qqhcn - Mathbox for Thierry Arnoux

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Theorem qqhcn 31342

Description: The ℚHom homomorphism is a continuous function. (Contributed by Thierry Arnoux, 9-Nov-2017.)

**Hypotheses**
Ref	Expression
qqhcn.q	⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
qqhcn.j	⊢ 𝐽 = (TopOpen‘𝑄)
qqhcn.z	⊢ 𝑍 = (ℤMod‘𝑅)
qqhcn.k	⊢ 𝐾 = (TopOpen‘𝑅)

**Assertion**
Ref	Expression
qqhcn	⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝐽 Cn 𝐾))

Proof of Theorem qqhcn Dummy variables 𝑒 𝑑 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 4156	. . . . . . . 8 ⊢ (NrmRing ∩ DivRing) ⊆ DivRing
2	1	sseli 3911	. . . . . . 7 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) → 𝑅 ∈ DivRing)
3	2	3ad2ant1 1130	. . . . . 6 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑅 ∈ DivRing)
4		simp3 1135	. . . . . 6 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (chr‘𝑅) = 0)
5		eqid 2798	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2798	. . . . . . 7 ⊢ (/_r‘𝑅) = (/_r‘𝑅)
7		eqid 2798	. . . . . . 7 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
8	5, 6, 7	qqhf 31337	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶(Base‘𝑅))
9	3, 4, 8	syl2anc 587	. . . . 5 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶(Base‘𝑅))
10		simpr 488	. . . . . . 7 ⊢ (((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ⁺)
11		qsscn 12347	. . . . . . . . . . . . . 14 ⊢ ℚ ⊆ ℂ
12		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑞 ∈ ℚ)
13	11, 12	sseldi 3913	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑞 ∈ ℂ)
14		0cn 10622	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℂ
15		eqid 2798	. . . . . . . . . . . . . . . 16 ⊢ (abs ∘ − ) = (abs ∘ − )
16	15	cnmetdval 23376	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℂ ∧ 𝑞 ∈ ℂ) → (0(abs ∘ − )𝑞) = (abs‘(0 − 𝑞)))
17	14, 16	mpan 689	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℂ → (0(abs ∘ − )𝑞) = (abs‘(0 − 𝑞)))
18		df-neg 10862	. . . . . . . . . . . . . . . 16 ⊢ -𝑞 = (0 − 𝑞)
19	18	fveq2i 6648	. . . . . . . . . . . . . . 15 ⊢ (abs‘-𝑞) = (abs‘(0 − 𝑞))
20	19	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℂ → (abs‘-𝑞) = (abs‘(0 − 𝑞)))
21		absneg 14629	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℂ → (abs‘-𝑞) = (abs‘𝑞))
22	17, 20, 21	3eqtr2d 2839	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℂ → (0(abs ∘ − )𝑞) = (abs‘𝑞))
23	13, 22	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (0(abs ∘ − )𝑞) = (abs‘𝑞))
24		zssq 12343	. . . . . . . . . . . . . . 15 ⊢ ℤ ⊆ ℚ
25		0z 11980	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℤ
26	24, 25	sselii 3912	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℚ
27	26	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 0 ∈ ℚ)
28	27, 12	ovresd 7295	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (0(abs ∘ − )𝑞))
29		eqid 2798	. . . . . . . . . . . . . 14 ⊢ (norm‘𝑅) = (norm‘𝑅)
30		qqhcn.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤMod‘𝑅)
31	29, 30	qqhnm 31341	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑞 ∈ ℚ) → ((norm‘𝑅)‘((ℚHom‘𝑅)‘𝑞)) = (abs‘𝑞))
32	31	adantlr 714	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((norm‘𝑅)‘((ℚHom‘𝑅)‘𝑞)) = (abs‘𝑞))
33	23, 28, 32	3eqtr4d 2843	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = ((norm‘𝑅)‘((ℚHom‘𝑅)‘𝑞)))
34	9	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (ℚHom‘𝑅):ℚ⟶(Base‘𝑅))
35	34, 27	ffvelrnd 6829	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘0) ∈ (Base‘𝑅))
36	34, 12	ffvelrnd 6829	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑞) ∈ (Base‘𝑅))
37	35, 36	ovresd 7295	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘0)((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))((ℚHom‘𝑅)‘𝑞)) = (((ℚHom‘𝑅)‘0)(dist‘𝑅)((ℚHom‘𝑅)‘𝑞)))
