Theorem zarcmplem 33827

Description: Lemma for zarcmp 33828. (Contributed by Thierry Arnoux, 2-Jul-2024.)

Hypotheses

Ref	Expression
zartop.1	⊢ 𝑆 = (Spec‘𝑅)
zartop.2	⊢ 𝐽 = (TopOpen‘𝑆)
zarcmplem.1	⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})

Assertion

Ref	Expression
zarcmplem	⊢ (𝑅 ∈ CRing → 𝐽 ∈ Comp)

Distinct variable groups:   𝑅,𝑖,𝑗   𝑖,𝐽,𝑗   𝑗,𝑉,𝑖

Allowed substitution hints:   𝑆(𝑖,𝑗)

Proof of Theorem zarcmplem

Dummy variables 𝑘 𝑥 𝑦 𝑎 𝑙 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		crngring 20272	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		zartop.1	. . . . 5 ⊢ 𝑆 = (Spec‘𝑅)
3		zartop.2	. . . . 5 ⊢ 𝐽 = (TopOpen‘𝑆)
4		eqid 2740	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
5	2, 3, 4	zar0ring 33824	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (♯‘(Base‘𝑅)) = 1) → 𝐽 = {∅})
6	1, 5	sylan 579	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) = 1) → 𝐽 = {∅})
7		0cmp 23423	. . 3 ⊢ {∅} ∈ Comp
8	6, 7	eqeltrdi 2852	. 2 ⊢ ((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) = 1) → 𝐽 ∈ Comp)
9	2, 3	zartop 33822	. . 3 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Top)
10		zarcmplem.1	. . . . . . . . . . . . . . 15 ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
11		fvex 6933	. . . . . . . . . . . . . . . 16 ⊢ (LIdeal‘𝑅) ∈ V
12	11	mptex 7260	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) ∈ V
13	10, 12	eqeltri 2840	. . . . . . . . . . . . . 14 ⊢ 𝑉 ∈ V
14		imaexg 7953	. . . . . . . . . . . . . 14 ⊢ (𝑉 ∈ V → (𝑉 “ (𝑎 supp (0g‘𝑅))) ∈ V)
15	13, 14	mp1i 13	. . . . . . . . . . . . 13 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑉 “ (𝑎 supp (0g‘𝑅))) ∈ V)
16		suppssdm 8218	. . . . . . . . . . . . . . 15 ⊢ (𝑎 supp (0g‘𝑅)) ⊆ dom 𝑎
17		imass2 6132	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 supp (0g‘𝑅)) ⊆ dom 𝑎 → (𝑉 “ (𝑎 supp (0g‘𝑅))) ⊆ (𝑉 “ dom 𝑎))
18	16, 17	mp1i 13	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑉 “ (𝑎 supp (0g‘𝑅))) ⊆ (𝑉 “ dom 𝑎))
19	10	funmpt2 6617	. . . . . . . . . . . . . . 15 ⊢ Fun 𝑉
20		ssidd 4032	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → dom 𝑎 ⊆ dom 𝑎)
21		simpllr 775	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) → 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥)))
22		fvexd 6935	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) → (Base‘𝑅) ∈ V)
23	13	cnvex 7965	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ◡𝑉 ∈ V
24	23	imaex 7954	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (◡𝑉 “ 𝑥) ∈ V
25	24	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) → (◡𝑉 “ 𝑥) ∈ V)
26	22, 25	elmapd 8898	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) → (𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥)) ↔ 𝑎:(◡𝑉 “ 𝑥)⟶(Base‘𝑅)))
27	21, 26	mpbid 232	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) → 𝑎:(◡𝑉 “ 𝑥)⟶(Base‘𝑅))
28	27	fdmd 6757	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) → dom 𝑎 = (◡𝑉 “ 𝑥))
29	28	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → dom 𝑎 = (◡𝑉 “ 𝑥))
30	20, 29	sseqtrd 4049	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → dom 𝑎 ⊆ (◡𝑉 “ 𝑥))
31		funimass2 6661	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝑉 ∧ dom 𝑎 ⊆ (◡𝑉 “ 𝑥)) → (𝑉 “ dom 𝑎) ⊆ 𝑥)
32	19, 30, 31	sylancr 586	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑉 “ dom 𝑎) ⊆ 𝑥)
33	18, 32	sstrd 4019	. . . . . . . . . . . . 13 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑉 “ (𝑎 supp (0g‘𝑅))) ⊆ 𝑥)
34	15, 33	elpwd 4628	. . . . . . . . . . . 12 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑉 “ (𝑎 supp (0g‘𝑅))) ∈ 𝒫 𝑥)
