Theorem zarcmplem 34513

Description: Lemma for zarcmp 34514. (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 20472	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		zartop.1	. . . . 5 ⊢ 𝑆 = (Spec‘𝑅)
3		zartop.2	. . . . 5 ⊢ 𝐽 = (TopOpen‘𝑆)
4		eqid 2761	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
5	2, 3, 4	zar0ring 34510	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (♯‘(Base‘𝑅)) = 1) → 𝐽 = {∅})
6	1, 5	sylan 592	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) = 1) → 𝐽 = {∅})
7		0cmp 23712	. . 3 ⊢ {∅} ∈ Comp
8	6, 7	eqeltrdi 2869	. 2 ⊢ ((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) = 1) → 𝐽 ∈ Comp)
9	2, 3	zartop 34508	. . 3 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Top)
10		zarcmplem.1	. . . . . . . . . . . . . . 15 ⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
11		fvex 6898	. . . . . . . . . . . . . . . 16 ⊢ (LIdeal‘𝑅) ∈ V
12	11	mptex 7229	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) ∈ V
13	10, 12	eqeltri 2857	. . . . . . . . . . . . . 14 ⊢ 𝑉 ∈ V
14		imaexg 7925	. . . . . . . . . . . . . 14 ⊢ (𝑉 ∈ V → (𝑉 “ (𝑎 supp (0g‘𝑅))) ∈ V)
15	13, 14	mp1i 14	. . . . . . . . . . . . 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 8194	. . . . . . . . . . . . . . 15 ⊢ (𝑎 supp (0g‘𝑅)) ⊆ dom 𝑎
17		imass2 6055	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 supp (0g‘𝑅)) ⊆ dom 𝑎 → (𝑉 “ (𝑎 supp (0g‘𝑅))) ⊆ (𝑉 “ dom 𝑎))
18	16, 17	mp1i 14	. . . . . . . . . . . . . 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 6579	. . . . . . . . . . . . . . 15 ⊢ Fun 𝑉
20		ssidd 3954	. . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . . . 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 6900	. . . . . . . . . . . . . . . . . . . 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 7937	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ◡𝑉 ∈ V
24	23	imaex 7926	. . . . . . . . . . . . . . . . . . . . 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 8860	. . . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . . 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 6720	. . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 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 3967	. . . . . . . . . . . . . . 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 6623	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝑉 ∧ dom 𝑎 ⊆ (◡𝑉 “ 𝑥)) → (𝑉 “ dom 𝑎) ⊆ 𝑥)
32	19, 30, 31	sylancr 599	. . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . 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 4563	. . . . . . . . . . . 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 788	. . . . . . . . . . . . . 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 9361	. . . . . . . . . . . . 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 9307	. . . . . . . . . . . . 13 ⊢ ((Fun 𝑉 ∧ (𝑎 supp (0g‘𝑅)) ∈ Fin) → (𝑉 “ (𝑎 supp (0g‘𝑅))) ∈ Fin)
38	19, 36, 37	sylancr 599	. . . . . . . . . . . 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 4146	. . . . . . . . . . 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 4910	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑉 “ (𝑎 supp (0g‘𝑅))) → ∩ 𝑦 = ∩ (𝑉 “ (𝑎 supp (0g‘𝑅))))
41	40	eqeq2d 2772	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑉 “ (𝑎 supp (0g‘𝑅))) → (∅ = ∩ 𝑦 ↔ ∅ = ∩ (𝑉 “ (𝑎 supp (0g‘𝑅)))))
42	41	adantl 487	. . . . . . . . . . 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 3973	. . . . . . . . . . . . . 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 6198	. . . . . . . . . . . . . 14 ⊢ (◡𝑉 “ 𝑥) ⊆ dom 𝑉
45	43, 44	sstrdi 3943	. . . . . . . . . . . . 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 33304	. . . . . . . . . . . . 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 599	. . . . . . . . . . . 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 787	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → 𝑅 ∈ CRing)
49	48	crngringd 20473	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → 𝑅 ∈ Ring)
50	49	ad4antr 745	. . . . . . . . . . . . 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 6898	. . . . . . . . . . . . . . . 16 ⊢ (PrmIdeal‘𝑅) ∈ V