38		inss1 4155	. . . . . . . . . . . . . . . . 17 ⊢ (NrmRing ∩ DivRing) ⊆ NrmRing
39	38	sseli 3911	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) → 𝑅 ∈ NrmRing)
40	39	3ad2ant1 1130	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑅 ∈ NrmRing)
41	40	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ NrmRing)
42		nrgngp 23268	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
43	41, 42	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ NrmGrp)
44		eqid 2798	. . . . . . . . . . . . . 14 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
45		eqid 2798	. . . . . . . . . . . . . 14 ⊢ (dist‘𝑅) = (dist‘𝑅)
46	29, 5, 44, 45	ngpdsr 23211	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ NrmGrp ∧ ((ℚHom‘𝑅)‘0) ∈ (Base‘𝑅) ∧ ((ℚHom‘𝑅)‘𝑞) ∈ (Base‘𝑅)) → (((ℚHom‘𝑅)‘0)(dist‘𝑅)((ℚHom‘𝑅)‘𝑞)) = ((norm‘𝑅)‘(((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘0))))
47	43, 35, 36, 46	syl3anc 1368	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘0)(dist‘𝑅)((ℚHom‘𝑅)‘𝑞)) = ((norm‘𝑅)‘(((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘0))))
48	3	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ DivRing)
49	4	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (chr‘𝑅) = 0)
50	5, 6, 7	qqh0 31335	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘0) = (0_g‘𝑅))
51	48, 49, 50	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘0) = (0_g‘𝑅))
52	51	oveq2d 7151	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘0)) = (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)(0_g‘𝑅)))
53		ngpgrp 23205	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ Grp)
54	43, 53	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ Grp)
55		eqid 2798	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
56	5, 55, 44	grpsubid1 18176	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Grp ∧ ((ℚHom‘𝑅)‘𝑞) ∈ (Base‘𝑅)) → (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)(0_g‘𝑅)) = ((ℚHom‘𝑅)‘𝑞))
57	54, 36, 56	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)(0_g‘𝑅)) = ((ℚHom‘𝑅)‘𝑞))
58	52, 57	eqtrd 2833	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘0)) = ((ℚHom‘𝑅)‘𝑞))
59	58	fveq2d 6649	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((norm‘𝑅)‘(((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘0))) = ((norm‘𝑅)‘((ℚHom‘𝑅)‘𝑞)))
60	37, 47, 59	3eqtrd 2837	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘0)((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))((ℚHom‘𝑅)‘𝑞)) = ((norm‘𝑅)‘((ℚHom‘𝑅)‘𝑞)))
61	33, 60	eqtr4d 2836	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (((ℚHom‘𝑅)‘0)((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))((ℚHom‘𝑅)‘𝑞)))
62	61	breq1d 5040	. . . . . . . . 9 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 ↔ (((ℚHom‘𝑅)‘0)((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
63	62	biimpd 232	. . . . . . . 8 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 → (((ℚHom‘𝑅)‘0)((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
64	63	ralrimiva 3149	. . . . . . 7 ⊢ (((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) → ∀𝑞 ∈ ℚ ((0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 → (((ℚHom‘𝑅)‘0)((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
65		breq2 5034	. . . . . . . 8 ⊢ (𝑑 = 𝑒 → ((0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 ↔ (0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒))
66	65	rspceaimv 3576	. . . . . . 7 ⊢ ((𝑒 ∈ ℝ⁺ ∧ ∀𝑞 ∈ ℚ ((0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 → (((ℚHom‘𝑅)‘0)((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))((ℚHom‘𝑅)‘𝑞)) < 𝑒)) → ∃𝑑 ∈ ℝ⁺ ∀𝑞 ∈ ℚ ((0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 → (((ℚHom‘𝑅)‘0)((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
