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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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