35		simpllr 775	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → 𝑎 finSupp (0g‘𝑅))
36	35	fsuppimpd 9439	. . . . . . . . . . . . 13 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑎 supp (0g‘𝑅)) ∈ Fin)
37		imafi 9381	. . . . . . . . . . . . 13 ⊢ ((Fun 𝑉 ∧ (𝑎 supp (0g‘𝑅)) ∈ Fin) → (𝑉 “ (𝑎 supp (0g‘𝑅))) ∈ Fin)
38	19, 36, 37	sylancr 586	. . . . . . . . . . . 12 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑉 “ (𝑎 supp (0g‘𝑅))) ∈ Fin)
39	34, 38	elind 4223	. . . . . . . . . . 11 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑉 “ (𝑎 supp (0g‘𝑅))) ∈ (𝒫 𝑥 ∩ Fin))
40		inteq 4973	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑉 “ (𝑎 supp (0g‘𝑅))) → ∩ 𝑦 = ∩ (𝑉 “ (𝑎 supp (0g‘𝑅))))
41	40	eqeq2d 2751	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑉 “ (𝑎 supp (0g‘𝑅))) → (∅ = ∩ 𝑦 ↔ ∅ = ∩ (𝑉 “ (𝑎 supp (0g‘𝑅)))))
42	41	adantl 481	. . . . . . . . . . 11 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ 𝑦 = (𝑉 “ (𝑎 supp (0g‘𝑅)))) → (∅ = ∩ 𝑦 ↔ ∅ = ∩ (𝑉 “ (𝑎 supp (0g‘𝑅)))))
43	16, 29	sseqtrid 4061	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑎 supp (0g‘𝑅)) ⊆ (◡𝑉 “ 𝑥))
44		cnvimass 6111	. . . . . . . . . . . . . 14 ⊢ (◡𝑉 “ 𝑥) ⊆ dom 𝑉
45	43, 44	sstrdi 4021	. . . . . . . . . . . . 13 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑎 supp (0g‘𝑅)) ⊆ dom 𝑉)
46		intimafv 32722	. . . . . . . . . . . . 13 ⊢ ((Fun 𝑉 ∧ (𝑎 supp (0g‘𝑅)) ⊆ dom 𝑉) → ∩ (𝑉 “ (𝑎 supp (0g‘𝑅))) = ∩ 𝑙 ∈ (𝑎 supp (0g‘𝑅))(𝑉‘𝑙))
47	19, 45, 46	sylancr 586	. . . . . . . . . . . 12 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → ∩ (𝑉 “ (𝑎 supp (0g‘𝑅))) = ∩ 𝑙 ∈ (𝑎 supp (0g‘𝑅))(𝑉‘𝑙))
48		simplll 774	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → 𝑅 ∈ CRing)
49	48	crngringd 20273	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → 𝑅 ∈ Ring)
50	49	ad4antr 731	. . . . . . . . . . . . 13 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → 𝑅 ∈ Ring)
51		fvex 6933	. . . . . . . . . . . . . . . 16 ⊢ (PrmIdeal‘𝑅) ∈ V
52	51	rabex 5357	. . . . . . . . . . . . . . 15 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ∈ V
53	52, 10	dmmpti 6724	. . . . . . . . . . . . . 14 ⊢ dom 𝑉 = (LIdeal‘𝑅)
54	45, 53	sseqtrdi 4059	. . . . . . . . . . . . 13 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑎 supp (0g‘𝑅)) ⊆ (LIdeal‘𝑅))
55		simp-7r 789	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (♯‘(Base‘𝑅)) ≠ 1)
56		simpllr 775	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g‘𝑅)) = ∅) → (1r‘𝑅) = (𝑅 Σg 𝑎))
57		eqid 2740	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0g‘𝑅) = (0g‘𝑅)
58		ringcmn 20305	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
59	1, 58	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ CMnd)
60	59	ad8antr 739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g‘𝑅)) = ∅) → 𝑅 ∈ CMnd)
61	24	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g‘𝑅)) = ∅) → (◡𝑉 “ 𝑥) ∈ V)
62	27	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g‘𝑅)) = ∅) → 𝑎:(◡𝑉 “ 𝑥)⟶(Base‘𝑅))
63		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g‘𝑅)) = ∅) → (𝑎 supp (0g‘𝑅)) = ∅)
64		ssidd 4032	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g‘𝑅)) = ∅) → ∅ ⊆ ∅)
65	63, 64	eqsstrd 4047	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g‘𝑅)) = ∅) → (𝑎 supp (0g‘𝑅)) ⊆ ∅)
66	35	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g‘𝑅)) = ∅) → 𝑎 finSupp (0g‘𝑅))
67	4, 57, 60, 61, 62, 65, 66	gsumres 19955	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g‘𝑅)) = ∅) → (𝑅 Σg (𝑎 ↾ ∅)) = (𝑅 Σg 𝑎))
68		res0 6013	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ↾ ∅) = ∅
69	68	oveq2i 7459	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 Σg (𝑎 ↾ ∅)) = (𝑅 Σg ∅)
