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Theorem rhmpreimacn 32506
Description: The function mapping a prime ideal to its preimage by a surjective ring homomorphism is continuous, when considering the Zariski topology. Corollary 1.2.3 of [EGA], p. 83. Notice that the direction of the continuous map 𝐺 is reverse: the original ring homomorphism 𝐹 goes from 𝑅 to 𝑆, but the continuous map 𝐺 goes from 𝐡 to 𝐴. This mapping is also called "induced map on prime spectra" or "pullback on primes". (Contributed by Thierry Arnoux, 8-Jul-2024.)
Hypotheses
Ref Expression
rhmpreimacn.t 𝑇 = (Specβ€˜π‘…)
rhmpreimacn.u π‘ˆ = (Specβ€˜π‘†)
rhmpreimacn.a 𝐴 = (PrmIdealβ€˜π‘…)
rhmpreimacn.b 𝐡 = (PrmIdealβ€˜π‘†)
rhmpreimacn.j 𝐽 = (TopOpenβ€˜π‘‡)
rhmpreimacn.k 𝐾 = (TopOpenβ€˜π‘ˆ)
rhmpreimacn.g 𝐺 = (𝑖 ∈ 𝐡 ↦ (◑𝐹 β€œ 𝑖))
rhmpreimacn.r (πœ‘ β†’ 𝑅 ∈ CRing)
rhmpreimacn.s (πœ‘ β†’ 𝑆 ∈ CRing)
rhmpreimacn.f (πœ‘ β†’ 𝐹 ∈ (𝑅 RingHom 𝑆))
rhmpreimacn.1 (πœ‘ β†’ ran 𝐹 = (Baseβ€˜π‘†))
Assertion
Ref Expression
rhmpreimacn (πœ‘ β†’ 𝐺 ∈ (𝐾 Cn 𝐽))
Distinct variable groups:   𝐴,𝑖   𝐡,𝑖   𝑖,𝐹   𝑖,𝐺   𝑖,𝐽   𝑅,𝑖   𝑆,𝑖   πœ‘,𝑖
Allowed substitution hints:   𝑇(𝑖)   π‘ˆ(𝑖)   𝐾(𝑖)

Proof of Theorem rhmpreimacn
Dummy variables 𝑗 π‘˜ π‘₯ 𝑏 π‘Ž 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rhmpreimacn.s . . 3 (πœ‘ β†’ 𝑆 ∈ CRing)
2 rhmpreimacn.u . . . 4 π‘ˆ = (Specβ€˜π‘†)
3 rhmpreimacn.k . . . 4 𝐾 = (TopOpenβ€˜π‘ˆ)
4 rhmpreimacn.b . . . 4 𝐡 = (PrmIdealβ€˜π‘†)
52, 3, 4zartopon 32498 . . 3 (𝑆 ∈ CRing β†’ 𝐾 ∈ (TopOnβ€˜π΅))
61, 5syl 17 . 2 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π΅))
7 rhmpreimacn.r . . 3 (πœ‘ β†’ 𝑅 ∈ CRing)
8 rhmpreimacn.t . . . 4 𝑇 = (Specβ€˜π‘…)
9 rhmpreimacn.j . . . 4 𝐽 = (TopOpenβ€˜π‘‡)
10 rhmpreimacn.a . . . 4 𝐴 = (PrmIdealβ€˜π‘…)
118, 9, 10zartopon 32498 . . 3 (𝑅 ∈ CRing β†’ 𝐽 ∈ (TopOnβ€˜π΄))
127, 11syl 17 . 2 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π΄))
131adantr 482 . . . 4 ((πœ‘ ∧ 𝑖 ∈ 𝐡) β†’ 𝑆 ∈ CRing)
14 rhmpreimacn.f . . . . 5 (πœ‘ β†’ 𝐹 ∈ (𝑅 RingHom 𝑆))
1514adantr 482 . . . 4 ((πœ‘ ∧ 𝑖 ∈ 𝐡) β†’ 𝐹 ∈ (𝑅 RingHom 𝑆))
16 simpr 486 . . . . 5 ((πœ‘ ∧ 𝑖 ∈ 𝐡) β†’ 𝑖 ∈ 𝐡)
1716, 4eleqtrdi 2848 . . . 4 ((πœ‘ ∧ 𝑖 ∈ 𝐡) β†’ 𝑖 ∈ (PrmIdealβ€˜π‘†))
1810rhmpreimaprmidl 32264 . . . 4 (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝑖 ∈ (PrmIdealβ€˜π‘†)) β†’ (◑𝐹 β€œ 𝑖) ∈ 𝐴)
1913, 15, 17, 18syl21anc 837 . . 3 ((πœ‘ ∧ 𝑖 ∈ 𝐡) β†’ (◑𝐹 β€œ 𝑖) ∈ 𝐴)
20 rhmpreimacn.g . . 3 𝐺 = (𝑖 ∈ 𝐡 ↦ (◑𝐹 β€œ 𝑖))
2119, 20fmptd 7067 . 2 (πœ‘ β†’ 𝐺:𝐡⟢𝐴)
224fvexi 6861 . . . . . . 7 𝐡 ∈ V
2322rabex 5294 . . . . . 6 {π‘˜ ∈ 𝐡 ∣ 𝑗 βŠ† π‘˜} ∈ V
24 sseq1 3974 . . . . . . . 8 (𝑙 = 𝑗 β†’ (𝑙 βŠ† π‘˜ ↔ 𝑗 βŠ† π‘˜))
2524rabbidv 3418 . . . . . . 7 (𝑙 = 𝑗 β†’ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜} = {π‘˜ ∈ 𝐡 ∣ 𝑗 βŠ† π‘˜})
2625cbvmptv 5223 . . . . . 6 (𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜}) = (𝑗 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑗 βŠ† π‘˜})
2723, 26fnmpti 6649 . . . . 5 (𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜}) Fn (LIdealβ€˜π‘†)
