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Theorem zarcmplem 33881
Description: Lemma for zarcmp 33882. (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) ∣ 𝑖𝑗})
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.
StepHypRef Expression
1 crngring 20243 . . . 4 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2 zartop.1 . . . . 5 𝑆 = (Spec‘𝑅)
3 zartop.2 . . . . 5 𝐽 = (TopOpen‘𝑆)
4 eqid 2736 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
52, 3, 4zar0ring 33878 . . . 4 ((𝑅 ∈ Ring ∧ (♯‘(Base‘𝑅)) = 1) → 𝐽 = {∅})
61, 5sylan 580 . . 3 ((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) = 1) → 𝐽 = {∅})
7 0cmp 23403 . . 3 {∅} ∈ Comp
86, 7eqeltrdi 2848 . 2 ((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) = 1) → 𝐽 ∈ Comp)
92, 3zartop 33876 . . 3 (𝑅 ∈ CRing → 𝐽 ∈ Top)
10 zarcmplem.1 . . . . . . . . . . . . . . 15 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
11 fvex 6918 . . . . . . . . . . . . . . . 16 (LIdeal‘𝑅) ∈ V
1211mptex 7244 . . . . . . . . . . . . . . 15 (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗}) ∈ V
1310, 12eqeltri 2836 . . . . . . . . . . . . . 14 𝑉 ∈ V
14 imaexg 7936 . . . . . . . . . . . . . 14 (𝑉 ∈ V → (𝑉 “ (𝑎 supp (0g𝑅))) ∈ V)
1513, 14mp1i 13 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑉 “ (𝑎 supp (0g𝑅))) ∈ V)
16 suppssdm 8203 . . . . . . . . . . . . . . 15 (𝑎 supp (0g𝑅)) ⊆ dom 𝑎
17 imass2 6119 . . . . . . . . . . . . . . 15 ((𝑎 supp (0g𝑅)) ⊆ dom 𝑎 → (𝑉 “ (𝑎 supp (0g𝑅))) ⊆ (𝑉 “ dom 𝑎))
1816, 17mp1i 13 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑉 “ (𝑎 supp (0g𝑅))) ⊆ (𝑉 “ dom 𝑎))
1910funmpt2 6604 . . . . . . . . . . . . . . 15 Fun 𝑉
20 ssidd 4006 . . . . . . . . . . . . . . . 16 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → dom 𝑎 ⊆ dom 𝑎)
21 simpllr 775 . . . . . . . . . . . . . . . . . . 19 (((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) → 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥)))
22 fvexd 6920 . . . . . . . . . . . . . . . . . . . 20 (((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) → (Base‘𝑅) ∈ V)
2313cnvex 7948 . . . . . . . . . . . . . . . . . . . . . 22 𝑉 ∈ V
2423imaex 7937 . . . . . . . . . . . . . . . . . . . . 21 (𝑉𝑥) ∈ V
2524a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) → (𝑉𝑥) ∈ V)
2622, 25elmapd 8881 . . . . . . . . . . . . . . . . . . 19 (((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) → (𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥)) ↔ 𝑎:(𝑉𝑥)⟶(Base‘𝑅)))
2721, 26mpbid 232 . . . . . . . . . . . . . . . . . 18 (((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) → 𝑎:(𝑉𝑥)⟶(Base‘𝑅))
2827fdmd 6745 . . . . . . . . . . . . . . . . 17 (((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) → dom 𝑎 = (𝑉𝑥))
2928adantr 480 . . . . . . . . . . . . . . . 16 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → dom 𝑎 = (𝑉𝑥))
3020, 29sseqtrd 4019 . . . . . . . . . . . . . . 15 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → dom 𝑎 ⊆ (𝑉𝑥))
31 funimass2 6648 . . . . . . . . . . . . . . 15 ((Fun 𝑉 ∧ dom 𝑎 ⊆ (𝑉𝑥)) → (𝑉 “ dom 𝑎) ⊆ 𝑥)
3219, 30, 31sylancr 587 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑉 “ dom 𝑎) ⊆ 𝑥)
3318, 32sstrd 3993 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑉 “ (𝑎 supp (0g𝑅))) ⊆ 𝑥)
3415, 33elpwd 4605 . . . . . . . . . . . 12 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑉 “ (𝑎 supp (0g𝑅))) ∈ 𝒫 𝑥)
35 simpllr 775 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → 𝑎 finSupp (0g𝑅))
3635fsuppimpd 9410 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑎 supp (0g𝑅)) ∈ Fin)
