Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  zarclsint Structured version   Visualization version   GIF version

Theorem zarclsint 32493
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 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
Assertion
Ref Expression
zarclsint ((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉 ∧ 𝑆 β‰  βˆ…) β†’ ∩ 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 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crngring 19983 . . . . . . 7 (𝑅 ∈ CRing β†’ 𝑅 ∈ Ring)
21ad4antr 731 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ 𝑅 ∈ Ring)
3 elpwi 4572 . . . . . . . . . . . 12 (π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…) β†’ π‘Ÿ βŠ† (LIdealβ€˜π‘…))
43adantl 483 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) β†’ π‘Ÿ βŠ† (LIdealβ€˜π‘…))
54adantr 482 . . . . . . . . . 10 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ π‘Ÿ βŠ† (LIdealβ€˜π‘…))
65sselda 3949 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ 𝑖 ∈ π‘Ÿ) β†’ 𝑖 ∈ (LIdealβ€˜π‘…))
7 eqid 2737 . . . . . . . . . 10 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
8 eqid 2737 . . . . . . . . . 10 (LIdealβ€˜π‘…) = (LIdealβ€˜π‘…)
97, 8lidlss 20696 . . . . . . . . 9 (𝑖 ∈ (LIdealβ€˜π‘…) β†’ 𝑖 βŠ† (Baseβ€˜π‘…))
106, 9syl 17 . . . . . . . 8 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ 𝑖 ∈ π‘Ÿ) β†’ 𝑖 βŠ† (Baseβ€˜π‘…))
1110ralrimiva 3144 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ βˆ€π‘– ∈ π‘Ÿ 𝑖 βŠ† (Baseβ€˜π‘…))
12 unissb 4905 . . . . . . 7 (βˆͺ π‘Ÿ βŠ† (Baseβ€˜π‘…) ↔ βˆ€π‘– ∈ π‘Ÿ 𝑖 βŠ† (Baseβ€˜π‘…))
1311, 12sylibr 233 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ βˆͺ π‘Ÿ βŠ† (Baseβ€˜π‘…))
14 eqid 2737 . . . . . . 7 (RSpanβ€˜π‘…) = (RSpanβ€˜π‘…)
1514, 7, 8rspcl 20708 . . . . . 6 ((𝑅 ∈ Ring ∧ βˆͺ π‘Ÿ βŠ† (Baseβ€˜π‘…)) β†’ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) ∈ (LIdealβ€˜π‘…))
162, 13, 15syl2anc 585 . . . . 5 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) ∈ (LIdealβ€˜π‘…))
17 sseq1 3974 . . . . . . . 8 (𝑖 = ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) β†’ (𝑖 βŠ† 𝑗 ↔ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗))
1817rabbidv 3418 . . . . . . 7 (𝑖 = ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗})
1918eqeq2d 2748 . . . . . 6 (𝑖 = ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) β†’ (∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} ↔ ∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗}))
2019adantl 483 . . . . 5 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ 𝑖 = ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ)) β†’ (∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} ↔ ∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗}))
21 simpr 486 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ 𝑆 = (𝑉 β€œ π‘Ÿ))
2221inteqd 4917 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ ∩ 𝑆 = ∩ (𝑉 β€œ π‘Ÿ))
23 zarclsx.1 . . . . . . . . . 10 𝑉 = (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
2423funmpt2 6545 . . . . . . . . 9 Fun 𝑉
2524a1i 11 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ Fun 𝑉)
26 fvex 6860 . . . . . . . . . . 11 (PrmIdealβ€˜π‘…) ∈ V
2726rabex 5294 . . . . . . . . . 10 {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} ∈ V
2827, 23dmmpti 6650 . . . . . . . . 9 dom 𝑉 = (LIdealβ€˜π‘…)
295, 28sseqtrrdi 4000 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ π‘Ÿ βŠ† dom 𝑉)