2814ad3antrrr 729 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) ∧ π‘Ž ∈ (LIdealβ€˜π‘…)) ∧ ((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž) = π‘₯) β†’ 𝐹 ∈ (𝑅 RingHom 𝑆))
29 rhmpreimacn.1 . . . . . . . . 9 (πœ‘ β†’ ran 𝐹 = (Baseβ€˜π‘†))
3029ad3antrrr 729 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) ∧ π‘Ž ∈ (LIdealβ€˜π‘…)) ∧ ((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž) = π‘₯) β†’ ran 𝐹 = (Baseβ€˜π‘†))
31 simplr 768 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) ∧ π‘Ž ∈ (LIdealβ€˜π‘…)) ∧ ((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž) = π‘₯) β†’ π‘Ž ∈ (LIdealβ€˜π‘…))
32 eqid 2737 . . . . . . . . 9 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
33 eqid 2737 . . . . . . . . 9 (LIdealβ€˜π‘…) = (LIdealβ€˜π‘…)
34 eqid 2737 . . . . . . . . 9 (LIdealβ€˜π‘†) = (LIdealβ€˜π‘†)
3532, 33, 34rhmimaidl 32246 . . . . . . . 8 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = (Baseβ€˜π‘†) ∧ π‘Ž ∈ (LIdealβ€˜π‘…)) β†’ (𝐹 β€œ π‘Ž) ∈ (LIdealβ€˜π‘†))
3628, 30, 31, 35syl3anc 1372 . . . . . . 7 ((((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) ∧ π‘Ž ∈ (LIdealβ€˜π‘…)) ∧ ((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž) = π‘₯) β†’ (𝐹 β€œ π‘Ž) ∈ (LIdealβ€˜π‘†))
37 fveqeq2 6856 . . . . . . . 8 (𝑏 = (𝐹 β€œ π‘Ž) β†’ (((𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜})β€˜π‘) = (◑𝐺 β€œ π‘₯) ↔ ((𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜})β€˜(𝐹 β€œ π‘Ž)) = (◑𝐺 β€œ π‘₯)))
3837adantl 483 . . . . . . 7 (((((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) ∧ π‘Ž ∈ (LIdealβ€˜π‘…)) ∧ ((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž) = π‘₯) ∧ 𝑏 = (𝐹 β€œ π‘Ž)) β†’ (((𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜})β€˜π‘) = (◑𝐺 β€œ π‘₯) ↔ ((𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜})β€˜(𝐹 β€œ π‘Ž)) = (◑𝐺 β€œ π‘₯)))
397ad3antrrr 729 . . . . . . . . 9 ((((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) ∧ π‘Ž ∈ (LIdealβ€˜π‘…)) ∧ ((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž) = π‘₯) β†’ 𝑅 ∈ CRing)
401ad3antrrr 729 . . . . . . . . 9 ((((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) ∧ π‘Ž ∈ (LIdealβ€˜π‘…)) ∧ ((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž) = π‘₯) β†’ 𝑆 ∈ CRing)
4124rabbidv 3418 . . . . . . . . . 10 (𝑙 = 𝑗 β†’ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜} = {π‘˜ ∈ 𝐴 ∣ 𝑗 βŠ† π‘˜})
4241cbvmptv 5223 . . . . . . . . 9 (𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜}) = (𝑗 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑗 βŠ† π‘˜})
438, 2, 10, 4, 9, 3, 20, 39, 40, 28, 30, 31, 42, 26rhmpreimacnlem 32505 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) ∧ π‘Ž ∈ (LIdealβ€˜π‘…)) ∧ ((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž) = π‘₯) β†’ ((𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜})β€˜(𝐹 β€œ π‘Ž)) = (◑𝐺 β€œ ((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž)))
44 simpr 486 . . . . . . . . 9 ((((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) ∧ π‘Ž ∈ (LIdealβ€˜π‘…)) ∧ ((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž) = π‘₯) β†’ ((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž) = π‘₯)