37 imafi 9354 . . . . . . . . . . . . 13 ((Fun 𝑉 ∧ (𝑎 supp (0g𝑅)) ∈ Fin) → (𝑉 “ (𝑎 supp (0g𝑅))) ∈ Fin)
3819, 36, 37sylancr 587 . . . . . . . . . . . 12 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑉 “ (𝑎 supp (0g𝑅))) ∈ Fin)
3934, 38elind 4199 . . . . . . . . . . 11 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑉 “ (𝑎 supp (0g𝑅))) ∈ (𝒫 𝑥 ∩ Fin))
40 inteq 4948 . . . . . . . . . . . . 13 (𝑦 = (𝑉 “ (𝑎 supp (0g𝑅))) → 𝑦 = (𝑉 “ (𝑎 supp (0g𝑅))))
4140eqeq2d 2747 . . . . . . . . . . . 12 (𝑦 = (𝑉 “ (𝑎 supp (0g𝑅))) → (∅ = 𝑦 ↔ ∅ = (𝑉 “ (𝑎 supp (0g𝑅)))))
4241adantl 481 . . . . . . . . . . 11 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ 𝑦 = (𝑉 “ (𝑎 supp (0g𝑅)))) → (∅ = 𝑦 ↔ ∅ = (𝑉 “ (𝑎 supp (0g𝑅)))))
4316, 29sseqtrid 4025 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑎 supp (0g𝑅)) ⊆ (𝑉𝑥))
44 cnvimass 6099 . . . . . . . . . . . . . 14 (𝑉𝑥) ⊆ dom 𝑉
4543, 44sstrdi 3995 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑎 supp (0g𝑅)) ⊆ dom 𝑉)
46 intimafv 32721 . . . . . . . . . . . . 13 ((Fun 𝑉 ∧ (𝑎 supp (0g𝑅)) ⊆ dom 𝑉) → (𝑉 “ (𝑎 supp (0g𝑅))) = 𝑙 ∈ (𝑎 supp (0g𝑅))(𝑉𝑙))
4719, 45, 46sylancr 587 . . . . . . . . . . . 12 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑉 “ (𝑎 supp (0g𝑅))) = 𝑙 ∈ (𝑎 supp (0g𝑅))(𝑉𝑙))
48 simplll 774 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → 𝑅 ∈ CRing)
4948crngringd 20244 . . . . . . . . . . . . . 14 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → 𝑅 ∈ Ring)
5049ad4antr 732 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → 𝑅 ∈ Ring)
51 fvex 6918 . . . . . . . . . . . . . . . 16 (PrmIdeal‘𝑅) ∈ V
5251rabex 5338 . . . . . . . . . . . . . . 15 {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} ∈ V
5352, 10dmmpti 6711 . . . . . . . . . . . . . 14 dom 𝑉 = (LIdeal‘𝑅)
5445, 53sseqtrdi 4023 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑎 supp (0g𝑅)) ⊆ (LIdeal‘𝑅))
55 simp-7r 789 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (♯‘(Base‘𝑅)) ≠ 1)
56 simpllr 775 . . . . . . . . . . . . . . . . . 18 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g𝑅)) = ∅) → (1r𝑅) = (𝑅 Σg 𝑎))
57 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (0g𝑅) = (0g𝑅)
58 ringcmn 20280 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
591, 58syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 ∈ CRing → 𝑅 ∈ CMnd)
6059ad8antr 740 . . . . . . . . . . . . . . . . . . . 20 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g𝑅)) = ∅) → 𝑅 ∈ CMnd)
6124a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g𝑅)) = ∅) → (𝑉𝑥) ∈ V)
6227ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g𝑅)) = ∅) → 𝑎:(𝑉𝑥)⟶(Base‘𝑅))
63 simpr 484 . . . . . . . . . . . . . . . . . . . . 21 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g𝑅)) = ∅) → (𝑎 supp (0g𝑅)) = ∅)
64 ssidd 4006 . . . . . . . . . . . . . . . . . . . . 21 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g𝑅)) = ∅) → ∅ ⊆ ∅)
6563, 64eqsstrd 4017 . . . . . . . . . . . . . . . . . . . 20 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g𝑅)) = ∅) → (𝑎 supp (0g𝑅)) ⊆ ∅)
6635adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g𝑅)) = ∅) → 𝑎 finSupp (0g𝑅))
674, 57, 60, 61, 62, 65, 66gsumres 19932 . . . . . . . . . . . . . . . . . . 19 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g𝑅)) = ∅) → (𝑅 Σg (𝑎 ↾ ∅)) = (𝑅 Σg 𝑎))
68 res0 6000 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ↾ ∅) = ∅
6968oveq2i 7443 . . . . . . . . . . . . . . . . . . . 20 (𝑅 Σg (𝑎 ↾ ∅)) = (𝑅 Σg ∅)
7057gsum0 18698 . . . . . . . . . . . . . . . . . . . 20 (𝑅 Σg ∅) = (0g𝑅)
