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Theorem zarcmplem 33842
Description: Lemma for zarcmp 33843. (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 20263 . . . 4 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2 zartop.1 . . . . 5 𝑆 = (Spec‘𝑅)
3 zartop.2 . . . . 5 𝐽 = (TopOpen‘𝑆)
4 eqid 2735 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
52, 3, 4zar0ring 33839 . . . 4 ((𝑅 ∈ Ring ∧ (♯‘(Base‘𝑅)) = 1) → 𝐽 = {∅})
61, 5sylan 580 . . 3 ((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) = 1) → 𝐽 = {∅})
7 0cmp 23418 . . 3 {∅} ∈ Comp
86, 7eqeltrdi 2847 . 2 ((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) = 1) → 𝐽 ∈ Comp)
92, 3zartop 33837 . . 3 (𝑅 ∈ CRing → 𝐽 ∈ Top)
10 zarcmplem.1 . . . . . . . . . . . . . . 15 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
11 fvex 6920 . . . . . . . . . . . . . . . 16 (LIdeal‘𝑅) ∈ V
1211mptex 7243 . . . . . . . . . . . . . . 15 (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗}) ∈ V
1310, 12eqeltri 2835 . . . . . . . . . . . . . 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 8201 . . . . . . . . . . . . . . 15 (𝑎 supp (0g𝑅)) ⊆ dom 𝑎
17 imass2 6123 . . . . . . . . . . . . . . 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 6607 . . . . . . . . . . . . . . 15 Fun 𝑉
20 ssidd 4019 . . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . . 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 6922 . . . . . . . . . . . . . . . . . . . 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 8879 . . . . . . . . . . . . . . . . . . 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 6747 . . . . . . . . . . . . . . . . 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 4036 . . . . . . . . . . . . . . 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 6651 . . . . . . . . . . . . . . 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 4006 . . . . . . . . . . . . 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 4611 . . . . . . . . . . . 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 776 . . . . . . . . . . . . . 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 9407 . . . . . . . . . . . . 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 9351 . . . . . . . . . . . . 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 4210 . . . . . . . . . . 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 4954 . . . . . . . . . . . . 13 (𝑦 = (𝑉 “ (𝑎 supp (0g𝑅))) → 𝑦 = (𝑉 “ (𝑎 supp (0g𝑅))))
4140eqeq2d 2746 . . . . . . . . . . . 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 4048 . . . . . . . . . . . . . 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 6102 . . . . . . . . . . . . . 14 (𝑉𝑥) ⊆ dom 𝑉
4543, 44sstrdi 4008 . . . . . . . . . . . . 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 32726 . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → 𝑅 ∈ CRing)
4948crngringd 20264 . . . . . . . . . . . . . 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 6920 . . . . . . . . . . . . . . . 16 (PrmIdeal‘𝑅) ∈ V
5251rabex 5345 . . . . . . . . . . . . . . 15 {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} ∈ V
5352, 10dmmpti 6713 . . . . . . . . . . . . . 14 dom 𝑉 = (LIdeal‘𝑅)
5445, 53sseqtrdi 4046 . . . . . . . . . . . . 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 790 . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . . . . 20 (0g𝑅) = (0g𝑅)
58 ringcmn 20296 . . . . . . . . . . . . . . . . . . . . . 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 4019 . . . . . . . . . . . . . . . . . . . . 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 4034 . . . . . . . . . . . . . . . . . . . 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 19946 . . . . . . . . . . . . . . . . . . 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 6004 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 ↾ ∅) = ∅
6968oveq2i 7442 . . . . . . . . . . . . . . . . . . . 20 (𝑅 Σg (𝑎 ↾ ∅)) = (𝑅 Σg ∅)