70	57	gsum0 18722	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 Σg ∅) = (0g‘𝑅)
71	69, 70	eqtri 2768	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 Σg (𝑎 ↾ ∅)) = (0g‘𝑅)
72	67, 71	eqtr3di 2795	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g‘𝑅)) = ∅) → (𝑅 Σg 𝑎) = (0g‘𝑅))
73	56, 72	eqtr2d 2781	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g‘𝑅)) = ∅) → (0g‘𝑅) = (1r‘𝑅))
74		eqid 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (1r‘𝑅) = (1r‘𝑅)
75	4, 57, 74	01eq0ring 20556	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ (0g‘𝑅) = (1r‘𝑅)) → (Base‘𝑅) = {(0g‘𝑅)})
76	50, 73, 75	syl2an2r 684	. . . . . . . . . . . . . . . 16 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g‘𝑅)) = ∅) → (Base‘𝑅) = {(0g‘𝑅)})
77	76	fveq2d 6924	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g‘𝑅)) = ∅) → (♯‘(Base‘𝑅)) = (♯‘{(0g‘𝑅)}))
78		fvex 6933	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑅) ∈ V
79		hashsng 14418	. . . . . . . . . . . . . . . 16 ⊢ ((0g‘𝑅) ∈ V → (♯‘{(0g‘𝑅)}) = 1)
80	78, 79	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (♯‘{(0g‘𝑅)}) = 1
81	77, 80	eqtrdi 2796	. . . . . . . . . . . . . 14 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g‘𝑅)) = ∅) → (♯‘(Base‘𝑅)) = 1)
82	55, 81	mteqand 3039	. . . . . . . . . . . . 13 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑎 supp (0g‘𝑅)) ≠ ∅)
83		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
84	10, 83	zarclsiin 33817	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ (𝑎 supp (0g‘𝑅)) ⊆ (LIdeal‘𝑅) ∧ (𝑎 supp (0g‘𝑅)) ≠ ∅) → ∩ 𝑙 ∈ (𝑎 supp (0g‘𝑅))(𝑉‘𝑙) = (𝑉‘((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅)))))
85	50, 54, 82, 84	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → ∩ 𝑙 ∈ (𝑎 supp (0g‘𝑅))(𝑉‘𝑙) = (𝑉‘((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅)))))
86		nfv 1913	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑙((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎))
87		nfra1 3290	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑙∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙
88	86, 87	nfan 1898	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑙(((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙)
89	54	sselda 4008	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ 𝑙 ∈ (𝑎 supp (0g‘𝑅))) → 𝑙 ∈ (LIdeal‘𝑅))
90		eqid 2740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
91	4, 90	lidlss 21245	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 ∈ (LIdeal‘𝑅) → 𝑙 ⊆ (Base‘𝑅))
92	89, 91	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ 𝑙 ∈ (𝑎 supp (0g‘𝑅))) → 𝑙 ⊆ (Base‘𝑅))
93	92	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑙 ∈ (𝑎 supp (0g‘𝑅)) → 𝑙 ⊆ (Base‘𝑅)))
94	88, 93	ralrimi 3263	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → ∀𝑙 ∈ (𝑎 supp (0g‘𝑅))𝑙 ⊆ (Base‘𝑅))
95		unissb 4963	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ (𝑎 supp (0g‘𝑅)) ⊆ (Base‘𝑅) ↔ ∀𝑙 ∈ (𝑎 supp (0g‘𝑅))𝑙 ⊆ (Base‘𝑅))
96	94, 95	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → ∪ (𝑎 supp (0g‘𝑅)) ⊆ (Base‘𝑅))
97	83, 4, 90	rspcl 21268	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ ∪ (𝑎 supp (0g‘𝑅)) ⊆ (Base‘𝑅)) → ((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅))) ∈ (LIdeal‘𝑅))
98	50, 96, 97	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → ((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅))) ∈ (LIdeal‘𝑅))
99	4, 90	lidlss 21245	. . . . . . . . . . . . . . . 16 ⊢ (((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅))) ∈ (LIdeal‘𝑅) → ((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅))) ⊆ (Base‘𝑅))