7169, 70eqtri 2764 . . . . . . . . . . . . . . . . . . 19 (𝑅 Σg (𝑎 ↾ ∅)) = (0g𝑅)
7267, 71eqtr3di 2791 . . . . . . . . . . . . . . . . . 18 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g𝑅)) = ∅) → (𝑅 Σg 𝑎) = (0g𝑅))
7356, 72eqtr2d 2777 . . . . . . . . . . . . . . . . 17 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g𝑅)) = ∅) → (0g𝑅) = (1r𝑅))
74 eqid 2736 . . . . . . . . . . . . . . . . . 18 (1r𝑅) = (1r𝑅)
754, 57, 7401eq0ring 20531 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ (0g𝑅) = (1r𝑅)) → (Base‘𝑅) = {(0g𝑅)})
7650, 73, 75syl2an2r 685 . . . . . . . . . . . . . . . 16 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g𝑅)) = ∅) → (Base‘𝑅) = {(0g𝑅)})
7776fveq2d 6909 . . . . . . . . . . . . . . 15 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g𝑅)) = ∅) → (♯‘(Base‘𝑅)) = (♯‘{(0g𝑅)}))
78 fvex 6918 . . . . . . . . . . . . . . . 16 (0g𝑅) ∈ V
79 hashsng 14409 . . . . . . . . . . . . . . . 16 ((0g𝑅) ∈ V → (♯‘{(0g𝑅)}) = 1)
8078, 79ax-mp 5 . . . . . . . . . . . . . . 15 (♯‘{(0g𝑅)}) = 1
8177, 80eqtrdi 2792 . . . . . . . . . . . . . 14 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ (𝑎 supp (0g𝑅)) = ∅) → (♯‘(Base‘𝑅)) = 1)
8255, 81mteqand 3032 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑎 supp (0g𝑅)) ≠ ∅)
83 eqid 2736 . . . . . . . . . . . . . 14 (RSpan‘𝑅) = (RSpan‘𝑅)
8410, 83zarclsiin 33871 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝑎 supp (0g𝑅)) ⊆ (LIdeal‘𝑅) ∧ (𝑎 supp (0g𝑅)) ≠ ∅) → 𝑙 ∈ (𝑎 supp (0g𝑅))(𝑉𝑙) = (𝑉‘((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅)))))
8550, 54, 82, 84syl3anc 1372 . . . . . . . . . . . 12 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → 𝑙 ∈ (𝑎 supp (0g𝑅))(𝑉𝑙) = (𝑉‘((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅)))))
86 nfv 1913 . . . . . . . . . . . . . . . . . . . 20 𝑙((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎))
87 nfra1 3283 . . . . . . . . . . . . . . . . . . . 20 𝑙𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙
8886, 87nfan 1898 . . . . . . . . . . . . . . . . . . 19 𝑙(((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙)
8954sselda 3982 . . . . . . . . . . . . . . . . . . . . 21 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ 𝑙 ∈ (𝑎 supp (0g𝑅))) → 𝑙 ∈ (LIdeal‘𝑅))
90 eqid 2736 . . . . . . . . . . . . . . . . . . . . . 22 (LIdeal‘𝑅) = (LIdeal‘𝑅)
914, 90lidlss 21223 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 ∈ (LIdeal‘𝑅) → 𝑙 ⊆ (Base‘𝑅))
9289, 91syl 17 . . . . . . . . . . . . . . . . . . . 20 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ 𝑙 ∈ (𝑎 supp (0g𝑅))) → 𝑙 ⊆ (Base‘𝑅))
9392ex 412 . . . . . . . . . . . . . . . . . . 19 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑙 ∈ (𝑎 supp (0g𝑅)) → 𝑙 ⊆ (Base‘𝑅)))
9488, 93ralrimi 3256 . . . . . . . . . . . . . . . . . 18 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → ∀𝑙 ∈ (𝑎 supp (0g𝑅))𝑙 ⊆ (Base‘𝑅))
95 unissb 4938 . . . . . . . . . . . . . . . . . 18 ( (𝑎 supp (0g𝑅)) ⊆ (Base‘𝑅) ↔ ∀𝑙 ∈ (𝑎 supp (0g𝑅))𝑙 ⊆ (Base‘𝑅))
9694, 95sylibr 234 . . . . . . . . . . . . . . . . 17 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑎 supp (0g𝑅)) ⊆ (Base‘𝑅))
9783, 4, 90rspcl 21246 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ (𝑎 supp (0g𝑅)) ⊆ (Base‘𝑅)) → ((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅))) ∈ (LIdeal‘𝑅))
9850, 96, 97syl2anc 584 . . . . . . . . . . . . . . . 16 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → ((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅))) ∈ (LIdeal‘𝑅))