7057gsum0 18710 . . . . . . . . . . . . . . . . . . . 20 (𝑅 Σg ∅) = (0g𝑅)
7169, 70eqtri 2763 . . . . . . . . . . . . . . . . . . 19 (𝑅 Σg (𝑎 ↾ ∅)) = (0g𝑅)
7267, 71eqtr3di 2790 . . . . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . . 18 (1r𝑅) = (1r𝑅)
754, 57, 7401eq0ring 20547 . . . . . . . . . . . . . . . . 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 6911 . . . . . . . . . . . . . . 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 6920 . . . . . . . . . . . . . . . 16 (0g𝑅) ∈ V
79 hashsng 14405 . . . . . . . . . . . . . . . 16 ((0g𝑅) ∈ V → (♯‘{(0g𝑅)}) = 1)
8078, 79ax-mp 5 . . . . . . . . . . . . . . 15 (♯‘{(0g𝑅)}) = 1
8177, 80eqtrdi 2791 . . . . . . . . . . . . . 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 3031 . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . 14 (RSpan‘𝑅) = (RSpan‘𝑅)
8410, 83zarclsiin 33832 . . . . . . . . . . . . 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 1370 . . . . . . . . . . . 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 1912 . . . . . . . . . . . . . . . . . . . 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 3282 . . . . . . . . . . . . . . . . . . . 20 𝑙𝑙 ∈ (𝑉𝑥)(𝑎𝑙) ∈ 𝑙
8886, 87nfan 1897 . . . . . . . . . . . . . . . . . . 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 3995 . . . . . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . . . . . . 22 (LIdeal‘𝑅) = (LIdeal‘𝑅)
914, 90lidlss 21240 . . . . . . . . . . . . . . . . . . . . 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 3255 . . . . . . . . . . . . . . . . . 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 4944 . . . . . . . . . . . . . . . . . 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 21263 . . . . . . . . . . . . . . . . 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 21240 . . . . . . . . . . . . . . . 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 21265 . . . . . . . . . . . . . . . . 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 6776 . . . . . . . . . . . . . . . . . . . . 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 6920 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘𝑅) ∈ V
106 ovex 7464 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 supp (0g𝑅)) ∈ V
107105, 106elmap 8910 . . . . . . . . . . . . . . . . . . . . 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 5151 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = (𝑎 ↾ (𝑎 supp (0g𝑅))) → (𝑏 finSupp (0g𝑅) ↔ (𝑎 ↾ (𝑎 supp (0g𝑅))) finSupp (0g𝑅)))
110 oveq2 7439 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = (𝑎 ↾ (𝑎 supp (0g𝑅))) → (𝑅 Σg 𝑏) = (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g𝑅)))))
111110eqeq2d 2746 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = (𝑎 ↾ (𝑎 supp (0g𝑅))) → ((1r𝑅) = (𝑅 Σg 𝑏) ↔ (1r𝑅) = (𝑅 Σg (𝑎 ↾ (𝑎 supp (0g𝑅))))))
112 fveq1 6906 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = (𝑎 ↾ (𝑎 supp (0g𝑅))) → (𝑏𝑘) = ((𝑎 ↾ (𝑎 supp (0g𝑅)))‘𝑘))
113112eleq1d 2824 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = (𝑎 ↾ (𝑎 supp (0g𝑅))) → ((𝑏𝑘) ∈ 𝑘 ↔ ((𝑎 ↾ (𝑎 supp (0g𝑅)))‘𝑘) ∈ 𝑘))
114113ralbidv 3176 . . . . . . . . . . . . . . . . . . . . . 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 1435 . . . . . . . . . . . . . . . . . . . . 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 6922 . . . . . . . . . . . . . . . . . . . . . 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 9431 . . . . . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . . . 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 4019 . . . . . . . . . . . . . . . . . . . . . . 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 19946 . . . . . . . . . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . . . . . . . . . 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 6927 . . . . . . . . . . . . . . . . . . . . . . 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 4048 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3995 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6907 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑙 = 𝑘 → (𝑎𝑙) = (𝑎𝑘))
