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Theorem zarclsint 31225
 Description: The intersection of a family of closed sets is closed in the Zariski topology. (Contributed by Thierry Arnoux, 16-Jun-2024.)
Hypothesis
Ref Expression
zarclsx.1 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
Assertion
Ref Expression
zarclsint ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉𝑆 ≠ ∅) → 𝑆 ∈ ran 𝑉)
Distinct variable groups:   𝑅,𝑖,𝑗   𝑆,𝑖   𝑖,𝑉
Allowed substitution hints:   𝑆(𝑗)   𝑉(𝑗)

Proof of Theorem zarclsint
Dummy variables 𝑙 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crngring 19305 . . . . . . 7 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
21ad4antr 731 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → 𝑅 ∈ Ring)
3 elpwi 4509 . . . . . . . . . . . 12 (𝑟 ∈ 𝒫 (LIdeal‘𝑅) → 𝑟 ⊆ (LIdeal‘𝑅))
43adantl 485 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) → 𝑟 ⊆ (LIdeal‘𝑅))
54adantr 484 . . . . . . . . . 10 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → 𝑟 ⊆ (LIdeal‘𝑅))
65sselda 3918 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) ∧ 𝑖𝑟) → 𝑖 ∈ (LIdeal‘𝑅))
7 eqid 2801 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
8 eqid 2801 . . . . . . . . . 10 (LIdeal‘𝑅) = (LIdeal‘𝑅)
97, 8lidlss 19979 . . . . . . . . 9 (𝑖 ∈ (LIdeal‘𝑅) → 𝑖 ⊆ (Base‘𝑅))
106, 9syl 17 . . . . . . . 8 ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) ∧ 𝑖𝑟) → 𝑖 ⊆ (Base‘𝑅))
1110ralrimiva 3152 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → ∀𝑖𝑟 𝑖 ⊆ (Base‘𝑅))
12 unissb 4835 . . . . . . 7 ( 𝑟 ⊆ (Base‘𝑅) ↔ ∀𝑖𝑟 𝑖 ⊆ (Base‘𝑅))
1311, 12sylibr 237 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → 𝑟 ⊆ (Base‘𝑅))
14 eqid 2801 . . . . . . 7 (RSpan‘𝑅) = (RSpan‘𝑅)
1514, 7, 8rspcl 19991 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑟 ⊆ (Base‘𝑅)) → ((RSpan‘𝑅)‘ 𝑟) ∈ (LIdeal‘𝑅))
162, 13, 15syl2anc 587 . . . . 5 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → ((RSpan‘𝑅)‘ 𝑟) ∈ (LIdeal‘𝑅))
17 sseq1 3943 . . . . . . . 8 (𝑖 = ((RSpan‘𝑅)‘ 𝑟) → (𝑖𝑗 ↔ ((RSpan‘𝑅)‘ 𝑟) ⊆ 𝑗))
1817rabbidv 3430 . . . . . . 7 (𝑖 = ((RSpan‘𝑅)‘ 𝑟) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘ 𝑟) ⊆ 𝑗})
1918eqeq2d 2812 . . . . . 6 (𝑖 = ((RSpan‘𝑅)‘ 𝑟) → ( 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} ↔ 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘ 𝑟) ⊆ 𝑗}))
2019adantl 485 . . . . 5 ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) ∧ 𝑖 = ((RSpan‘𝑅)‘ 𝑟)) → ( 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} ↔ 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘ 𝑟) ⊆ 𝑗}))
21 simpr 488 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → 𝑆 = (𝑉𝑟))
2221inteqd 4846 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → 𝑆 = (𝑉𝑟))
23 zarclsx.1 . . . . . . . . . 10 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
2423funmpt2 6367 . . . . . . . . 9 Fun 𝑉
2524a1i 11 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → Fun 𝑉)
26 fvex 6662 . . . . . . . . . . 11 (PrmIdeal‘𝑅) ∈ V
2726rabex 5202 . . . . . . . . . 10 {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} ∈ V