994, 90lidlss 21223 . . . . . . . . . . . . . . . 16 (((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅))) ∈ (LIdeal‘𝑅) → ((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅))) ⊆ (Base‘𝑅))
10098, 99syl 17 . . . . . . . . . . . . . . 15 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → ((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅))) ⊆ (Base‘𝑅))
10183, 4, 74rsp1 21248 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ Ring → ((RSpan‘𝑅)‘{(1r𝑅)}) = (Base‘𝑅))
10250, 101syl 17 . . . . . . . . . . . . . . . 16 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → ((RSpan‘𝑅)‘{(1r𝑅)}) = (Base‘𝑅))
10327adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → 𝑎:(𝑉𝑥)⟶(Base‘𝑅))
104103, 43fssresd 6774 . . . . . . . . . . . . . . . . . . . . 21 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑎 ↾ (𝑎 supp (0g𝑅))):(𝑎 supp (0g𝑅))⟶(Base‘𝑅))
105 fvex 6918 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘𝑅) ∈ V
106 ovex 7465 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 supp (0g𝑅)) ∈ V
107105, 106elmap 8912 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 ↾ (𝑎 supp (0g𝑅))) ∈ ((Base‘𝑅) ↑m (𝑎 supp (0g𝑅))) ↔ (𝑎 ↾ (𝑎 supp (0g𝑅))):(𝑎 supp (0g𝑅))⟶(Base‘𝑅))
108104, 107sylibr 234 . . . . . . . . . . . . . . . . . . . 20 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑎 ↾ (𝑎 supp (0g𝑅))) ∈ ((Base‘𝑅) ↑m (𝑎 supp (0g𝑅))))
109 breq1 5145 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = (𝑎 ↾ (𝑎 supp (0g𝑅))) → (𝑏 finSupp (0g𝑅) ↔ (𝑎 ↾ (𝑎 supp (0g𝑅))) finSupp (0g𝑅)))
110 oveq2 7440 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = (𝑎 ↾ (𝑎 supp (0g𝑅))) → (𝑅 Σg 𝑏) = (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g𝑅)))))
111110eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = (𝑎 ↾ (𝑎 supp (0g𝑅))) → ((1r𝑅) = (𝑅 Σg 𝑏) ↔ (1r𝑅) = (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g𝑅))))))
112 fveq1 6904 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = (𝑎 ↾ (𝑎 supp (0g𝑅))) → (𝑏𝑘) = ((𝑎 ↾ (𝑎 supp (0g𝑅)))‘𝑘))
113112eleq1d 2825 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = (𝑎 ↾ (𝑎 supp (0g𝑅))) → ((𝑏𝑘) ∈ 𝑘 ↔ ((𝑎 ↾ (𝑎 supp (0g𝑅)))‘𝑘) ∈ 𝑘))
114113ralbidv 3177 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = (𝑎 ↾ (𝑎 supp (0g𝑅))) → (∀𝑘 ∈ (𝑎 supp (0g𝑅))(𝑏𝑘) ∈ 𝑘 ↔ ∀𝑘 ∈ (𝑎 supp (0g𝑅))((𝑎 ↾ (𝑎 supp (0g𝑅)))‘𝑘) ∈ 𝑘))
115109, 111, 1143anbi123d 1437 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = (𝑎 ↾ (𝑎 supp (0g𝑅))) → ((𝑏 finSupp (0g𝑅) ∧ (1r𝑅) = (𝑅 Σg 𝑏) ∧ ∀𝑘 ∈ (𝑎 supp (0g𝑅))(𝑏𝑘) ∈ 𝑘) ↔ ((𝑎 ↾ (𝑎 supp (0g𝑅))) finSupp (0g𝑅) ∧ (1r𝑅) = (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g𝑅)))) ∧ ∀𝑘 ∈ (𝑎 supp (0g𝑅))((𝑎 ↾ (𝑎 supp (0g𝑅)))‘𝑘) ∈ 𝑘)))
116115adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ 𝑏 = (𝑎 ↾ (𝑎 supp (0g𝑅)))) → ((𝑏 finSupp (0g𝑅) ∧ (1r𝑅) = (𝑅 Σg 𝑏) ∧ ∀𝑘 ∈ (𝑎 supp (0g𝑅))(𝑏𝑘) ∈ 𝑘) ↔ ((𝑎 ↾ (𝑎 supp (0g𝑅))) finSupp (0g𝑅) ∧ (1r𝑅) = (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g𝑅)))) ∧ ∀𝑘 ∈ (𝑎 supp (0g𝑅))((𝑎 ↾ (𝑎 supp (0g𝑅)))‘𝑘) ∈ 𝑘)))
117 fvexd 6920 . . . . . . . . . . . . . . . . . . . . . 22 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (0g𝑅) ∈ V)
11835, 117fsuppres 9434 . . . . . . . . . . . . . . . . . . . . 21 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑎 ↾ (𝑎 supp (0g𝑅))) finSupp (0g𝑅))
119 simplr 768 . . . . . . . . . . . . . . . . . . . . . 22 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (1r𝑅) = (𝑅 Σg 𝑎))
12050, 58syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → 𝑅 ∈ CMnd)
12124a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑉𝑥) ∈ V)
122 ssidd 4006 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑎 supp (0g𝑅)) ⊆ (𝑎 supp (0g𝑅)))