130 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑙 = 𝑘𝑙 = 𝑘)
131129, 130eleq12d 2833 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3614 . . . . . . . . . . . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . . . . . . . . . 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 3144 . . . . . . . . . . . . . . . . . . . . 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 1127 . . . . . . . . . . . . . . . . . . . 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 3624 . . . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . . . . 20 (.r𝑅) = (.r𝑅)
14183, 4, 57, 140, 50, 54elrspunidl 33436 . . . . . . . . . . . . . . . . . . 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 4814 . . . . . . . . . . . . . . . . 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 21267 . . . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . . . 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 4035 . . . . . . . . . . . . . . 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 4013 . . . . . . . . . . . . . 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 6911 . . . . . . . . . . . . 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 21261 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ Ring → (Base‘𝑅) ∈ (LIdeal‘𝑅))
1501, 149syl 17 . . . . . . . . . . . . . . . 16 (𝑅 ∈ CRing → (Base‘𝑅) ∈ (LIdeal‘𝑅))
15110, 4zarcls1 33830 . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . 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 2780 . . . . . . . . . . 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 3624 . . . . . . . . . 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 1348 . . . . . . . 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 20280 . . . . . . . . . . 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 769 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → 𝑥 ∈ 𝒫 (Clsd‘𝐽))
163 eqid 2735 . . . . . . . . . . . . . . . . . . 19 (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
1642, 3, 163, 10zartopn 33836 . . . . . . . . . . . . . . . . . 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 4622 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → 𝒫 ran 𝑉 = 𝒫 (Clsd‘𝐽))
168162, 167eleqtrrd 2842 . . . . . . . . . . . . . 14 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → 𝑥 ∈ 𝒫 ran 𝑉)
169168elpwid 4614 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → 𝑥 ⊆ ran 𝑉)
170 intimafv 32726 . . . . . . . . . . . . . . 15 ((Fun 𝑉 ∧ (𝑉𝑥) ⊆ dom 𝑉) → (𝑉 “ (𝑉𝑥)) = 𝑙 ∈ (𝑉𝑥)(𝑉𝑙))
17119, 44, 170mp2an 692 . . . . . . . . . . . . . 14 (𝑉 “ (𝑉𝑥)) = 𝑙 ∈ (𝑉𝑥)(𝑉𝑙)
172 funimacnv 6649 . . . . . . . . . . . . . . . . 17 (Fun 𝑉 → (𝑉 “ (𝑉𝑥)) = (𝑥 ∩ ran 𝑉))
17319, 172ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑉 “ (𝑉𝑥)) = (𝑥 ∩ ran 𝑉)
174 dfss2 3981 . . . . . . . . . . . . . . . . 17 (𝑥 ⊆ ran 𝑉 ↔ (𝑥 ∩ ran 𝑉) = 𝑥)
175174biimpi 216 . . . . . . . . . . . . . . . 16 (𝑥 ⊆ ran 𝑉 → (𝑥 ∩ ran 𝑉) = 𝑥)
176173, 175eqtrid 2787 . . . . . . . . . . . . . . 15 (𝑥 ⊆ ran 𝑉 → (𝑉 “ (𝑉𝑥)) = 𝑥)
177176inteqd 4956 . . . . . . . . . . . . . 14 (𝑥 ⊆ ran 𝑉 (𝑉 “ (𝑉𝑥)) = 𝑥)
178171, 177eqtr3id 2789 . . . . . . . . . . . . 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 4046 . . . . . . . . . . . . 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 4954 . . . . . . . . . . . . . . . . . 18 (𝑥 = ∅ → 𝑥 = ∅)
184 int0 4967 . . . . . . . . . . . . . . . . . 18 ∅ = V
185183, 184eqtrdi 2791 . . . . . . . . . . . . . . . . 17 (𝑥 = ∅ → 𝑥 = V)
186 vn0 4351 . . . . . . . . . . . . . . . . . 18 V ≠ ∅
187 neeq1 3001 . . . . . . . . . . . . . . . . . 18 ( 𝑥 = V → ( 𝑥 ≠ ∅ ↔ V ≠ ∅))
188186, 187mpbiri 258 . . . . . . . . . . . . . . . . 17 ( 𝑥 = V → 𝑥 ≠ ∅)