100	98, 99	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → ((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅))) ⊆ (Base‘𝑅))
101	83, 4, 74	rsp1 21270	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Ring → ((RSpan‘𝑅)‘{(1r‘𝑅)}) = (Base‘𝑅))
102	50, 101	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → ((RSpan‘𝑅)‘{(1r‘𝑅)}) = (Base‘𝑅))
103	27	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → 𝑎:(◡𝑉 “ 𝑥)⟶(Base‘𝑅))
104	103, 43	fssresd 6788	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑎 ↾ (𝑎 supp (0g‘𝑅))):(𝑎 supp (0g‘𝑅))⟶(Base‘𝑅))
105		fvex 6933	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝑅) ∈ V
106		ovex 7481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 supp (0g‘𝑅)) ∈ V
107	105, 106	elmap 8929	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 ↾ (𝑎 supp (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m (𝑎 supp (0g‘𝑅))) ↔ (𝑎 ↾ (𝑎 supp (0g‘𝑅))):(𝑎 supp (0g‘𝑅))⟶(Base‘𝑅))
108	104, 107	sylibr 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑎 ↾ (𝑎 supp (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m (𝑎 supp (0g‘𝑅))))
109		breq1 5169	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = (𝑎 ↾ (𝑎 supp (0g‘𝑅))) → (𝑏 finSupp (0g‘𝑅) ↔ (𝑎 ↾ (𝑎 supp (0g‘𝑅))) finSupp (0g‘𝑅)))
110		oveq2 7456	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = (𝑎 ↾ (𝑎 supp (0g‘𝑅))) → (𝑅 Σg 𝑏) = (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g‘𝑅)))))
111	110	eqeq2d 2751	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = (𝑎 ↾ (𝑎 supp (0g‘𝑅))) → ((1r‘𝑅) = (𝑅 Σg 𝑏) ↔ (1r‘𝑅) = (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g‘𝑅))))))
112		fveq1 6919	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = (𝑎 ↾ (𝑎 supp (0g‘𝑅))) → (𝑏‘𝑘) = ((𝑎 ↾ (𝑎 supp (0g‘𝑅)))‘𝑘))
113	112	eleq1d 2829	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = (𝑎 ↾ (𝑎 supp (0g‘𝑅))) → ((𝑏‘𝑘) ∈ 𝑘 ↔ ((𝑎 ↾ (𝑎 supp (0g‘𝑅)))‘𝑘) ∈ 𝑘))
114	113	ralbidv 3184	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = (𝑎 ↾ (𝑎 supp (0g‘𝑅))) → (∀𝑘 ∈ (𝑎 supp (0g‘𝑅))(𝑏‘𝑘) ∈ 𝑘 ↔ ∀𝑘 ∈ (𝑎 supp (0g‘𝑅))((𝑎 ↾ (𝑎 supp (0g‘𝑅)))‘𝑘) ∈ 𝑘))
115	109, 111, 114	3anbi123d 1436	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = (𝑎 ↾ (𝑎 supp (0g‘𝑅))) → ((𝑏 finSupp (0g‘𝑅) ∧ (1r‘𝑅) = (𝑅 Σg 𝑏) ∧ ∀𝑘 ∈ (𝑎 supp (0g‘𝑅))(𝑏‘𝑘) ∈ 𝑘) ↔ ((𝑎 ↾ (𝑎 supp (0g‘𝑅))) finSupp (0g‘𝑅) ∧ (1r‘𝑅) = (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g‘𝑅)))) ∧ ∀𝑘 ∈ (𝑎 supp (0g‘𝑅))((𝑎 ↾ (𝑎 supp (0g‘𝑅)))‘𝑘) ∈ 𝑘)))
116	115	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ 𝑏 = (𝑎 ↾ (𝑎 supp (0g‘𝑅)))) → ((𝑏 finSupp (0g‘𝑅) ∧ (1r‘𝑅) = (𝑅 Σg 𝑏) ∧ ∀𝑘 ∈ (𝑎 supp (0g‘𝑅))(𝑏‘𝑘) ∈ 𝑘) ↔ ((𝑎 ↾ (𝑎 supp (0g‘𝑅))) finSupp (0g‘𝑅) ∧ (1r‘𝑅) = (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g‘𝑅)))) ∧ ∀𝑘 ∈ (𝑎 supp (0g‘𝑅))((𝑎 ↾ (𝑎 supp (0g‘𝑅)))‘𝑘) ∈ 𝑘)))
117		fvexd 6935	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (0g‘𝑅) ∈ V)
118	35, 117	fsuppres 9462	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑎 ↾ (𝑎 supp (0g‘𝑅))) finSupp (0g‘𝑅))
119		simplr 768	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (1r‘𝑅) = (𝑅 Σg 𝑎))
120	50, 58	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → 𝑅 ∈ CMnd)
121	24	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (◡𝑉 “ 𝑥) ∈ V)
122		ssidd 4032	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑎 supp (0g‘𝑅)) ⊆ (𝑎 supp (0g‘𝑅)))
123	4, 57, 120, 121, 103, 122, 35	gsumres 19955	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g‘𝑅)))) = (𝑅 Σg 𝑎))
124	119, 123	eqtr4d 2783	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (1r‘𝑅) = (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g‘𝑅)))))