52	51	rabex 5300	. . . . . . . . . . . . . . 15 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} ∈ V
53	52, 10	dmmpti 6683	. . . . . . . . . . . . . 14 ⊢ dom 𝑉 = (LIdeal‘𝑅)
54	45, 53	sseqtrdi 3971	. . . . . . . . . . . . 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 802	. . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0g‘𝑅) = (0g‘𝑅)
58		ringcmn 20511	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
59	1, 58	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ CMnd)
60	59	ad8antr 753	. . . . . . . . . . . . . . . . . . . 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 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)) = ∅) → 𝑎:(◡𝑉 “ 𝑥)⟶(Base‘𝑅))
63		simpr 490	. . . . . . . . . . . . . . . . . . . . 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 3954	. . . . . . . . . . . . . . . . . . . . 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 3965	. . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . 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 20127	. . . . . . . . . . . . . . . . . . 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 5974	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ↾ ∅) = ∅
69	68	oveq2i 7431	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 Σg (𝑎 ↾ ∅)) = (𝑅 Σg ∅)
70	57	gsum0 18873	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 Σg ∅) = (0g‘𝑅)
71	69, 70	eqtri 2784	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 Σg (𝑎 ↾ ∅)) = (0g‘𝑅)
72	67, 71	eqtr3di 2811	. . . . . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . 18 ⊢ (1r‘𝑅) = (1r‘𝑅)
75	4, 57, 74	01eq0ring 20781	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ (0g‘𝑅) = (1r‘𝑅)) → (Base‘𝑅) = {(0g‘𝑅)})
76	50, 73, 75	syl2an2r 698	. . . . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . . . 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 6898	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝑅) ∈ V
79		hashsng 14513	. . . . . . . . . . . . . . . 16 ⊢ ((0g‘𝑅) ∈ V → (♯‘{(0g‘𝑅)}) = 1)
80	78, 79	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (♯‘{(0g‘𝑅)}) = 1
81	77, 80	eqtrdi 2812	. . . . . . . . . . . . . 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 3047	. . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . 14 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
84	10, 83	zarclsiin 34503	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 1947	. . . . . . . . . . . . . . . . . . . 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 3287	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑙∀𝑙 ∈ (◡𝑉 “ 𝑥)(𝑎‘𝑙) ∈ 𝑙
88	86, 87	nfan 1932	. . . . . . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
91	4, 90	lidlss 21490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 ∈ (LIdeal‘𝑅) → 𝑙 ⊆ (Base‘𝑅))
92	89, 91	syl 18	. . . . . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . . . . 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 3261	. . . . . . . . . . . . . . . . . 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 4901	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ (𝑎 supp (0g‘𝑅)) ⊆ (Base‘𝑅) ↔ ∀𝑙 ∈ (𝑎 supp (0g‘𝑅))𝑙 ⊆ (Base‘𝑅))
96	94, 95	sylibr 237	. . . . . . . . . . . . . . . . 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 21518	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ ∪ (𝑎 supp (0g‘𝑅)) ⊆ (Base‘𝑅)) → ((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅))) ∈ (LIdeal‘𝑅))
98	50, 96, 97	syl2anc 596	. . . . . . . . . . . . . . . 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 21490	. . . . . . . . . . . . . . . 16 ⊢ (((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅))) ∈ (LIdeal‘𝑅) → ((RSpan‘𝑅)‘∪ (𝑎 supp (0g‘𝑅))) ⊆ (Base‘𝑅))
100	98, 99	syl 18	. . . . . . . . . . . . . . 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 21520	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Ring → ((RSpan‘𝑅)‘{(1r‘𝑅)}) = (Base‘𝑅))
102	50, 101	syl 18	. . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . 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 6749	. . . . . . . . . . . . . . . . . . . . 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 6898	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝑅) ∈ V
106		ovex 7453	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 supp (0g‘𝑅)) ∈ V
107	105, 106	elmap 8899	. . . . . . . . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . . . . . . . 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 5106	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = (𝑎 ↾ (𝑎 supp (0g‘𝑅))) → (𝑏 finSupp (0g‘𝑅) ↔ (𝑎 ↾ (𝑎 supp (0g‘𝑅))) finSupp (0g‘𝑅)))
110		oveq2 7428	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = (𝑎 ↾ (𝑎 supp (0g‘𝑅))) → (𝑅 Σg 𝑏) = (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g‘𝑅)))))
111	110	eqeq2d 2772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = (𝑎 ↾ (𝑎 supp (0g‘𝑅))) → ((1r‘𝑅) = (𝑅 Σg 𝑏) ↔ (1r‘𝑅) = (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g‘𝑅))))))
112		fveq1 6884	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = (𝑎 ↾ (𝑎 supp (0g‘𝑅))) → (𝑏‘𝑘) = ((𝑎 ↾ (𝑎 supp (0g‘𝑅)))‘𝑘))
113	112	eleq1d 2846	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = (𝑎 ↾ (𝑎 supp (0g‘𝑅))) → ((𝑏‘𝑘) ∈ 𝑘 ↔ ((𝑎 ↾ (𝑎 supp (0g‘𝑅)))‘𝑘) ∈ 𝑘))
114	113	ralbidv 3186	. . . . . . . . . . . . . . . . . . . . . 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 1464	. . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . 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 6900	. . . . . . . . . . . . . . . . . . . . . 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 9385	. . . . . . . . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . . . 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 3954	. . . . . . . . . . . . . . . . . . . . . . 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 20127	. . . . . . . . . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . . . . . . . . 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 3973	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑙 = 𝑘 → (𝑎‘𝑙) = (𝑎‘𝑘))
130		id 23	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑙 = 𝑘 → 𝑙 = 𝑘)
131	129, 130	eleq12d 2855	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑙 = 𝑘 → ((𝑎‘𝑙) ∈ 𝑙 ↔ (𝑎‘𝑘) ∈ 𝑘))
132	131	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3569	. . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . 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 665	. . . . . . . . . . . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . . . . . . . . . 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 3155	. . . . . . . . . . . . . . . . . . . . 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 1146	. . . . . . . . . . . . . . . . . . . 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 3579	. . . . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . . . 20 ⊢ (.r‘𝑅) = (.r‘𝑅)
141	83, 4, 57, 140, 50, 54	elrspunidl 33978	. . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . 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 4747	. . . . . . . . . . . . . . . . 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 21522	. . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . 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 3966	. . . . . . . . . . . . . . 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 3948	. . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . 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 21513	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ (LIdeal‘𝑅))
150	1, 149	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) ∈ (LIdeal‘𝑅))
151	10, 4	zarcls1 34501	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (LIdeal‘𝑅)) → ((𝑉‘(Base‘𝑅)) = ∅ ↔ (Base‘𝑅) = (Base‘𝑅)))
152	150, 151	mpdan 700	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ CRing → ((𝑉‘(Base‘𝑅)) = ∅ ↔ (Base‘𝑅) = (Base‘𝑅)))
153	4, 152	mpbiri 261	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → (𝑉‘(Base‘𝑅)) = ∅)
154	153	ad7antr 751	. . . . . . . . . . . . 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 2796	. . . . . . . . . . . 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 2801	. . . . . . . . . . 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 3579	. . . . . . . . . 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 440	. . . . . . . . 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 1368	. . . . . . . 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 20494	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