189185, 188syl 17 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → 𝑥 ≠ ∅)
190189necon2i 2973 . . . . . . . . . . . . . . 15 ( 𝑥 = ∅ → 𝑥 ≠ ∅)
191190adantl 481 . . . . . . . . . . . . . 14 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → 𝑥 ≠ ∅)
192 preiman0 32725 . . . . . . . . . . . . . 14 ((Fun 𝑉𝑥 ⊆ ran 𝑉𝑥 ≠ ∅) → (𝑉𝑥) ≠ ∅)
193182, 169, 191, 192syl3anc 1370 . . . . . . . . . . . . 13 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → (𝑉𝑥) ≠ ∅)
19410, 83zarclsiin 33832 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝑉𝑥) ⊆ (LIdeal‘𝑅) ∧ (𝑉𝑥) ≠ ∅) → 𝑙 ∈ (𝑉𝑥)(𝑉𝑙) = (𝑉‘((RSpan‘𝑅)‘ (𝑉𝑥))))
19549, 181, 193, 194syl3anc 1370 . . . . . . . . . . . 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 2783 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → (𝑉‘((RSpan‘𝑅)‘ (𝑉𝑥))) = ∅)
198181sselda 3995 . . . . . . . . . . . . . . . 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 3144 . . . . . . . . . . . . . 14 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → ∀𝑙 ∈ (𝑉𝑥)𝑙 ⊆ (Base‘𝑅))
201 unissb 4944 . . . . . . . . . . . . . 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 21263 . . . . . . . . . . . . 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 33830 . . . . . . . . . . . 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 2842 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → (1r𝑅) ∈ ((RSpan‘𝑅)‘ (𝑉𝑥)))
20983, 4, 57, 140, 49, 181elrspunidl 33436 . . . . . . . . 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 3160 . . . . . . 7 ((((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) ∧ 𝑥 = ∅) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∅ = 𝑦)
212 0ex 5313 . . . . . . . 8 ∅ ∈ V
213 vex 3482 . . . . . . . 8 𝑥 ∈ V
214 elfi 9451 . . . . . . . 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 2952 . . . 4 (((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → (¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅))
219218ralrimiva 3144 . . 3 ((𝑅 ∈ CRing ∧ (♯‘(Base‘𝑅)) ≠ 1) → ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅))
220 cmpfi 23432 . . . 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 3027 1 (𝑅 ∈ CRing → 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   cuni 4912   cint 4951   ciin 4997   class class class wbr 5148  cmpt 5231  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  Fun wfun 6557  wf 6559  cfv 6563  (class class class)co 7431   supp csupp 8184  m cmap 8865  Fincfn 8984   finSupp cfsupp 9399  ficfi 9448  1c1 11154  chash 14366  Basecbs 17245  .rcmulr 17299  TopOpenctopn 17468  0gc0g 17486   Σg cgsu 17487  CMndccmn 19813  1rcur 20199  Ringcrg 20251  CRingccrg 20252  LIdealclidl 21234  RSpancrsp 21235  Topctop 22915  TopOnctopon 22932  Clsdccld 23040  Compccmp 23410  PrmIdealcprmidl 33443  Speccrspec 33823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679  ax-ac2 10501  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-addf 11232  ax-mulf 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-disj 5116  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-rpss 7742  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-oi 9548  df-r1 9802  df-rank 9803  df-dju 9939  df-card 9977  df-ac 10154  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  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 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-lsm 19669  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-rhm 20489  df-nzr 20530  df-subrng 20563  df-subrg 20587  df-lmod 20877  df-lss 20948  df-lsp 20988  df-lmhm 21039  df-lbs 21092  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-rsp 21237  df-lpidl 21350  df-cnfld 21383  df-zring 21476  df-zrh 21532  df-dsmm 21770  df-frlm 21785  df-uvc 21821  df-top 22916  df-topon 22933  df-cld 23043  df-cmp 23411  df-prmidl 33444  df-mxidl 33468  df-idlsrg 33509  df-rspec 33824
This theorem is referenced by:  zarcmp  33843
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