1234, 57, 120, 121, 103, 122, 35gsumres 19932 . . . . . . . . . . . . . . . . . . . . . 22 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g𝑅)))) = (𝑅 Σg 𝑎))
124119, 123eqtr4d 2779 . . . . . . . . . . . . . . . . . . . . 21 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (1r𝑅) = (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g𝑅)))))
125 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ 𝑘 ∈ (𝑎 supp (0g𝑅))) → 𝑘 ∈ (𝑎 supp (0g𝑅)))
126125fvresd 6925 . . . . . . . . . . . . . . . . . . . . . . 23 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ 𝑘 ∈ (𝑎 supp (0g𝑅))) → ((𝑎 ↾ (𝑎 supp (0g𝑅)))‘𝑘) = (𝑎𝑘))
12716, 28sseqtrid 4025 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) → (𝑎 supp (0g𝑅)) ⊆ (𝑉𝑥))
128127sselda 3982 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ 𝑘 ∈ (𝑎 supp (0g𝑅))) → 𝑘 ∈ (𝑉𝑥))
129 fveq2 6905 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑙 = 𝑘 → (𝑎𝑙) = (𝑎𝑘))
130 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑙 = 𝑘𝑙 = 𝑘)
131129, 130eleq12d 2834 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑙 = 𝑘 → ((𝑎𝑙) ∈ 𝑙 ↔ (𝑎𝑘) ∈ 𝑘))
132131adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ 𝑘 ∈ (𝑎 supp (0g𝑅))) ∧ 𝑙 = 𝑘) → ((𝑎𝑙) ∈ 𝑙 ↔ (𝑎𝑘) ∈ 𝑘))
133128, 132rspcdv 3613 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ 𝑘 ∈ (𝑎 supp (0g𝑅))) → (∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙 → (𝑎𝑘) ∈ 𝑘))
134133imp 406 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ 𝑘 ∈ (𝑎 supp (0g𝑅))) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑎𝑘) ∈ 𝑘)
135134an32s 652 . . . . . . . . . . . . . . . . . . . . . . 23 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ 𝑘 ∈ (𝑎 supp (0g𝑅))) → (𝑎𝑘) ∈ 𝑘)
136126, 135eqeltrd 2840 . . . . . . . . . . . . . . . . . . . . . 22 (((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) ∧ 𝑘 ∈ (𝑎 supp (0g𝑅))) → ((𝑎 ↾ (𝑎 supp (0g𝑅)))‘𝑘) ∈ 𝑘)
137136ralrimiva 3145 . . . . . . . . . . . . . . . . . . . . 21 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → ∀𝑘 ∈ (𝑎 supp (0g𝑅))((𝑎 ↾ (𝑎 supp (0g𝑅)))‘𝑘) ∈ 𝑘)
138118, 124, 1373jca 1128 . . . . . . . . . . . . . . . . . . . 20 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → ((𝑎 ↾ (𝑎 supp (0g𝑅))) finSupp (0g𝑅) ∧ (1r𝑅) = (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g𝑅)))) ∧ ∀𝑘 ∈ (𝑎 supp (0g𝑅))((𝑎 ↾ (𝑎 supp (0g𝑅)))‘𝑘) ∈ 𝑘))
139108, 116, 138rspcedvd 3623 . . . . . . . . . . . . . . . . . . 19 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → ∃𝑏 ∈ ((Base‘𝑅) ↑m (𝑎 supp (0g𝑅)))(𝑏 finSupp (0g𝑅) ∧ (1r𝑅) = (𝑅 Σg 𝑏) ∧ ∀𝑘 ∈ (𝑎 supp (0g𝑅))(𝑏𝑘) ∈ 𝑘))
140 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (.r𝑅) = (.r𝑅)
14183, 4, 57, 140, 50, 54elrspunidl 33457 . . . . . . . . . . . . . . . . . . 19 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → ((1r𝑅) ∈ ((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅))) ↔ ∃𝑏 ∈ ((Base‘𝑅) ↑m (𝑎 supp (0g𝑅)))(𝑏 finSupp (0g𝑅) ∧ (1r𝑅) = (𝑅 Σg 𝑏) ∧ ∀𝑘 ∈ (𝑎 supp (0g𝑅))(𝑏𝑘) ∈ 𝑘)))
142139, 141mpbird 257 . . . . . . . . . . . . . . . . . 18 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (1r𝑅) ∈ ((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅))))
143142snssd 4808 . . . . . . . . . . . . . . . . 17 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → {(1r𝑅)} ⊆ ((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅))))
14483, 90rspssp 21250 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ ((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅))) ∈ (LIdeal‘𝑅) ∧ {(1r𝑅)} ⊆ ((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅)))) → ((RSpan‘𝑅)‘{(1r𝑅)}) ⊆ ((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅))))
