Theorem rhmpreimacn 34517

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 𝐵 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 = (Spec‘𝑅)
rhmpreimacn.u	⊢ 𝑈 = (Spec‘𝑆)
rhmpreimacn.a	⊢ 𝐴 = (PrmIdeal‘𝑅)
rhmpreimacn.b	⊢ 𝐵 = (PrmIdeal‘𝑆)
rhmpreimacn.j	⊢ 𝐽 = (TopOpen‘𝑇)
rhmpreimacn.k	⊢ 𝐾 = (TopOpen‘𝑈)
rhmpreimacn.g	⊢ 𝐺 = (𝑖 ∈ 𝐵 ↦ (◡𝐹 “ 𝑖))
rhmpreimacn.r	⊢ (𝜑 → 𝑅 ∈ CRing)
rhmpreimacn.s	⊢ (𝜑 → 𝑆 ∈ CRing)
rhmpreimacn.f	⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))
rhmpreimacn.1	⊢ (𝜑 → ran 𝐹 = (Base‘𝑆))

Assertion

Ref	Expression
rhmpreimacn	⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))

Distinct variable groups:   𝑅,𝑖   𝑖,𝐽   𝑆,𝑖   𝜑,𝑖   𝑖,𝐺   𝐵,𝑖   𝐴,𝑖   𝑖,𝐹

Allowed substitution hints:   𝑇(𝑖)   𝑈(𝑖)   𝐾(𝑖)