30 intimafv 31666 . . . . . . . 8 ((Fun 𝑉 ∧ π‘Ÿ βŠ† dom 𝑉) β†’ ∩ (𝑉 β€œ π‘Ÿ) = ∩ 𝑙 ∈ π‘Ÿ (π‘‰β€˜π‘™))
3125, 29, 30syl2anc 585 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ ∩ (𝑉 β€œ π‘Ÿ) = ∩ 𝑙 ∈ π‘Ÿ (π‘‰β€˜π‘™))
3222, 31eqtrd 2777 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ ∩ 𝑆 = ∩ 𝑙 ∈ π‘Ÿ (π‘‰β€˜π‘™))
33 simplr 768 . . . . . . . . . 10 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ π‘Ÿ = βˆ…) β†’ 𝑆 = (𝑉 β€œ π‘Ÿ))
34 simpr 486 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ π‘Ÿ = βˆ…) β†’ π‘Ÿ = βˆ…)
3534imaeq2d 6018 . . . . . . . . . . 11 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ π‘Ÿ = βˆ…) β†’ (𝑉 β€œ π‘Ÿ) = (𝑉 β€œ βˆ…))
36 ima0 6034 . . . . . . . . . . 11 (𝑉 β€œ βˆ…) = βˆ…
3735, 36eqtrdi 2793 . . . . . . . . . 10 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ π‘Ÿ = βˆ…) β†’ (𝑉 β€œ π‘Ÿ) = βˆ…)
3833, 37eqtrd 2777 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ π‘Ÿ = βˆ…) β†’ 𝑆 = βˆ…)
39 simp-4r 783 . . . . . . . . . 10 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ π‘Ÿ = βˆ…) β†’ 𝑆 β‰  βˆ…)
4039neneqd 2949 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ π‘Ÿ = βˆ…) β†’ Β¬ 𝑆 = βˆ…)
4138, 40pm2.65da 816 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ Β¬ π‘Ÿ = βˆ…)
4241neqned 2951 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ π‘Ÿ β‰  βˆ…)
4323, 14zarclsiin 32492 . . . . . . 7 ((𝑅 ∈ Ring ∧ π‘Ÿ βŠ† (LIdealβ€˜π‘…) ∧ π‘Ÿ β‰  βˆ…) β†’ ∩ 𝑙 ∈ π‘Ÿ (π‘‰β€˜π‘™) = (π‘‰β€˜((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ)))
442, 5, 42, 43syl3anc 1372 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ ∩ 𝑙 ∈ π‘Ÿ (π‘‰β€˜π‘™) = (π‘‰β€˜((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ)))
4523a1i 11 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ 𝑉 = (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗}))
4618adantl 483 . . . . . . 7 ((((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) ∧ 𝑖 = ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ)) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗})
4726rabex 5294 . . . . . . . 8 {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗} ∈ V
4847a1i 11 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗} ∈ V)
4945, 46, 16, 48fvmptd 6960 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ (π‘‰β€˜((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ)) = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗})
5032, 44, 493eqtrd 2781 . . . . 5 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ ∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ ((RSpanβ€˜π‘…)β€˜βˆͺ π‘Ÿ) βŠ† 𝑗})
5116, 20, 50rspcedvd 3586 . . . 4 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ βˆƒπ‘– ∈ (LIdealβ€˜π‘…)∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
52 intex 5299 . . . . . . . 8 (𝑆 β‰  βˆ… ↔ ∩ 𝑆 ∈ V)
5352biimpi 215 . . . . . . 7 (𝑆 β‰  βˆ… β†’ ∩ 𝑆 ∈ V)
54533ad2ant3 1136 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉 ∧ 𝑆 β‰  βˆ…) β†’ ∩ 𝑆 ∈ V)
5523elrnmpt 5916 . . . . . 6 (∩ 𝑆 ∈ V β†’ (∩ 𝑆 ∈ ran 𝑉 ↔ βˆƒπ‘– ∈ (LIdealβ€˜π‘…)∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗}))
5654, 55syl 17 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉 ∧ 𝑆 β‰  βˆ…) β†’ (∩ 𝑆 ∈ ran 𝑉 ↔ βˆƒπ‘– ∈ (LIdealβ€˜π‘…)∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗}))
5756ad5ant123 1365 . . . 4 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ (∩ 𝑆 ∈ ran 𝑉 ↔ βˆƒπ‘– ∈ (LIdealβ€˜π‘…)∩ 𝑆 = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗}))
5851, 57mpbird 257 . . 3 (((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ ∩ 𝑆 ∈ ran 𝑉)