161	49, 160	syl 18	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → (1r‘𝑅) ∈ (Base‘𝑅))
162		simplr 781	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → 𝑥 ∈ 𝒫 (Clsd‘𝐽))
163		eqid 2761	. . . . . . . . . . . . . . . . . . 19 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
164	2, 3, 163, 10	zartopn 34507	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) ∧ ran 𝑉 = (Clsd‘𝐽)))
165	164	simprd 501	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ CRing → ran 𝑉 = (Clsd‘𝐽))
166	48, 165	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ran 𝑉 = (Clsd‘𝐽))
167	166	pweqd 4574	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → 𝒫 ran 𝑉 = 𝒫 (Clsd‘𝐽))
168	162, 167	eleqtrrd 2864	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → 𝑥 ∈ 𝒫 ran 𝑉)
169	168	elpwid 4566	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → 𝑥 ⊆ ran 𝑉)
170		intimafv 33304	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝑉 ∧ (◡𝑉 “ 𝑥) ⊆ dom 𝑉) → ∩ (𝑉 “ (◡𝑉 “ 𝑥)) = ∩ 𝑙 ∈ (◡𝑉 “ 𝑥)(𝑉‘𝑙))
171	19, 44, 170	mp2an 705	. . . . . . . . . . . . . 14 ⊢ ∩ (𝑉 “ (◡𝑉 “ 𝑥)) = ∩ 𝑙 ∈ (◡𝑉 “ 𝑥)(𝑉‘𝑙)
172		funimacnv 6621	. . . . . . . . . . . . . . . . 17 ⊢ (Fun 𝑉 → (𝑉 “ (◡𝑉 “ 𝑥)) = (𝑥 ∩ ran 𝑉))
173	19, 172	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 “ (◡𝑉 “ 𝑥)) = (𝑥 ∩ ran 𝑉)
174		dfss2 3917	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ⊆ ran 𝑉 ↔ (𝑥 ∩ ran 𝑉) = 𝑥)
175	174	biimpi 219	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ ran 𝑉 → (𝑥 ∩ ran 𝑉) = 𝑥)
176	173, 175	eqtrid 2808	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ ran 𝑉 → (𝑉 “ (◡𝑉 “ 𝑥)) = 𝑥)
177	176	inteqd 4912	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ ran 𝑉 → ∩ (𝑉 “ (◡𝑉 “ 𝑥)) = ∩ 𝑥)
178	171, 177	eqtr3id 2810	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ ran 𝑉 → ∩ 𝑙 ∈ (◡𝑉 “ 𝑥)(𝑉‘𝑙) = ∩ 𝑥)
179	169, 178	syl 18	. . . . . . . . . . . 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 3971	. . . . . . . . . . . . 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 4910	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ∅ → ∩ 𝑥 = ∩ ∅)
184		int0 4922	. . . . . . . . . . . . . . . . . 18 ⊢ ∩ ∅ = V
185	183, 184	eqtrdi 2812	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ∅ → ∩ 𝑥 = V)
186		vn0 4291	. . . . . . . . . . . . . . . . . 18 ⊢ V ≠ ∅
187		neeq1 3018	. . . . . . . . . . . . . . . . . 18 ⊢ (∩ 𝑥 = V → (∩ 𝑥 ≠ ∅ ↔ V ≠ ∅))
188	186, 187	mpbiri 261	. . . . . . . . . . . . . . . . 17 ⊢ (∩ 𝑥 = V → ∩ 𝑥 ≠ ∅)
189	185, 188	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ∅ → ∩ 𝑥 ≠ ∅)
190	189	necon2i 2990	. . . . . . . . . . . . . . 15 ⊢ (∩ 𝑥 = ∅ → 𝑥 ≠ ∅)
191	190	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → 𝑥 ≠ ∅)
192		preiman0 33303	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝑉 ∧ 𝑥 ⊆ ran 𝑉 ∧ 𝑥 ≠ ∅) → (◡𝑉 “ 𝑥) ≠ ∅)
193	182, 169, 191, 192	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → (◡𝑉 “ 𝑥) ≠ ∅)
194	10, 83	zarclsiin 34503	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ∩ 𝑙 ∈ (◡𝑉 “ 𝑥)(𝑉‘𝑙) = (𝑉‘((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥))))
196		simpr 490	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ∩ 𝑥 = ∅)
197	179, 195, 196	3eqtr3d 2804	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → (𝑉‘((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥))) = ∅)
198	181	sselda 3931	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑙 ∈ (◡𝑉 “ 𝑥)) → 𝑙 ∈ (LIdeal‘𝑅))
199	198, 91	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) ∧ 𝑙 ∈ (◡𝑉 “ 𝑥)) → 𝑙 ⊆ (Base‘𝑅))
200	199	ralrimiva 3155	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ∀𝑙 ∈ (◡𝑉 “ 𝑥)𝑙 ⊆ (Base‘𝑅))
201		unissb 4901	. . . . . . . . . . . . . 14 ⊢ (∪ (◡𝑉 “ 𝑥) ⊆ (Base‘𝑅) ↔ ∀𝑙 ∈ (◡𝑉 “ 𝑥)𝑙 ⊆ (Base‘𝑅))
202	200, 201	sylibr 237	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ∪ (◡𝑉 “ 𝑥) ⊆ (Base‘𝑅))
203	83, 4, 90	rspcl 21518	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ ∪ (◡𝑉 “ 𝑥) ⊆ (Base‘𝑅)) → ((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥)) ∈ (LIdeal‘𝑅))
204	49, 202, 203	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥)) ∈ (LIdeal‘𝑅))