67	10, 64, 66	syl2anc 587	. . . . . 6 ⊢ (((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) → ∃𝑑 ∈ ℝ⁺ ∀𝑞 ∈ ℚ ((0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 → (((ℚHom‘𝑅)‘0)((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
68	67	ralrimiva 3149	. . . . 5 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑞 ∈ ℚ ((0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 → (((ℚHom‘𝑅)‘0)((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
69		qqhcn.q	. . . . . . . 8 ⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
70		cnfldxms 23382	. . . . . . . . 9 ⊢ ℂ_fld ∈ ∞MetSp
71		qex 12348	. . . . . . . . 9 ⊢ ℚ ∈ V
72		ressxms 23132	. . . . . . . . 9 ⊢ ((ℂ_fld ∈ ∞MetSp ∧ ℚ ∈ V) → (ℂ_fld ↾_s ℚ) ∈ ∞MetSp)
73	70, 71, 72	mp2an 691	. . . . . . . 8 ⊢ (ℂ_fld ↾_s ℚ) ∈ ∞MetSp
74	69, 73	eqeltri 2886	. . . . . . 7 ⊢ 𝑄 ∈ ∞MetSp
75	69	qrngbas 26203	. . . . . . . 8 ⊢ ℚ = (Base‘𝑄)
76		cnfldds 20101	. . . . . . . . . 10 ⊢ (abs ∘ − ) = (dist‘ℂ_fld)
77	69, 76	ressds 16678	. . . . . . . . 9 ⊢ (ℚ ∈ V → (abs ∘ − ) = (dist‘𝑄))
78	71, 77	ax-mp 5	. . . . . . . 8 ⊢ (abs ∘ − ) = (dist‘𝑄)
79	75, 78	xmsxmet2 23066	. . . . . . 7 ⊢ (𝑄 ∈ ∞MetSp → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (∞Met‘ℚ))
80	74, 79	mp1i 13	. . . . . 6 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (∞Met‘ℚ))
81		ngpxms 23207	. . . . . . . . 9 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
82	39, 42, 81	3syl 18	. . . . . . . 8 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) → 𝑅 ∈ ∞MetSp)
83	82	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑅 ∈ ∞MetSp)
84	5, 45	xmsxmet2 23066	. . . . . . 7 ⊢ (𝑅 ∈ ∞MetSp → ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) ∈ (∞Met‘(Base‘𝑅)))
85	83, 84	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) ∈ (∞Met‘(Base‘𝑅)))
86	26	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 0 ∈ ℚ)
87		qqhcn.j	. . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘𝑄)
88	78	reseq1i 5814	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (ℚ × ℚ)) = ((dist‘𝑄) ↾ (ℚ × ℚ))
89	87, 75, 88	xmstopn 23058	. . . . . . . 8 ⊢ (𝑄 ∈ ∞MetSp → 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℚ × ℚ))))
90	74, 89	ax-mp 5	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℚ × ℚ)))
91		eqid 2798	. . . . . . 7 ⊢ (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))
92	90, 91	metcnp 23148	. . . . . 6 ⊢ ((((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (∞Met‘ℚ) ∧ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) ∈ (∞Met‘(Base‘𝑅)) ∧ 0 ∈ ℚ) → ((ℚHom‘𝑅) ∈ ((𝐽 CnP (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))‘0) ↔ ((ℚHom‘𝑅):ℚ⟶(Base‘𝑅) ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑞 ∈ ℚ ((0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 → (((ℚHom‘𝑅)‘0)((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))((ℚHom‘𝑅)‘𝑞)) < 𝑒))))
93	80, 85, 86, 92	syl3anc 1368	. . . . 5 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅) ∈ ((𝐽 CnP (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))‘0) ↔ ((ℚHom‘𝑅):ℚ⟶(Base‘𝑅) ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑞 ∈ ℚ ((0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 → (((ℚHom‘𝑅)‘0)((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))((ℚHom‘𝑅)‘𝑞)) < 𝑒))))
94	9, 68, 93	mpbir2and 712	. . . 4 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ ((𝐽 CnP (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))‘0))
95		qqhcn.k	. . . . . . . 8 ⊢ 𝐾 = (TopOpen‘𝑅)
96		eqid 2798	. . . . . . . 8 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
97	95, 5, 96	xmstopn 23058	. . . . . . 7 ⊢ (𝑅 ∈ ∞MetSp → 𝐾 = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
98	83, 97	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝐾 = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