59 fvexd 6862 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ (LIdealβ€˜π‘…) ∈ V)
6024a1i 11 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ Fun 𝑉)
61 simplr 768 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ 𝑆 βŠ† ran 𝑉)
6227, 23fnmpti 6649 . . . . . . . 8 𝑉 Fn (LIdealβ€˜π‘…)
63 fnima 6636 . . . . . . . 8 (𝑉 Fn (LIdealβ€˜π‘…) β†’ (𝑉 β€œ (LIdealβ€˜π‘…)) = ran 𝑉)
6462, 63ax-mp 5 . . . . . . 7 (𝑉 β€œ (LIdealβ€˜π‘…)) = ran 𝑉
6561, 64sseqtrrdi 4000 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ 𝑆 βŠ† (𝑉 β€œ (LIdealβ€˜π‘…)))
66 ssimaexg 6932 . . . . . 6 (((LIdealβ€˜π‘…) ∈ V ∧ Fun 𝑉 ∧ 𝑆 βŠ† (𝑉 β€œ (LIdealβ€˜π‘…))) β†’ βˆƒπ‘Ÿ(π‘Ÿ βŠ† (LIdealβ€˜π‘…) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)))
6759, 60, 65, 66syl3anc 1372 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ βˆƒπ‘Ÿ(π‘Ÿ βŠ† (LIdealβ€˜π‘…) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)))
68 vex 3452 . . . . . . . . . 10 π‘Ÿ ∈ V
6968a1i 11 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ βŠ† (LIdealβ€˜π‘…)) β†’ π‘Ÿ ∈ V)
70 simpr 486 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ βŠ† (LIdealβ€˜π‘…)) β†’ π‘Ÿ βŠ† (LIdealβ€˜π‘…))
7169, 70elpwd 4571 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) ∧ π‘Ÿ βŠ† (LIdealβ€˜π‘…)) β†’ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…))
7271ex 414 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ (π‘Ÿ βŠ† (LIdealβ€˜π‘…) β†’ π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)))
7372anim1d 612 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ ((π‘Ÿ βŠ† (LIdealβ€˜π‘…) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ (π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ))))
7473eximdv 1921 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ (βˆƒπ‘Ÿ(π‘Ÿ βŠ† (LIdealβ€˜π‘…) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)) β†’ βˆƒπ‘Ÿ(π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ))))
7567, 74mpd 15 . . . 4 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ βˆƒπ‘Ÿ(π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)))
76 df-rex 3075 . . . 4 (βˆƒπ‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)𝑆 = (𝑉 β€œ π‘Ÿ) ↔ βˆƒπ‘Ÿ(π‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…) ∧ 𝑆 = (𝑉 β€œ π‘Ÿ)))
7775, 76sylibr 233 . . 3 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ βˆƒπ‘Ÿ ∈ 𝒫 (LIdealβ€˜π‘…)𝑆 = (𝑉 β€œ π‘Ÿ))
7858, 77r19.29a 3160 . 2 (((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉) ∧ 𝑆 β‰  βˆ…) β†’ ∩ 𝑆 ∈ ran 𝑉)
79783impa 1111 1 ((𝑅 ∈ CRing ∧ 𝑆 βŠ† ran 𝑉 ∧ 𝑆 β‰  βˆ…) β†’ ∩ 𝑆 ∈ ran 𝑉)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  {crab 3410  Vcvv 3448   βŠ† wss 3915  βˆ…c0 4287  π’« cpw 4565  βˆͺ cuni 4870  βˆ© cint 4912  βˆ© ciin 4960   ↦ cmpt 5193  dom cdm 5638  ran crn 5639   β€œ cima 5641  Fun wfun 6495   Fn wfn 6496  β€˜cfv 6501  Basecbs 17090  Ringcrg 19971  CRingccrg 19972  LIdealclidl 20647  RSpancrsp 20648  PrmIdealcprmidl 32247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-mulr 17154  df-sca 17156  df-vsca 17157  df-ip 17158  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-grp 18758  df-minusg 18759  df-sbg 18760  df-subg 18932  df-mgp 19904  df-ur 19921  df-ring 19973  df-cring 19974  df-subrg 20236  df-lmod 20340  df-lss 20409  df-lsp 20449  df-sra 20649  df-rgmod 20650  df-lidl 20651  df-rsp 20652  df-prmidl 32248
This theorem is referenced by:  zartopn  32496
  Copyright terms: Public domain W3C validator