14550, 98, 143, 144syl3anc 1372 . . . . . . . . . . . . . . . 16 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → ((RSpan‘𝑅)‘{(1r𝑅)}) ⊆ ((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅))))
146102, 145eqsstrrd 4018 . . . . . . . . . . . . . . 15 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (Base‘𝑅) ⊆ ((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅))))
147100, 146eqssd 4000 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → ((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅))) = (Base‘𝑅))
148147fveq2d 6909 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑉‘((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅)))) = (𝑉‘(Base‘𝑅)))
14990, 4lidl1 21244 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ Ring → (Base‘𝑅) ∈ (LIdeal‘𝑅))
1501, 149syl 17 . . . . . . . . . . . . . . . 16 (𝑅 ∈ CRing → (Base‘𝑅) ∈ (LIdeal‘𝑅))
15110, 4zarcls1 33869 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ CRing ∧ (Base‘𝑅) ∈ (LIdeal‘𝑅)) → ((𝑉‘(Base‘𝑅)) = ∅ ↔ (Base‘𝑅) = (Base‘𝑅)))
152150, 151mpdan 687 . . . . . . . . . . . . . . 15 (𝑅 ∈ CRing → ((𝑉‘(Base‘𝑅)) = ∅ ↔ (Base‘𝑅) = (Base‘𝑅)))
1534, 152mpbiri 258 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing → (𝑉‘(Base‘𝑅)) = ∅)
154153ad7antr 738 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑉‘(Base‘𝑅)) = ∅)
155148, 154eqtrd 2776 . . . . . . . . . . . 12 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → (𝑉‘((RSpan‘𝑅)‘ (𝑎 supp (0g𝑅)))) = ∅)
15647, 85, 1553eqtrrd 2781 . . . . . . . . . . 11 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → ∅ = (𝑉 “ (𝑎 supp (0g𝑅))))
15739, 42, 156rspcedvd 3623 . . . . . . . . . 10 ((((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ 𝑎 finSupp (0g𝑅)) ∧ (1r𝑅) = (𝑅 Σg 𝑎)) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∅ = 𝑦)
158157exp41 434 . . . . . . . . 9 (((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) → (𝑎 finSupp (0g𝑅) → ((1r𝑅) = (𝑅 Σg 𝑎) → (∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∅ = 𝑦))))
1591583imp2 1349 . . . . . . . 8 ((((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))) ∧ (𝑎 finSupp (0g𝑅) ∧ (1r𝑅) = (𝑅 Σg 𝑎) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∅ = 𝑦)
1604, 74ringidcl 20263 . . . . . . . . . . 11 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
16149, 160syl 17 . . . . . . . . . 10 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → (1r𝑅) ∈ (Base‘𝑅))
162 simplr 768 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → 𝑥 ∈ 𝒫 (Clsd‘𝐽))
163 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
1642, 3, 163, 10zartopn 33875 . . . . . . . . . . . . . . . . . 18 (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) ∧ ran 𝑉 = (Clsd‘𝐽)))
165164simprd 495 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ CRing → ran 𝑉 = (Clsd‘𝐽))
16648, 165syl 17 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → ran 𝑉 = (Clsd‘𝐽))
167166pweqd 4616 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → 𝒫 ran 𝑉 = 𝒫 (Clsd‘𝐽))
168162, 167eleqtrrd 2843 . . . . . . . . . . . . . 14 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → 𝑥 ∈ 𝒫 ran 𝑉)
169168elpwid 4608 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → 𝑥 ⊆ ran 𝑉)
170 intimafv 32721 . . . . . . . . . . . . . . 15 ((Fun 𝑉 ∧ (𝑉𝑥) ⊆ dom 𝑉) → (𝑉 “ (𝑉𝑥)) = 𝑙 ∈ (𝑉𝑥)(𝑉𝑙))
17119, 44, 170mp2an 692 . . . . . . . . . . . . . 14 (𝑉 “ (𝑉𝑥)) = 𝑙 ∈ (𝑉𝑥)(𝑉𝑙)
172 funimacnv 6646 . . . . . . . . . . . . . . . . 17 (Fun 𝑉 → (𝑉 “ (𝑉𝑥)) = (𝑥 ∩ ran 𝑉))