125		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ 𝑘 ∈ (𝑎 supp (0g‘𝑅))) → 𝑘 ∈ (𝑎 supp (0g‘𝑅)))
126	125	fvresd 6940	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ 𝑘 ∈ (𝑎 supp (0g‘𝑅))) → ((𝑎 ↾ (𝑎 supp (0g‘𝑅)))‘𝑘) = (𝑎‘𝑘))
127	16, 28	sseqtrid 4061	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) → (𝑎 supp (0g‘𝑅)) ⊆ (◡𝑉 “ 𝑥))
128	127	sselda 4008	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ 𝑘 ∈ (𝑎 supp (0g‘𝑅))) → 𝑘 ∈ (◡𝑉 “ 𝑥))
129		fveq2 6920	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑙 = 𝑘 → (𝑎‘𝑙) = (𝑎‘𝑘))
130		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑙 = 𝑘 → 𝑙 = 𝑘)
131	129, 130	eleq12d 2838	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑙 = 𝑘 → ((𝑎‘𝑙) ∈ 𝑙 ↔ (𝑎‘𝑘) ∈ 𝑘))
132	131	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ 𝑘 ∈ (𝑎 supp (0g‘𝑅))) ∧ 𝑙 = 𝑘) → ((𝑎‘𝑙) ∈ 𝑙 ↔ (𝑎‘𝑘) ∈ 𝑘))
133	128, 132	rspcdv 3627	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ 𝑘 ∈ (𝑎 supp (0g‘𝑅))) → (∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙 → (𝑎‘𝑘) ∈ 𝑘))
134	133	imp 406	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ 𝑘 ∈ (𝑎 supp (0g‘𝑅))) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑎‘𝑘) ∈ 𝑘)
135	134	an32s 651	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ 𝑘 ∈ (𝑎 supp (0g‘𝑅))) → (𝑎‘𝑘) ∈ 𝑘)
136	126, 135	eqeltrd 2844	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) ∧ 𝑘 ∈ (𝑎 supp (0g‘𝑅))) → ((𝑎 ↾ (𝑎 supp (0g‘𝑅)))‘𝑘) ∈ 𝑘)
137	136	ralrimiva 3152	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → ∀𝑘 ∈ (𝑎 supp (0g‘𝑅))((𝑎 ↾ (𝑎 supp (0g‘𝑅)))‘𝑘) ∈ 𝑘)
138	118, 124, 137	3jca 1128	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → ((𝑎 ↾ (𝑎 supp (0g‘𝑅))) finSupp (0g‘𝑅) ∧ (1r‘𝑅) = (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g‘𝑅)))) ∧ ∀𝑘 ∈ (𝑎 supp (0g‘𝑅))((𝑎 ↾ (𝑎 supp (0g‘𝑅)))‘𝑘) ∈ 𝑘))
139	108, 116, 138	rspcedvd 3637	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → ∃𝑏 ∈ ((Base‘𝑅) ↑m (𝑎 supp (0g‘𝑅)))(𝑏 finSupp (0g‘𝑅) ∧ (1r‘𝑅) = (𝑅 Σg 𝑏) ∧ ∀𝑘 ∈ (𝑎 supp (0g‘𝑅))(𝑏‘𝑘) ∈ 𝑘))
140		eqid 2740	. . . . . . . . . . . . . . . . . . . 20 ⊢ (.r‘𝑅) = (.r‘𝑅)
141	83, 4, 57, 140, 50, 54	elrspunidl 33421	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → ((1r‘𝑅) ∈ ((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅))) ↔ ∃𝑏 ∈ ((Base‘𝑅) ↑m (𝑎 supp (0g‘𝑅)))(𝑏 finSupp (0g‘𝑅) ∧ (1r‘𝑅) = (𝑅 Σg 𝑏) ∧ ∀𝑘 ∈ (𝑎 supp (0g‘𝑅))(𝑏‘𝑘) ∈ 𝑘)))
142	139, 141	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (1r‘𝑅) ∈ ((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅))))
143	142	snssd 4834	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → {(1r‘𝑅)} ⊆ ((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅))))
144	83, 90	rspssp 21272	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ ((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅))) ∈ (LIdeal‘𝑅) ∧ {(1r‘𝑅)} ⊆ ((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅)))) → ((RSpan‘𝑅)‘{(1r‘𝑅)}) ⊆ ((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅))))
145	50, 98, 143, 144	syl3anc 1371	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → ((RSpan‘𝑅)‘{(1r‘𝑅)}) ⊆ ((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅))))
146	102, 145	eqsstrrd 4048	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (Base‘𝑅) ⊆ ((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅))))
147	100, 146	eqssd 4026	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → ((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅))) = (Base‘𝑅))
148	147	fveq2d 6924	. . . . . . . . . . . . 13 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑉‘((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅)))) = (𝑉‘(Base‘𝑅)))