4544imaeq2d 6018 . . . . . . . 8 ((((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) ∧ π‘Ž ∈ (LIdealβ€˜π‘…)) ∧ ((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž) = π‘₯) β†’ (◑𝐺 β€œ ((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž)) = (◑𝐺 β€œ π‘₯))
4643, 45eqtrd 2777 . . . . . . 7 ((((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) ∧ π‘Ž ∈ (LIdealβ€˜π‘…)) ∧ ((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž) = π‘₯) β†’ ((𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜})β€˜(𝐹 β€œ π‘Ž)) = (◑𝐺 β€œ π‘₯))
4736, 38, 46rspcedvd 3586 . . . . . 6 ((((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) ∧ π‘Ž ∈ (LIdealβ€˜π‘…)) ∧ ((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž) = π‘₯) β†’ βˆƒπ‘ ∈ (LIdealβ€˜π‘†)((𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜})β€˜π‘) = (◑𝐺 β€œ π‘₯))
4810fvexi 6861 . . . . . . . . 9 𝐴 ∈ V
4948rabex 5294 . . . . . . . 8 {π‘˜ ∈ 𝐴 ∣ 𝑗 βŠ† π‘˜} ∈ V
5049, 42fnmpti 6649 . . . . . . 7 (𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜}) Fn (LIdealβ€˜π‘…)
51 simpr 486 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) β†’ π‘₯ ∈ (Clsdβ€˜π½))
527adantr 482 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) β†’ 𝑅 ∈ CRing)
538, 9, 10, 42zartopn 32496 . . . . . . . . . 10 (𝑅 ∈ CRing β†’ (𝐽 ∈ (TopOnβ€˜π΄) ∧ ran (𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜}) = (Clsdβ€˜π½)))
5453simprd 497 . . . . . . . . 9 (𝑅 ∈ CRing β†’ ran (𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜}) = (Clsdβ€˜π½))
5552, 54syl 17 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) β†’ ran (𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜}) = (Clsdβ€˜π½))
5651, 55eleqtrrd 2841 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) β†’ π‘₯ ∈ ran (𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜}))
57 fvelrnb 6908 . . . . . . . 8 ((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜}) Fn (LIdealβ€˜π‘…) β†’ (π‘₯ ∈ ran (𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜}) ↔ βˆƒπ‘Ž ∈ (LIdealβ€˜π‘…)((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž) = π‘₯))
5857biimpa 478 . . . . . . 7 (((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜}) Fn (LIdealβ€˜π‘…) ∧ π‘₯ ∈ ran (𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})) β†’ βˆƒπ‘Ž ∈ (LIdealβ€˜π‘…)((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž) = π‘₯)
5950, 56, 58sylancr 588 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) β†’ βˆƒπ‘Ž ∈ (LIdealβ€˜π‘…)((𝑙 ∈ (LIdealβ€˜π‘…) ↦ {π‘˜ ∈ 𝐴 ∣ 𝑙 βŠ† π‘˜})β€˜π‘Ž) = π‘₯)
6047, 59r19.29a 3160 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) β†’ βˆƒπ‘ ∈ (LIdealβ€˜π‘†)((𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜})β€˜π‘) = (◑𝐺 β€œ π‘₯))
61 fvelrnb 6908 . . . . . 6 ((𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜}) Fn (LIdealβ€˜π‘†) β†’ ((◑𝐺 β€œ π‘₯) ∈ ran (𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜}) ↔ βˆƒπ‘ ∈ (LIdealβ€˜π‘†)((𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜})β€˜π‘) = (◑𝐺 β€œ π‘₯)))
