Theorem rhmpreimacn 32853

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:   𝐴,𝑖   𝐵,𝑖   𝑖,𝐹   𝑖,𝐺   𝑖,𝐽   𝑅,𝑖   𝑆,𝑖   𝜑,𝑖

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

Proof of Theorem rhmpreimacn

Dummy variables 𝑗 𝑘 𝑥 𝑏 𝑎 𝑙 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 32845	. . 3 ⊢ (𝑆 ∈ CRing → 𝐾 ∈ (TopOn‘𝐵))
6	1, 5	syl 17	. 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 32845	. . 3 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ (TopOn‘𝐴))
12	7, 11	syl 17	. 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐴))
13	1	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐵) → 𝑆 ∈ CRing)
14		rhmpreimacn.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))
15	14	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐵) → 𝐹 ∈ (𝑅 RingHom 𝑆))
16		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐵) → 𝑖 ∈ 𝐵)
17	16, 4	eleqtrdi 2843	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐵) → 𝑖 ∈ (PrmIdeal‘𝑆))
18	10	rhmpreimaprmidl 32558	. . . 4 ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝑖 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝑖) ∈ 𝐴)
19	13, 15, 17, 18	syl21anc 836	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐵) → (◡𝐹 “ 𝑖) ∈ 𝐴)
20		rhmpreimacn.g	. . 3 ⊢ 𝐺 = (𝑖 ∈ 𝐵 ↦ (◡𝐹 “ 𝑖))
21	19, 20	fmptd 7110	. 2 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
22	4	fvexi 6902	. . . . . . 7 ⊢ 𝐵 ∈ V
23	22	rabex 5331	. . . . . 6 ⊢ {𝑘 ∈ 𝐵 ∣ 𝑗 ⊆ 𝑘} ∈ V
24		sseq1 4006	. . . . . . . 8 ⊢ (𝑙 = 𝑗 → (𝑙 ⊆ 𝑘 ↔ 𝑗 ⊆ 𝑘))
25	24	rabbidv 3440	. . . . . . 7 ⊢ (𝑙 = 𝑗 → {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘} = {𝑘 ∈ 𝐵 ∣ 𝑗 ⊆ 𝑘})
26	25	cbvmptv 5260	. . . . . 6 ⊢ (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) = (𝑗 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑗 ⊆ 𝑘})
27	23, 26	fnmpti 6690	. . . . 5 ⊢ (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) Fn (LIdeal‘𝑆)
28	14	ad3antrrr 728	. . . . . . . 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 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → ran 𝐹 = (Base‘𝑆))
31		simplr 767	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → 𝑎 ∈ (LIdeal‘𝑅))
32		eqid 2732	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
33		eqid 2732	. . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
34		eqid 2732	. . . . . . . . 9 ⊢ (LIdeal‘𝑆) = (LIdeal‘𝑆)
35	32, 33, 34	rhmimaidl 32538	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = (Base‘𝑆) ∧ 𝑎 ∈ (LIdeal‘𝑅)) → (𝐹 “ 𝑎) ∈ (LIdeal‘𝑆))
36	28, 30, 31, 35	syl3anc 1371	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → (𝐹 “ 𝑎) ∈ (LIdeal‘𝑆))
37		fveqeq2 6897	. . . . . . . 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 482	. . . . . . 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 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → 𝑅 ∈ CRing)
40	1	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) ∧ 𝑎 ∈ (LIdeal‘𝑅)) ∧ ((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥) → 𝑆 ∈ CRing)
41	24	rabbidv 3440	. . . . . . . . . 10 ⊢ (𝑙 = 𝑗 → {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘} = {𝑘 ∈ 𝐴 ∣ 𝑗 ⊆ 𝑘})
42	41	cbvmptv 5260	. . . . . . . . 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 32852	. . . . . . . 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 485	. . . . . . . . 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 6057	. . . . . . . 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 2772	. . . . . . 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 3614	. . . . . 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 6902	. . . . . . . . 9 ⊢ 𝐴 ∈ V
49	48	rabex 5331	. . . . . . . 8 ⊢ {𝑘 ∈ 𝐴 ∣ 𝑗 ⊆ 𝑘} ∈ V