2827, 23dmmpti 6468 . . . . . . . . 9 dom 𝑉 = (LIdeal‘𝑅)
295, 28sseqtrrdi 3969 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → 𝑟 ⊆ dom 𝑉)
30 intimafv 30473 . . . . . . . 8 ((Fun 𝑉𝑟 ⊆ dom 𝑉) → (𝑉𝑟) = 𝑙𝑟 (𝑉𝑙))
3125, 29, 30syl2anc 587 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → (𝑉𝑟) = 𝑙𝑟 (𝑉𝑙))
3222, 31eqtrd 2836 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → 𝑆 = 𝑙𝑟 (𝑉𝑙))
33 simplr 768 . . . . . . . . . 10 ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) ∧ 𝑟 = ∅) → 𝑆 = (𝑉𝑟))
34 simpr 488 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) ∧ 𝑟 = ∅) → 𝑟 = ∅)
3534imaeq2d 5900 . . . . . . . . . . 11 ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) ∧ 𝑟 = ∅) → (𝑉𝑟) = (𝑉 “ ∅))
36 ima0 5916 . . . . . . . . . . 11 (𝑉 “ ∅) = ∅
3735, 36eqtrdi 2852 . . . . . . . . . 10 ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) ∧ 𝑟 = ∅) → (𝑉𝑟) = ∅)
3833, 37eqtrd 2836 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) ∧ 𝑟 = ∅) → 𝑆 = ∅)
39 simp-4r 783 . . . . . . . . . 10 ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) ∧ 𝑟 = ∅) → 𝑆 ≠ ∅)
4039neneqd 2995 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) ∧ 𝑟 = ∅) → ¬ 𝑆 = ∅)
4138, 40pm2.65da 816 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → ¬ 𝑟 = ∅)
4241neqned 2997 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → 𝑟 ≠ ∅)
4323, 14zarclsiin 31224 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑟 ≠ ∅) → 𝑙𝑟 (𝑉𝑙) = (𝑉‘((RSpan‘𝑅)‘ 𝑟)))
442, 5, 42, 43syl3anc 1368 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → 𝑙𝑟 (𝑉𝑙) = (𝑉‘((RSpan‘𝑅)‘ 𝑟)))
4523a1i 11 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗}))
4618adantl 485 . . . . . . 7 ((((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) ∧ 𝑖 = ((RSpan‘𝑅)‘ 𝑟)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘ 𝑟) ⊆ 𝑗})
4726rabex 5202 . . . . . . . 8 {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘ 𝑟) ⊆ 𝑗} ∈ V
4847a1i 11 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘ 𝑟) ⊆ 𝑗} ∈ V)
4945, 46, 16, 48fvmptd 6756 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → (𝑉‘((RSpan‘𝑅)‘ 𝑟)) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘ 𝑟) ⊆ 𝑗})
5032, 44, 493eqtrd 2840 . . . . 5 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ ((RSpan‘𝑅)‘ 𝑟) ⊆ 𝑗})
5116, 20, 50rspcedvd 3577 . . . 4 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → ∃𝑖 ∈ (LIdeal‘𝑅) 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
52 intex 5207 . . . . . . . 8 (𝑆 ≠ ∅ ↔ 𝑆 ∈ V)
5352biimpi 219 . . . . . . 7 (𝑆 ≠ ∅ → 𝑆 ∈ V)
54533ad2ant3 1132 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉𝑆 ≠ ∅) → 𝑆 ∈ V)
5523elrnmpt 5796 . . . . . 6 ( 𝑆 ∈ V → ( 𝑆 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅) 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗}))
5654, 55syl 17 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉𝑆 ≠ ∅) → ( 𝑆 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅) 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗}))
5756ad5ant123 1361 . . . 4 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → ( 𝑆 ∈ ran 𝑉 ↔ ∃𝑖 ∈ (LIdeal‘𝑅) 𝑆 = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗}))