149	90, 4	lidl1 21266	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ (LIdeal‘𝑅))
150	1, 149	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) ∈ (LIdeal‘𝑅))
151	10, 4	zarcls1 33815	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (LIdeal‘𝑅)) → ((𝑉‘(Base‘𝑅)) = ∅ ↔ (Base‘𝑅) = (Base‘𝑅)))
152	150, 151	mpdan 686	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ CRing → ((𝑉‘(Base‘𝑅)) = ∅ ↔ (Base‘𝑅) = (Base‘𝑅)))
153	4, 152	mpbiri 258	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → (𝑉‘(Base‘𝑅)) = ∅)
154	153	ad7antr 737	. . . . . . . . . . . . 13 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑉‘(Base‘𝑅)) = ∅)
155	148, 154	eqtrd 2780	. . . . . . . . . . . 12 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → (𝑉‘((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅)))) = ∅)
156	47, 85, 155	3eqtrrd 2785	. . . . . . . . . . 11 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → ∅ = ∩ (𝑉 “ (𝑎 supp (0g‘𝑅))))
157	39, 42, 156	rspcedvd 3637	. . . . . . . . . 10 ⊢ ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ 𝑎 finSupp (0g‘𝑅)) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∅ = ∩ 𝑦)
158	157	exp41 434	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) → (𝑎 finSupp (0g‘𝑅) → ((1r‘𝑅) = (𝑅 Σg 𝑎) → (∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∅ = ∩ 𝑦))))
159	158	3imp2 1349	. . . . . . . 8 ⊢ ((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))) ∧ (𝑎 finSupp (0g‘𝑅) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∅ = ∩ 𝑦)
160	4, 74	ringidcl 20289	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
161	49, 160	syl 17	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → (1r‘𝑅) ∈ (Base‘𝑅))
162		simplr 768	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → 𝑥 ∈ 𝒫 (Clsd‘𝐽))
163		eqid 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
164	2, 3, 163, 10	zartopn 33821	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) ∧ ran 𝑉 = (Clsd‘𝐽)))
165	164	simprd 495	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ CRing → ran 𝑉 = (Clsd‘𝐽))
166	48, 165	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ran 𝑉 = (Clsd‘𝐽))
167	166	pweqd 4639	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → 𝒫 ran 𝑉 = 𝒫 (Clsd‘𝐽))
168	162, 167	eleqtrrd 2847	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → 𝑥 ∈ 𝒫 ran 𝑉)
169	168	elpwid 4631	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → 𝑥 ⊆ ran 𝑉)
170		intimafv 32722	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝑉 ∧ (◡𝑉 “ 𝑥) ⊆ dom 𝑉) → ∩ (𝑉 “ (◡𝑉 “ 𝑥)) = ∩ 𝑙 ∈ (◡𝑉 “ 𝑥)(𝑉‘𝑙))
171	19, 44, 170	mp2an 691	. . . . . . . . . . . . . 14 ⊢ ∩ (𝑉 “ (◡𝑉 “ 𝑥)) = ∩ 𝑙 ∈ (◡𝑉 “ 𝑥)(𝑉‘𝑙)
172		funimacnv 6659	. . . . . . . . . . . . . . . . 17 ⊢ (Fun 𝑉 → (𝑉 “ (◡𝑉 “ 𝑥)) = (𝑥 ∩ ran 𝑉))
173	19, 172	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 “ (◡𝑉 “ 𝑥)) = (𝑥 ∩ ran 𝑉)
174		dfss2 3994	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ⊆ ran 𝑉 ↔ (𝑥 ∩ ran 𝑉) = 𝑥)
175	174	biimpi 216	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ ran 𝑉 → (𝑥 ∩ ran 𝑉) = 𝑥)
176	173, 175	eqtrid 2792	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ ran 𝑉 → (𝑉 “ (◡𝑉 “ 𝑥)) = 𝑥)
177	176	inteqd 4975	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ ran 𝑉 → ∩ (𝑉 “ (◡𝑉 “ 𝑥)) = ∩ 𝑥)
178	171, 177	eqtr3id 2794	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ ran 𝑉 → ∩ 𝑙 ∈ (◡𝑉 “ 𝑥)(𝑉‘𝑙) = ∩ 𝑥)
179	169, 178	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ∩ 𝑙 ∈ (◡𝑉 “ 𝑥)(𝑉‘𝑙) = ∩ 𝑥)
180	44	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → (◡𝑉 “ 𝑥) ⊆ dom 𝑉)
181	180, 53	sseqtrdi 4059	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → (◡𝑉 “ 𝑥) ⊆ (LIdeal‘𝑅))