50	49, 42	fnmpti 6690	. . . . . . 7 ⊢ (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) Fn (LIdeal‘𝑅)
51		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ (Clsd‘𝐽))
52	7	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑅 ∈ CRing)
53	8, 9, 10, 42	zartopn 32843	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘𝐴) ∧ ran (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐽)))
54	53	simprd 496	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → ran (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐽))
55	52, 54	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → ran (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐽))
56	51, 55	eleqtrrd 2836	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ∈ ran (𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘}))
57		fvelrnb 6949	. . . . . . . 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 477	. . . . . . 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 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → ∃𝑎 ∈ (LIdeal‘𝑅)((𝑙 ∈ (LIdeal‘𝑅) ↦ {𝑘 ∈ 𝐴 ∣ 𝑙 ⊆ 𝑘})‘𝑎) = 𝑥)
60	47, 59	r19.29a 3162	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → ∃𝑏 ∈ (LIdeal‘𝑆)((𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘})‘𝑏) = (◡𝐺 “ 𝑥))
61		fvelrnb 6949	. . . . . 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 478	. . . . 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 587	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (◡𝐺 “ 𝑥) ∈ ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}))
64	2, 3, 4, 26	zartopn 32843	. . . . . . 7 ⊢ (𝑆 ∈ CRing → (𝐾 ∈ (TopOn‘𝐵) ∧ ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐾)))
65	64	simprd 496	. . . . . 6 ⊢ (𝑆 ∈ CRing → ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐾))
66	1, 65	syl 17	. . . . 5 ⊢ (𝜑 → ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐾))
67	66	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → ran (𝑙 ∈ (LIdeal‘𝑆) ↦ {𝑘 ∈ 𝐵 ∣ 𝑙 ⊆ 𝑘}) = (Clsd‘𝐾))
68	63, 67	eleqtrd 2835	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Clsd‘𝐽)) → (◡𝐺 “ 𝑥) ∈ (Clsd‘𝐾))
69	68	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Clsd‘𝐽)(◡𝐺 “ 𝑥) ∈ (Clsd‘𝐾))
70		iscncl 22764	. . 3 ⊢ ((𝐾 ∈ (TopOn‘𝐵) ∧ 𝐽 ∈ (TopOn‘𝐴)) → (𝐺 ∈ (𝐾 Cn 𝐽) ↔ (𝐺:𝐵⟶𝐴 ∧ ∀𝑥 ∈ (Clsd‘𝐽)(◡𝐺 “ 𝑥) ∈ (Clsd‘𝐾))))
71	70	biimpar 478	. 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 837	1 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  {crab 3432   ⊆ wss 3947   ↦ cmpt 5230  ^◡ccnv 5674  ran crn 5676   “ cima 5678   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7405  Basecbs 17140  TopOpenctopn 17363  CRingccrg 20050   RingHom crh 20240  LIdealclidl 20775  TopOnctopon 22403  Clsdccld 22511   Cn ccn 22719  PrmIdealcprmidl 32541  Speccrspec 32830

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-ac2 10454  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-rpss 7709  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-ac 10107  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-sca 17209  df-vsca 17210  df-ip 17211  df-tset 17212  df-ple 17213  df-rest 17364  df-topn 17365  df-0g 17383  df-mre 17526  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-submnd 18668  df-grp 18818  df-minusg 18819  df-sbg 18820  df-subg 18997  df-ghm 19084  df-cntz 19175  df-lsm 19498  df-cmn 19644  df-abl 19645  df-mgp 19982  df-ur 19999  df-ring 20051  df-cring 20052  df-rnghom 20243  df-subrg 20353  df-lmod 20465  df-lss 20535  df-lsp 20575  df-sra 20777  df-rgmod 20778  df-lidl 20779  df-rsp 20780  df-lpidl 20873  df-top 22387  df-topon 22404  df-cld 22514  df-cn 22722  df-prmidl 32542  df-mxidl 32564  df-idlsrg 32603  df-rspec 32831

This theorem is referenced by: (None)