5851, 57mpbird 260 . . 3 (((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ∈ 𝒫 (LIdeal‘𝑅)) ∧ 𝑆 = (𝑉𝑟)) → 𝑆 ∈ ran 𝑉)
59 fvexd 6664 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → (LIdeal‘𝑅) ∈ V)
6024a1i 11 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → Fun 𝑉)
61 simplr 768 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ ran 𝑉)
6227, 23fnmpti 6467 . . . . . . . 8 𝑉 Fn (LIdeal‘𝑅)
63 fnima 6454 . . . . . . . 8 (𝑉 Fn (LIdeal‘𝑅) → (𝑉 “ (LIdeal‘𝑅)) = ran 𝑉)
6462, 63ax-mp 5 . . . . . . 7 (𝑉 “ (LIdeal‘𝑅)) = ran 𝑉
6561, 64sseqtrrdi 3969 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ (𝑉 “ (LIdeal‘𝑅)))
66 ssimaexg 6728 . . . . . 6 (((LIdeal‘𝑅) ∈ V ∧ Fun 𝑉𝑆 ⊆ (𝑉 “ (LIdeal‘𝑅))) → ∃𝑟(𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑆 = (𝑉𝑟)))
6759, 60, 65, 66syl3anc 1368 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ∃𝑟(𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑆 = (𝑉𝑟)))
68 vex 3447 . . . . . . . . . 10 𝑟 ∈ V
6968a1i 11 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ⊆ (LIdeal‘𝑅)) → 𝑟 ∈ V)
70 simpr 488 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ⊆ (LIdeal‘𝑅)) → 𝑟 ⊆ (LIdeal‘𝑅))
7169, 70elpwd 4508 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) ∧ 𝑟 ⊆ (LIdeal‘𝑅)) → 𝑟 ∈ 𝒫 (LIdeal‘𝑅))
7271ex 416 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → (𝑟 ⊆ (LIdeal‘𝑅) → 𝑟 ∈ 𝒫 (LIdeal‘𝑅)))
7372anim1d 613 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ((𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑆 = (𝑉𝑟)) → (𝑟 ∈ 𝒫 (LIdeal‘𝑅) ∧ 𝑆 = (𝑉𝑟))))
7473eximdv 1918 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → (∃𝑟(𝑟 ⊆ (LIdeal‘𝑅) ∧ 𝑆 = (𝑉𝑟)) → ∃𝑟(𝑟 ∈ 𝒫 (LIdeal‘𝑅) ∧ 𝑆 = (𝑉𝑟))))
7567, 74mpd 15 . . . 4 (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ∃𝑟(𝑟 ∈ 𝒫 (LIdeal‘𝑅) ∧ 𝑆 = (𝑉𝑟)))
76 df-rex 3115 . . . 4 (∃𝑟 ∈ 𝒫 (LIdeal‘𝑅)𝑆 = (𝑉𝑟) ↔ ∃𝑟(𝑟 ∈ 𝒫 (LIdeal‘𝑅) ∧ 𝑆 = (𝑉𝑟)))
7775, 76sylibr 237 . . 3 (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → ∃𝑟 ∈ 𝒫 (LIdeal‘𝑅)𝑆 = (𝑉𝑟))
7858, 77r19.29a 3251 . 2 (((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉) ∧ 𝑆 ≠ ∅) → 𝑆 ∈ ran 𝑉)
79783impa 1107 1 ((𝑅 ∈ CRing ∧ 𝑆 ⊆ ran 𝑉𝑆 ≠ ∅) → 𝑆 ∈ ran 𝑉)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2112   ≠ wne 2990  ∀wral 3109  ∃wrex 3110  {crab 3113  Vcvv 3444   ⊆ wss 3884  ∅c0 4246  𝒫 cpw 4500  ∪ cuni 4803  ∩ cint 4841  ∩ ciin 4885   ↦ cmpt 5113  dom cdm 5523  ran crn 5524   “ cima 5526  Fun wfun 6322   Fn wfn 6323  ‘cfv 6328  Basecbs 16478  Ringcrg 19293  CRingccrg 19294  LIdealclidl 19938  RSpancrsp 19939  PrmIdealcprmidl 31018 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-sca 16576  df-vsca 16577  df-ip 16578  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-grp 18101  df-minusg 18102  df-sbg 18103  df-subg 18271  df-mgp 19236  df-ur 19248  df-ring 19295  df-cring 19296  df-subrg 19529  df-lmod 19632  df-lss 19700  df-lsp 19740  df-sra 19940  df-rgmod 19941  df-lidl 19942  df-rsp 19943  df-prmidl 31019 This theorem is referenced by:  zartopn  31228
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