqtrrdi 2848	1 ⊢ (𝜑 → ∪ 𝑍 ∈ 𝑃)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 {crab 3390 ⊆ wss 3890 ∅c0 4274 ∪ cuni 4851 ∩ cint 4890 Or wor 5529 ‘cfv 6490 (class class class)co 7358 [⊊] crpss 7667 Basecbs 17168 +_gcplusg 17209 ._rcmulr 17210 0_gc0g 17391 1_rcur 20151 Ringcrg 20203 LIdealclidl 21194

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-rpss 7668 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-er 8634 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-sca 17225 df-vsca 17226 df-ip 17227 df-0g 17393 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19088 df-cmn 19746 df-abl 19747 df-mgp 20111 df-rng 20123 df-ur 20152 df-ring 20205 df-subrg 20536 df-lmod 20846 df-lss 20916 df-sra 21158 df-rgmod 21159 df-lidl 21196

This theorem is referenced by: ssmxidl 33554

W3C validator