99	98	oveq2d 7151	. . . . 5 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (𝐽 CnP 𝐾) = (𝐽 CnP (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))))
100	99	fveq1d 6647	. . . 4 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → ((𝐽 CnP 𝐾)‘0) = ((𝐽 CnP (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))‘0))
101	94, 100	eleqtrrd 2893	. . 3 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ ((𝐽 CnP 𝐾)‘0))
102		cnfldtgp 23474	. . . . . 6 ⊢ ℂ_fld ∈ TopGrp
103		qsubdrg 20143	. . . . . . . 8 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s ℚ) ∈ DivRing)
104	103	simpli 487	. . . . . . 7 ⊢ ℚ ∈ (SubRing‘ℂ_fld)
105		subrgsubg 19534	. . . . . . 7 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) → ℚ ∈ (SubGrp‘ℂ_fld))
106	104, 105	ax-mp 5	. . . . . 6 ⊢ ℚ ∈ (SubGrp‘ℂ_fld)
107	69	subgtgp 22710	. . . . . 6 ⊢ ((ℂ_fld ∈ TopGrp ∧ ℚ ∈ (SubGrp‘ℂ_fld)) → 𝑄 ∈ TopGrp)
108	102, 106, 107	mp2an 691	. . . . 5 ⊢ 𝑄 ∈ TopGrp
109		tgptmd 22684	. . . . 5 ⊢ (𝑄 ∈ TopGrp → 𝑄 ∈ TopMnd)
110	108, 109	mp1i 13	. . . 4 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑄 ∈ TopMnd)
111		nrgtrg 23296	. . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing)
112		trgtmd2 22774	. . . . 5 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopMnd)
113	40, 111, 112	3syl 18	. . . 4 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑅 ∈ TopMnd)
114	5, 6, 7, 69	qqhghm 31339	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
115	3, 4, 114	syl2anc 587	. . . 4 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
116	75, 87, 95	ghmcnp 22720	. . . 4 ⊢ ((𝑄 ∈ TopMnd ∧ 𝑅 ∈ TopMnd ∧ (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅)) → ((ℚHom‘𝑅) ∈ ((𝐽 CnP 𝐾)‘0) ↔ (0 ∈ ℚ ∧ (ℚHom‘𝑅) ∈ (𝐽 Cn 𝐾))))
117	110, 113, 115, 116	syl3anc 1368	. . 3 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅) ∈ ((𝐽 CnP 𝐾)‘0) ↔ (0 ∈ ℚ ∧ (ℚHom‘𝑅) ∈ (𝐽 Cn 𝐾))))
118	101, 117	mpbid 235	. 2 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (0 ∈ ℚ ∧ (ℚHom‘𝑅) ∈ (𝐽 Cn 𝐾)))
119	118	simprd 499	1 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ∩ cin 3880 class class class wbr 5030 × cxp 5517 ↾ cres 5521 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 < clt 10664 − cmin 10859 -cneg 10860 ℤcz 11969 ℚcq 12336 ℝ⁺crp 12377 abscabs 14585 Basecbs 16475 ↾_s cress 16476 distcds 16566 TopOpenctopn 16687 0_gc0g 16705 Grpcgrp 18095 -_gcsg 18097 SubGrpcsubg 18265 GrpHom cghm 18347 /_rcdvr 19428 DivRingcdr 19495 SubRingcsubrg 19524 ∞Metcxmet 20076 MetOpencmopn 20081 ℂ_fldccnfld 20091 ℤRHomczrh 20193 ℤModczlm 20194 chrcchr 20195 Cn ccn 21829 CnP ccnp 21830 TopMndctmd 22675 TopGrpctgp 22676 TopRingctrg 22761 ∞MetSpcxms 22924 normcnm 23183 NrmGrpcngp 23184 NrmRingcnrg 23186 NrmModcnlm 23187 ℚHomcqqh 31323

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 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

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 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-gcd 15834 df-numer 16065 df-denom 16066 df-gz 16256 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-plusf 17843 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-cntz 18439 df-od 18648 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-rnghom 19463 df-drng 19497 df-subrg 19526 df-abv 19581 df-lmod 19629 df-scaf 19630 df-sra 19937 df-rgmod 19938 df-nzr 20024 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-zring 20164 df-zrh 20197 df-zlm 20198 df-chr 20199 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cn 21832 df-cnp 21833 df-tx 22167 df-hmeo 22360 df-tmd 22677 df-tgp 22678 df-trg 22765 df-xms 22927 df-ms 22928 df-tms 22929 df-nm 23189 df-ngp 23190 df-nrg 23192 df-nlm 23193 df-qqh 31324

This theorem is referenced by: rrhqima 31365

W3C validator