182	19	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → Fun 𝑉)
183		inteq 4973	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → ∩ 𝑥 = ∩ ∅)
184		int0 4986	. . . . . . . . . . . . . . . . . 18 ⊢ ∩ ∅ = V
185	183, 184	eqtrdi 2796	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → ∩ 𝑥 = V)
186		vn0 4368	. . . . . . . . . . . . . . . . . 18 ⊢ V ≠ ∅
187		neeq1 3009	. . . . . . . . . . . . . . . . . 18 ⊢ (∩ 𝑥 = V → (∩ 𝑥 ≠ ∅ ↔ V ≠ ∅))
188	186, 187	mpbiri 258	. . . . . . . . . . . . . . . . 17 ⊢ (∩ 𝑥 = V → ∩ 𝑥 ≠ ∅)
189	185, 188	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → ∩ 𝑥 ≠ ∅)
190	189	necon2i 2981	. . . . . . . . . . . . . . 15 ⊢ (∩ 𝑥 = ∅ → 𝑥 ≠ ∅)
191	190	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → 𝑥 ≠ ∅)
192		preiman0 32721	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝑉 ∧ 𝑥 ⊆ ran 𝑉 ∧ 𝑥 ≠ ∅) → (◡𝑉 “ 𝑥) ≠ ∅)
193	182, 169, 191, 192	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → (◡𝑉 “ 𝑥) ≠ ∅)
194	10, 83	zarclsiin 33817	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ (◡𝑉 “ 𝑥) ⊆ (LIdeal‘𝑅) ∧ (◡𝑉 “ 𝑥) ≠ ∅) → ∩ 𝑙 ∈ (◡𝑉 “ 𝑥)(𝑉‘𝑙) = (𝑉‘((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥))))
195	49, 181, 193, 194	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ∩ 𝑙 ∈ (◡𝑉 “ 𝑥)(𝑉‘𝑙) = (𝑉‘((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥))))
196		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ∩ 𝑥 = ∅)
197	179, 195, 196	3eqtr3d 2788	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → (𝑉‘((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥))) = ∅)
198	181	sselda 4008	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑙 ∈ (◡𝑉 “ 𝑥)) → 𝑙 ∈ (LIdeal‘𝑅))
199	198, 91	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑙 ∈ (◡𝑉 “ 𝑥)) → 𝑙 ⊆ (Base‘𝑅))
200	199	ralrimiva 3152	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ∀𝑙 ∈ (◡𝑉 “ 𝑥)𝑙 ⊆ (Base‘𝑅))
201		unissb 4963	. . . . . . . . . . . . . 14 ⊢ (∪ (◡𝑉 “ 𝑥) ⊆ (Base‘𝑅) ↔ ∀𝑙 ∈ (◡𝑉 “ 𝑥)𝑙 ⊆ (Base‘𝑅))
202	200, 201	sylibr 234	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ∪ (◡𝑉 “ 𝑥) ⊆ (Base‘𝑅))
203	83, 4, 90	rspcl 21268	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ ∪ (◡𝑉 “ 𝑥) ⊆ (Base‘𝑅)) → ((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥)) ∈ (LIdeal‘𝑅))
204	49, 202, 203	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥)) ∈ (LIdeal‘𝑅))
205	10, 4	zarcls1 33815	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ ((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥)) ∈ (LIdeal‘𝑅)) → ((𝑉‘((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥))) = ∅ ↔ ((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥)) = (Base‘𝑅)))
206	48, 204, 205	syl2anc 583	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ((𝑉‘((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥))) = ∅ ↔ ((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥)) = (Base‘𝑅)))
207	197, 206	mpbid 232	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥)) = (Base‘𝑅))
208	161, 207	eleqtrrd 2847	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → (1r‘𝑅) ∈ ((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥)))
209	83, 4, 57, 140, 49, 181	elrspunidl 33421	. . . . . . . . 9 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ((1r‘𝑅) ∈ ((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥)) ↔ ∃𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))(𝑎 finSupp (0g‘𝑅) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙)))
210	208, 209	mpbid 232	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ∃𝑎 ∈ ((Base‘𝑅) ↑m (◡𝑉 “ 𝑥))(𝑎 finSupp (0g‘𝑅) ∧ (1r‘𝑅) = (𝑅 Σg 𝑎) ∧ ∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙))