205	10, 4	zarcls1 34501	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 235	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ((RSpan‘𝑅)‘∪ (◡𝑉 “ 𝑥)) = (Base‘𝑅))
208	161, 207	eleqtrrd 2864	. . . . . . . . 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 33978	. . . . . . . . 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 235	. . . . . . . 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 3171	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∅ = ∩ 𝑦)
212		0ex 5261	. . . . . . . 8 ⊢ ∅ ∈ V
213		vex 3455	. . . . . . . 8 ⊢ 𝑥 ∈ V
214		elfi 9405	. . . . . . . 8 ⊢ ((∅ ∈ V ∧ 𝑥 ∈ V) → (∅ ∈ (fi‘𝑥) ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∅ = ∩ 𝑦))
215	212, 213, 214	mp2an 705	. . . . . . 7 ⊢ (∅ ∈ (fi‘𝑥) ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∅ = ∩ 𝑦)
216	211, 215	sylibr 237	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ ∩ 𝑥 = ∅) → ∅ ∈ (fi‘𝑥))
217	216	ex 418	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → (∩ 𝑥 = ∅ → ∅ ∈ (fi‘𝑥)))
218	217	necon3bd 2970	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → (¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅))
219	218	ralrimiva 3155	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) → ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅))
220		cmpfi 23726	. . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)))
221	220	biimpar 483	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → ∩ 𝑥 ≠ ∅)) → 𝐽 ∈ Comp)
222	9, 219, 221	syl2an2r 698	. 2 ⊢ ((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) → 𝐽 ∈ Comp)
223	8, 222	pm2.61dane 3043	1 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Comp)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  {crab 3413  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  ∪ cuni 4867  ∩ cint 4907  ∩ ciin 4952   class class class wbr 5103   ↦ cmpt 5186  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   ↾ cres 5653   “ cima 5654  Fun wfun 6532  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   supp csupp 8177   ↑_m cmap 8847  Fincfn 8973   finSupp cfsupp 9353  ficfi 9402  1c1 11201  ♯chash 14474  Basecbs 17387  ._rcmulr 17429  TopOpenctopn 17592  0_gc0g 17610   Σ_g cgsu 17611  CMndccmn 19994  1_rcur 20407  Ringcrg 20459  CRingccrg 20460  LIdealclidl 21484  RSpancrsp 21485  PrmIdealcprmidl 21616  Topctop 23211  TopOnctopon 23228  Clsdccld 23334  Compccmp 23704  Speccrspec 34494

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-reg 9586  ax-inf2 9642  ax-ac2 10541  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-addf 11279  ax-mulf 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-disj 5071  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-rpss 7739  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8178  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-er 8717  df-map 8849  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fsupp 9354  df-fi 9403  df-sup 9434  df-oi 9504  df-r1 9768  df-rank 9769  df-scott 9929  df-dju 9982  df-card 10020  df-ac 10195  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-fzo 13789  df-seq 14145  df-hash 14475  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-starv 17443  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-unif 17451  df-hom 17452  df-cco 17453  df-rest 17593  df-topn 17594  df-0g 17612  df-gsum 17613  df-prds 17618  df-pws 17620  df-mre 17756  df-mrc 17757  df-acs 17759  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-mhm 18978  df-submnd 18979  df-grp 19147  df-minusg 19148  df-sbg 19149  df-mulg 19278  df-subg 19333  df-ghm 19428  df-cntz 19531  df-lsm 19850  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-ring 20461  df-cring 20462  df-rhm 20702  df-nzr 20763  df-subrng 20798  df-subrg 20822  df-lmod 21137  df-lss 21207  df-lsp 21247  df-lmhm 21297  df-lbs 21350  df-sra 21448  df-rgmod 21449  df-lidl 21486  df-rsp 21487  df-prmidl 21617  df-lpidl 21646  df-cnfld 21679  df-zring 21753  df-zrh 21809  df-dsmm 22038  df-frlm 22053  df-uvc 22089  df-top 23212  df-topon 23229  df-cld 23337  df-cmp 23705  df-mxidl 33985  df-idlsrg 34033  df-rspec 34495

This theorem is used by:  zarcmp  34514