Proof of Theorem rhmpreimacn

Dummy variables 𝑎 𝑏 𝑘 𝑙 𝑗 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rhmpreimacn.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ CRing)
2		rhmpreimacn.u	. . . 4 ⊢ 𝑈 = (Spec‘𝑆)
3		rhmpreimacn.k	. . . 4 ⊢ 𝐾 = (TopOpen‘𝑈)
4		rhmpreimacn.b	. . . 4 ⊢ 𝐵 = (PrmIdeal‘𝑆)
5	2, 3, 4	zartopon 34509	. . 3 ⊢ (𝑆 ∈ CRing → 𝐾 ∈ (TopOn‘𝐵))
6	1, 5	syl 18	. 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝐵))
7		rhmpreimacn.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
8		rhmpreimacn.t	. . . 4 ⊢ 𝑇 = (Spec‘𝑅)
9		rhmpreimacn.j	. . . 4 ⊢ 𝐽 = (TopOpen‘𝑇)
10		rhmpreimacn.a	. . . 4 ⊢ 𝐴 = (PrmIdeal‘𝑅)
11	8, 9, 10	zartopon 34509	. . 3 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘𝐴))
12	7, 11	syl 18	. 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐴))
13	1	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐵) → 𝑆 ∈ CRing)
14		rhmpreimacn.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))
15	14	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐵) → 𝐹 ∈ (𝑅 RingHom 𝑆))
16		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐵) → 𝑖 ∈ 𝐵)
17	16, 4	eleqtrdi 2870	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐵) → 𝑖 ∈ (PrmIdeal‘𝑆))
18	10	rhmpreimaprmidl 21635	. . . 4 ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝑖 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝑖) ∈ 𝐴)
19	13, 15, 17, 18	syl21anc 851	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐵) → (◡𝐹 “ 𝑖) ∈ 𝐴)
20		rhmpreimacn.g	. . 3 ⊢ 𝐺 = (𝑖 ∈ 𝐵 ↦ (◡𝐹 “ 𝑖))
21	19, 20	fmptd 7114	. 2 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
22	4	fvexi 6899	. . . . . . 7 ⊢ 𝐵 ∈ V
23	22	rabex 5299	. . . . . 6 ⊢ {𝑘 ∈ 𝐵 ∣ 𝑗 ⊆ 𝑘} ∈ V
24		sseq1 3933	. . . . . . . 8 ⊢ (𝑙 = 𝑗 → (𝑙 ⊆ 𝑘 ↔ 𝑗 ⊆ 𝑘))
25	24	rabbidv 3419	. . . . . . 7 ⊢ (𝑙 = 𝑗 → {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘} = {𝑘 ∈ 𝐵 ∣ 𝑗 ⊆ 𝑘})
26	25	cbvmptv 5208	. . . . . 6 ⊢ (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) = (𝑗 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑗 ⊆ 𝑘})
27	23, 26	fnmpti 6682	. . . . 5 ⊢ (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) Fn (LIdeal‘𝑆)
28	14	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → 𝐹 ∈ (𝑅 RingHom 𝑆))
29		rhmpreimacn.1	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐹 = (Base‘𝑆))
30	29	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → ran 𝐹 = (Base‘𝑆))
31		simplr 781	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → 𝑎 ∈ (LIdeal‘𝑅))
32		eqid 2760	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
33		eqid 2760	. . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
34		eqid 2760	. . . . . . . . 9 ⊢ (LIdeal‘𝑆) = (LIdeal‘𝑆)
35	32, 33, 34	rhmimaidl 33982	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = (Base‘𝑆) ∧ 𝑎 ∈ (LIdeal‘𝑅)) → (𝐹 “ 𝑎) ∈ (LIdeal‘𝑆))
36	28, 30, 31, 35	syl3anc 1398	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → (𝐹 “ 𝑎) ∈ (LIdeal‘𝑆))
37		fveqeq2 6894	. . . . . . . 8 ⊢ (𝑏 = (𝐹 “ 𝑎) → (((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘𝑏) = (◡𝐺 “ 𝑥) ↔ ((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘(𝐹 “ 𝑎)) = (◡𝐺 “ 𝑥)))
38	37	adantl 487	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) ∧ 𝑏 = (𝐹 “ 𝑎)) → (((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘𝑏) = (◡𝐺 “ 𝑥) ↔ ((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘(𝐹 “ 𝑎)) = (◡𝐺 “ 𝑥)))
39	7	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → 𝑅 ∈ CRing)
40	1	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → 𝑆 ∈ CRing)
41	24	rabbidv 3419	. . . . . . . . . 10 ⊢ (𝑙 = 𝑗 → {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘} = {𝑘 ∈ 𝐴 ∣ 𝑗 ⊆ 𝑘})
42	41	cbvmptv 5208	. . . . . . . . 9 ⊢ (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) = (𝑗 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑗 ⊆ 𝑘})
43	8, 2, 10, 4, 9, 3, 20, 39, 40, 28, 30, 31, 42, 26	rhmpreimacnlem 34516	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → ((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘(𝐹 “ 𝑎)) = (◡𝐺 “ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎)))
44		simpr 490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥)
45	44	imaeq2d 6051	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → (◡𝐺 “ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎)) = (◡𝐺 “ 𝑥))
46	43, 45	eqtrd 2795	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → ((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘(𝐹 “ 𝑎)) = (◡𝐺 “ 𝑥))
47	36, 38, 46	rspcedvd 3578	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → ∃𝑏 ∈ (LIdeal‘𝑆)((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘𝑏) = (◡𝐺 “ 𝑥))
48	10	fvexi 6899	. . . . . . . . 9 ⊢ 𝐴 ∈ V