17319, 172ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑉 “ (𝑉𝑥)) = (𝑥 ∩ ran 𝑉)
174 dfss2 3968 . . . . . . . . . . . . . . . . 17 (𝑥 ⊆ ran 𝑉 ↔ (𝑥 ∩ ran 𝑉) = 𝑥)
175174biimpi 216 . . . . . . . . . . . . . . . 16 (𝑥 ⊆ ran 𝑉 → (𝑥 ∩ ran 𝑉) = 𝑥)
176173, 175eqtrid 2788 . . . . . . . . . . . . . . 15 (𝑥 ⊆ ran 𝑉 → (𝑉 “ (𝑉𝑥)) = 𝑥)
177176inteqd 4950 . . . . . . . . . . . . . 14 (𝑥 ⊆ ran 𝑉 (𝑉 “ (𝑉𝑥)) = 𝑥)
178171, 177eqtr3id 2790 . . . . . . . . . . . . 13 (𝑥 ⊆ ran 𝑉 𝑙 ∈ (𝑉𝑥)(𝑉𝑙) = 𝑥)
179169, 178syl 17 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → 𝑙 ∈ (𝑉𝑥)(𝑉𝑙) = 𝑥)
18044a1i 11 . . . . . . . . . . . . . 14 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → (𝑉𝑥) ⊆ dom 𝑉)
181180, 53sseqtrdi 4023 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → (𝑉𝑥) ⊆ (LIdeal‘𝑅))
18219a1i 11 . . . . . . . . . . . . . 14 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → Fun 𝑉)
183 inteq 4948 . . . . . . . . . . . . . . . . . 18 (𝑥 = ∅ → 𝑥 = ∅)
184 int0 4961 . . . . . . . . . . . . . . . . . 18 ∅ = V
185183, 184eqtrdi 2792 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → 𝑥 = V)
186 vn0 4344 . . . . . . . . . . . . . . . . . 18 V ≠ ∅
187 neeq1 3002 . . . . . . . . . . . . . . . . . 18 ( 𝑥 = V → ( 𝑥 ≠ ∅ ↔ V ≠ ∅))
188186, 187mpbiri 258 . . . . . . . . . . . . . . . . 17 ( 𝑥 = V → 𝑥 ≠ ∅)
189185, 188syl 17 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → 𝑥 ≠ ∅)
190189necon2i 2974 . . . . . . . . . . . . . . 15 ( 𝑥 = ∅ → 𝑥 ≠ ∅)
191190adantl 481 . . . . . . . . . . . . . 14 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → 𝑥 ≠ ∅)
192 preiman0 32720 . . . . . . . . . . . . . 14 ((Fun 𝑉𝑥 ⊆ ran 𝑉𝑥 ≠ ∅) → (𝑉𝑥) ≠ ∅)
193182, 169, 191, 192syl3anc 1372 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → (𝑉𝑥) ≠ ∅)
19410, 83zarclsiin 33871 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝑉𝑥) ⊆ (LIdeal‘𝑅) ∧ (𝑉𝑥) ≠ ∅) → 𝑙 ∈ (𝑉𝑥)(𝑉𝑙) = (𝑉‘((RSpan‘𝑅)‘ (𝑉𝑥))))
19549, 181, 193, 194syl3anc 1372 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → 𝑙 ∈ (𝑉𝑥)(𝑉𝑙) = (𝑉‘((RSpan‘𝑅)‘ (𝑉𝑥))))
196 simpr 484 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → 𝑥 = ∅)
197179, 195, 1963eqtr3d 2784 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → (𝑉‘((RSpan‘𝑅)‘ (𝑉𝑥))) = ∅)
198181sselda 3982 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑙 ∈ (𝑉𝑥)) → 𝑙 ∈ (LIdeal‘𝑅))
199198, 91syl 17 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) ∧ 𝑙 ∈ (𝑉𝑥)) → 𝑙 ⊆ (Base‘𝑅))
200199ralrimiva 3145 . . . . . . . . . . . . . 14 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → ∀𝑙 ∈ (𝑉𝑥)𝑙 ⊆ (Base‘𝑅))
201 unissb 4938 . . . . . . . . . . . . . 14 ( (𝑉𝑥) ⊆ (Base‘𝑅) ↔ ∀𝑙 ∈ (𝑉𝑥)𝑙 ⊆ (Base‘𝑅))
202200, 201sylibr 234 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → (𝑉𝑥) ⊆ (Base‘𝑅))
20383, 4, 90rspcl 21246 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝑉𝑥) ⊆ (Base‘𝑅)) → ((RSpan‘𝑅)‘ (𝑉𝑥)) ∈ (LIdeal‘𝑅))
20449, 202, 203syl2anc 584 . . . . . . . . . . . 12 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → ((RSpan‘𝑅)‘ (𝑉𝑥)) ∈ (LIdeal‘𝑅))
20510, 4zarcls1 33869 . . . . . . . . . . . 12 ((𝑅 ∈ CRing ∧ ((RSpan‘𝑅)‘ (𝑉𝑥)) ∈ (LIdeal‘𝑅)) → ((𝑉‘((RSpan‘𝑅)‘ (𝑉𝑥))) = ∅ ↔ ((RSpan‘𝑅)‘ (𝑉𝑥)) = (Base‘𝑅)))
20648, 204, 205syl2anc 584 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → ((𝑉‘((RSpan‘𝑅)‘ (𝑉𝑥))) = ∅ ↔ ((RSpan‘𝑅)‘ (𝑉𝑥)) = (Base‘𝑅)))
207197, 206mpbid 232 . . . . . . . . . 10 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → ((RSpan‘𝑅)‘ (𝑉𝑥)) = (Base‘𝑅))
208161, 207eleqtrrd 2843 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → (1r𝑅) ∈ ((RSpan‘𝑅)‘ (𝑉𝑥)))