6261biimpar 479 . . . . 5 (((𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜}) Fn (LIdealβ€˜π‘†) ∧ βˆƒπ‘ ∈ (LIdealβ€˜π‘†)((𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜})β€˜π‘) = (◑𝐺 β€œ π‘₯)) β†’ (◑𝐺 β€œ π‘₯) ∈ ran (𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜}))
6327, 60, 62sylancr 588 . . . 4 ((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) β†’ (◑𝐺 β€œ π‘₯) ∈ ran (𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜}))
642, 3, 4, 26zartopn 32496 . . . . . . 7 (𝑆 ∈ CRing β†’ (𝐾 ∈ (TopOnβ€˜π΅) ∧ ran (𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜}) = (Clsdβ€˜πΎ)))
6564simprd 497 . . . . . 6 (𝑆 ∈ CRing β†’ ran (𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜}) = (Clsdβ€˜πΎ))
661, 65syl 17 . . . . 5 (πœ‘ β†’ ran (𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜}) = (Clsdβ€˜πΎ))
6766adantr 482 . . . 4 ((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) β†’ ran (𝑙 ∈ (LIdealβ€˜π‘†) ↦ {π‘˜ ∈ 𝐡 ∣ 𝑙 βŠ† π‘˜}) = (Clsdβ€˜πΎ))
6863, 67eleqtrd 2840 . . 3 ((πœ‘ ∧ π‘₯ ∈ (Clsdβ€˜π½)) β†’ (◑𝐺 β€œ π‘₯) ∈ (Clsdβ€˜πΎ))
6968ralrimiva 3144 . 2 (πœ‘ β†’ βˆ€π‘₯ ∈ (Clsdβ€˜π½)(◑𝐺 β€œ π‘₯) ∈ (Clsdβ€˜πΎ))
70 iscncl 22636 . . 3 ((𝐾 ∈ (TopOnβ€˜π΅) ∧ 𝐽 ∈ (TopOnβ€˜π΄)) β†’ (𝐺 ∈ (𝐾 Cn 𝐽) ↔ (𝐺:𝐡⟢𝐴 ∧ βˆ€π‘₯ ∈ (Clsdβ€˜π½)(◑𝐺 β€œ π‘₯) ∈ (Clsdβ€˜πΎ))))
7170biimpar 479 . 2 (((𝐾 ∈ (TopOnβ€˜π΅) ∧ 𝐽 ∈ (TopOnβ€˜π΄)) ∧ (𝐺:𝐡⟢𝐴 ∧ βˆ€π‘₯ ∈ (Clsdβ€˜π½)(◑𝐺 β€œ π‘₯) ∈ (Clsdβ€˜πΎ))) β†’ 𝐺 ∈ (𝐾 Cn 𝐽))
726, 12, 21, 69, 71syl22anc 838 1 (πœ‘ β†’ 𝐺 ∈ (𝐾 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  βˆƒwrex 3074  {crab 3410   βŠ† wss 3915   ↦ cmpt 5193  β—‘ccnv 5637  ran crn 5639   β€œ cima 5641   Fn wfn 6496  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  Basecbs 17090  TopOpenctopn 17310  CRingccrg 19972   RingHom crh 20152  LIdealclidl 20647  TopOnctopon 22275  Clsdccld 22383   Cn ccn 22591  PrmIdealcprmidl 32247  Speccrspec 32483
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-ac2 10406  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-tp 4596  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-se 5594  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-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-rpss 7665  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-ac 10059  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-9 12230  df-n0 12421  df-z 12507  df-dec 12626  df-uz 12771  df-fz 13432  df-struct 17026  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-tset 17159  df-ple 17160  df-rest 17311  df-topn 17312  df-0g 17330  df-mre 17473  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-mhm 18608  df-submnd 18609  df-grp 18758  df-minusg 18759  df-sbg 18760  df-subg 18932  df-ghm 19013  df-cntz 19104  df-lsm 19425  df-cmn 19571  df-abl 19572  df-mgp 19904  df-ur 19921  df-ring 19973  df-cring 19974  df-rnghom 20155  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-lpidl 20729  df-top 22259  df-topon 22276  df-cld 22386  df-cn 22594  df-prmidl 32248  df-mxidl 32269  df-idlsrg 32283  df-rspec 32484
This theorem is referenced by: (None)
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