49	48	rabex 5299	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐴 ∣ 𝑗 ⊆ 𝑘} ∈ V
50	49, 42	fnmpti 6682	. . . . . . 7 ⊢ (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) Fn (LIdeal‘𝑅)
51		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (Clsd‘𝐽))
52	7	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑅 ∈ CRing)
53	8, 9, 10, 42	zartopn 34507	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝐴) ∧ ran (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐽)))
54	53	simprd 501	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → ran (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐽))
55	52, 54	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → ran (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐽))
56	51, 55	eleqtrrd 2863	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ ran (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}))
57		fvelrnb 6945	. . . . . . . 8 ⊢ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) Fn (LIdeal‘𝑅) → (𝑥 ∈ ran (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) ↔ ∃𝑎 ∈ (LIdeal‘𝑅)((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥))
58	57	biimpa 482	. . . . . . 7 ⊢ (((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) Fn (LIdeal‘𝑅) ∧ 𝑥 ∈ ran (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})) → ∃𝑎 ∈ (LIdeal‘𝑅)((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥)
59	50, 56, 58	sylancr 599	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → ∃𝑎 ∈ (LIdeal‘𝑅)((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥)
60	47, 59	r19.29a 3170	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → ∃𝑏 ∈ (LIdeal‘𝑆)((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘𝑏) = (◡𝐺 “ 𝑥))
61		fvelrnb 6945	. . . . . 6 ⊢ ((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) Fn (LIdeal‘𝑆) → ((◡𝐺 “ 𝑥) ∈ ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) ↔ ∃𝑏 ∈ (LIdeal‘𝑆)((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘𝑏) = (◡𝐺 “ 𝑥)))
62	61	biimpar 483	. . . . 5 ⊢ (((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) Fn (LIdeal‘𝑆) ∧ ∃𝑏 ∈ (LIdeal‘𝑆)((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘𝑏) = (◡𝐺 “ 𝑥)) → (◡𝐺 “ 𝑥) ∈ ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}))
63	27, 60, 62	sylancr 599	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (◡𝐺 “ 𝑥) ∈ ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}))
64	2, 3, 4, 26	zartopn 34507	. . . . . . 7 ⊢ (𝑆 ∈ CRing → (𝐾 ∈ (TopOn‘𝐵) ∧ ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐾)))
65	64	simprd 501	. . . . . 6 ⊢ (𝑆 ∈ CRing → ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐾))
66	1, 65	syl 18	. . . . 5 ⊢ (𝜑 → ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐾))
67	66	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐾))
68	63, 67	eleqtrd 2862	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (◡𝐺 “ 𝑥) ∈ (Clsd‘𝐾))
69	68	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Clsd‘𝐽)(◡𝐺 “ 𝑥) ∈ (Clsd‘𝐾))
70		iscncl 23587	. . 3 ⊢ ((𝐾 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐴)) → (𝐺 ∈ (𝐾 Cn 𝐽) ↔ (𝐺:𝐵⟶𝐴 ∧ ∀𝑥 ∈ (Clsd‘𝐽)(◡𝐺 “ 𝑥) ∈ (Clsd‘𝐾))))
71	70	biimpar 483	. 2 ⊢ (((𝐾 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐴)) ∧ (𝐺:𝐵⟶𝐴 ∧ ∀𝑥 ∈ (Clsd‘𝐽)(◡𝐺 “ 𝑥) ∈ (Clsd‘𝐾))) → 𝐺 ∈ (𝐾 Cn 𝐽))
72	6, 12, 21, 69, 71	syl22anc 852	1 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412   ⊆ wss 3895   ↦ cmpt 5185  ^◡ccnv 5649  ran crn 5651   “ cima 5653   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  TopOpenctopn 17592  CRingccrg 20460   RingHom crh 20699  LIdealclidl 21484  PrmIdealcprmidl 21616  TopOnctopon 23228  Clsdccld 23334   Cn ccn 23542  Speccrspec 34494

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-ac2 10541  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-rpss 7739  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-oadd 8480  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-dju 9982  df-card 10020  df-ac 10195  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-rest 17593  df-topn 17594  df-0g 17612  df-mre 17756  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-mhm 18978  df-submnd 18979  df-grp 19147  df-minusg 19148  df-sbg 19149  df-subg 19333  df-ghm 19428  df-cntz 19531  df-lsm 19850  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-ring 20461  df-cring 20462  df-rhm 20702  df-subrg 20822  df-lmod 21137  df-lss 21207  df-lsp 21247  df-sra 21448  df-rgmod 21449  df-lidl 21486  df-rsp 21487  df-prmidl 21617  df-lpidl 21646  df-top 23212  df-topon 23229  df-cld 23337  df-cn 23545  df-mxidl 33985  df-idlsrg 34033  df-rspec 34495

This theorem is used by: (None)