20983, 4, 57, 140, 49, 181elrspunidl 33457 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → ((1r𝑅) ∈ ((RSpan‘𝑅)‘ (𝑉𝑥)) ↔ ∃𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))(𝑎 finSupp (0g𝑅) ∧ (1r𝑅) = (𝑅 Σg 𝑎) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙)))
210208, 209mpbid 232 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → ∃𝑎 ∈ ((Base‘𝑅) ↑m (𝑉𝑥))(𝑎 finSupp (0g𝑅) ∧ (1r𝑅) = (𝑅 Σg 𝑎) ∧ ∀𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙))
211159, 210r19.29a 3161 . . . . . . 7 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∅ = 𝑦)
212 0ex 5306 . . . . . . . 8 ∅ ∈ V
213 vex 3483 . . . . . . . 8 𝑥 ∈ V
214 elfi 9454 . . . . . . . 8 ((∅ ∈ V ∧ 𝑥 ∈ V) → (∅ ∈ (fi‘𝑥) ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∅ = 𝑦))
215212, 213, 214mp2an 692 . . . . . . 7 (∅ ∈ (fi‘𝑥) ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∅ = 𝑦)
216211, 215sylibr 234 . . . . . 6 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → ∅ ∈ (fi‘𝑥))
217216ex 412 . . . . 5 (((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → ( 𝑥 = ∅ → ∅ ∈ (fi‘𝑥)))
218217necon3bd 2953 . . . 4 (((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → (¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅))
219218ralrimiva 3145 . . 3 ((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) → ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅))
220 cmpfi 23417 . . . 4 (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅)))
221220biimpar 477 . . 3 ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅)) → 𝐽 ∈ Comp)
2229, 219, 221syl2an2r 685 . 2 ((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) → 𝐽 ∈ Comp)
2238, 222pm2.61dane 3028 1 (𝑅 ∈ CRing → 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  {crab 3435  Vcvv 3479  cin 3949  wss 3950  c0 4332  𝒫 cpw 4599  {csn 4625   cuni 4906   cint 4945   ciin 4991   class class class wbr 5142  cmpt 5224  ccnv 5683  dom cdm 5684  ran crn 5685  cres 5686  cima 5687  Fun wfun 6554  wf 6556  cfv 6560  (class class class)co 7432   supp csupp 8186  m cmap 8867  Fincfn 8986   finSupp cfsupp 9402  ficfi 9451  1c1 11157  chash 14370  Basecbs 17248  .rcmulr 17299  TopOpenctopn 17467  0gc0g 17485   Σg cgsu 17486  CMndccmn 19799  1rcur 20179  Ringcrg 20231  CRingccrg 20232  LIdealclidl 21217  RSpancrsp 21218  Topctop 22900  TopOnctopon 22917  Clsdccld 23025  Compccmp 23395  PrmIdealcprmidl 33464  Speccrspec 33862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-reg 9633  ax-inf2 9682  ax-ac2 10504  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-addf 11235  ax-mulf 11236
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-disj 5110  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-rpss 7744  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-fi 9452  df-sup 9483  df-oi 9551  df-r1 9805  df-rank 9806  df-dju 9942  df-card 9980  df-ac 10157  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-ghm 19232  df-cntz 19336  df-lsm 19655  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-cring 20234  df-rhm 20473  df-nzr 20514  df-subrng 20547  df-subrg 20571  df-lmod 20861  df-lss 20931  df-lsp 20971  df-lmhm 21022  df-lbs 21075  df-sra 21173  df-rgmod 21174  df-lidl 21219  df-rsp 21220  df-lpidl 21333  df-cnfld 21366  df-zring 21459  df-zrh 21515  df-dsmm 21753  df-frlm 21768  df-uvc 21804  df-top 22901  df-topon 22918  df-cld 23028  df-cmp 23396  df-prmidl 33465  df-mxidl 33489  df-idlsrg 33530  df-rspec 33863
This theorem is referenced by:  zarcmp  33882
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