211	159, 210	r19.29a 3168	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∅ = ∩ 𝑦)
212		0ex 5325	. . . . . . . 8 ⊢ ∅ ∈ V
213		vex 3492	. . . . . . . 8 ⊢ 𝑥 ∈ V
214		elfi 9482	. . . . . . . 8 ⊢ ((∅ ∈ V ∧ 𝑥 ∈ V) → (∅ ∈ (fi‘𝑥) ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∅ = ∩ 𝑦))
215	212, 213, 214	mp2an 691	. . . . . . 7 ⊢ (∅ ∈ (fi‘𝑥) ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∅ = ∩ 𝑦)
216	211, 215	sylibr 234	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ∅ ∈ (fi‘𝑥))
217	216	ex 412	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → (∩ 𝑥 = ∅ → ∅ ∈ (fi‘𝑥)))
218	217	necon3bd 2960	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → (¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅))
219	218	ralrimiva 3152	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) → ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅))
220		cmpfi 23437	. . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)))
221	220	biimpar 477	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)) → 𝐽 ∈ Comp)
222	9, 219, 221	syl2an2r 684	. 2 ⊢ ((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) → 𝐽 ∈ Comp)
223	8, 222	pm2.61dane 3035	1 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Comp)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  {crab 3443  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  {csn 4648  ∪ cuni 4931  ∩ cint 4970  ∩ ciin 5016   class class class wbr 5166   ↦ cmpt 5249  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   ↾ cres 5702   “ cima 5703  Fun wfun 6567  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   supp csupp 8201   ↑_m cmap 8884  Fincfn 9003   finSupp cfsupp 9431  ficfi 9479  1c1 11185  ♯chash 14379  Basecbs 17258  ._rcmulr 17312  TopOpenctopn 17481  0_gc0g 17499   Σ_g cgsu 17500  CMndccmn 19822  1_rcur 20208  Ringcrg 20260  CRingccrg 20261  LIdealclidl 21239  RSpancrsp 21240  Topctop 22920  TopOnctopon 22937  Clsdccld 23045  Compccmp 23415  PrmIdealcprmidl 33428  Speccrspec 33808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-reg 9661  ax-inf2 9710  ax-ac2 10532  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-addf 11263  ax-mulf 11264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-disj 5134  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-rpss 7758  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-er 8763  df-map 8886  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-fi 9480  df-sup 9511  df-oi 9579  df-r1 9833  df-rank 9834  df-dju 9970  df-card 10008  df-ac 10185  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-fzo 13712  df-seq 14053  df-hash 14380  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-starv 17326  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-unif 17334  df-hom 17335  df-cco 17336  df-rest 17482  df-topn 17483  df-0g 17501  df-gsum 17502  df-prds 17507  df-pws 17509  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-mulg 19108  df-subg 19163  df-ghm 19253  df-cntz 19357  df-lsm 19678  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-cring 20263  df-rhm 20498  df-nzr 20539  df-subrng 20572  df-subrg 20597  df-lmod 20882  df-lss 20953  df-lsp 20993  df-lmhm 21044  df-lbs 21097  df-sra 21195  df-rgmod 21196  df-lidl 21241  df-rsp 21242  df-lpidl 21355  df-cnfld 21388  df-zring 21481  df-zrh 21537  df-dsmm 21775  df-frlm 21790  df-uvc 21826  df-top 22921  df-topon 22938  df-cld 23048  df-cmp 23416  df-prmidl 33429  df-mxidl 33453  df-idlsrg 33494  df-rspec 33809

This theorem is referenced by:  zarcmp  33828