qqhcn - Mathbox for Thierry Arnoux

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Theorem qqhcn 33435

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 4229	. . . . . . . 8 ⊢ (NrmRing ∩ DivRing) ⊆ DivRing
2	1	sseli 3978	. . . . . . 7 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) → 𝑅 ∈ DivRing)
3	2	3ad2ant1 1132	. . . . . 6 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑅 ∈ DivRing)
4		simp3 1137	. . . . . 6 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (chr‘𝑅) = 0)
5		eqid 2731	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2731	. . . . . . 7 ⊢ (/_r‘𝑅) = (/_r‘𝑅)
7		eqid 2731	. . . . . . 7 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
8	5, 6, 7	qqhf 33430	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶(Base‘𝑅))
9	3, 4, 8	syl2anc 583	. . . . 5 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶(Base‘𝑅))
10		simpr 484	. . . . . . 7 ⊢ (((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ⁺)
11		qsscn 12951	. . . . . . . . . . . . . 14 ⊢ ℚ ⊆ ℂ
12		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑞 ∈ ℚ)
13	11, 12	sselid 3980	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑞 ∈ ℂ)
14		0cn 11213	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℂ
15		eqid 2731	. . . . . . . . . . . . . . . 16 ⊢ (abs ∘ − ) = (abs ∘ − )
16	15	cnmetdval 24607	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℂ ∧ 𝑞 ∈ ℂ) → (0(abs ∘ − )𝑞) = (abs‘(0 − 𝑞)))
17	14, 16	mpan 687	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℂ → (0(abs ∘ − )𝑞) = (abs‘(0 − 𝑞)))
18		df-neg 11454	. . . . . . . . . . . . . . . 16 ⊢ -𝑞 = (0 − 𝑞)
19	18	fveq2i 6894	. . . . . . . . . . . . . . 15 ⊢ (abs‘-𝑞) = (abs‘(0 − 𝑞))
20	19	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℂ → (abs‘-𝑞) = (abs‘(0 − 𝑞)))
21		absneg 15231	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℂ → (abs‘-𝑞) = (abs‘𝑞))
22	17, 20, 21	3eqtr2d 2777	. . . . . . . . . . . . 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 12947	. . . . . . . . . . . . . . 15 ⊢ ℤ ⊆ ℚ
25		0z 12576	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℤ
26	24, 25	sselii 3979	. . . . . . . . . . . . . 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 7578	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (0(abs ∘ − )𝑞))
29		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (norm‘𝑅) = (norm‘𝑅)
30		qqhcn.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤMod‘𝑅)
31	29, 30	qqhnm 33434	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑞 ∈ ℚ) → ((norm‘𝑅)‘((ℚHom‘𝑅)‘𝑞)) = (abs‘𝑞))
32	31	adantlr 712	. . . . . . . . . . . 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 2781	. . . . . . . . . . 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 723	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (ℚHom‘𝑅):ℚ⟶(Base‘𝑅))
35	34, 27	ffvelcdmd 7087	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘0) ∈ (Base‘𝑅))
36	34, 12	ffvelcdmd 7087	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑞) ∈ (Base‘𝑅))
37	35, 36	ovresd 7578	. . . . . . . . . . . 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 4228	. . . . . . . . . . . . . . . . 17 ⊢ (NrmRing ∩ DivRing) ⊆ NrmRing
39	38	sseli 3978	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) → 𝑅 ∈ NrmRing)
40	39	3ad2ant1 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑅 ∈ NrmRing)
41	40	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ NrmRing)
42		nrgngp 24499	. . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . 14 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
45		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (dist‘𝑅) = (dist‘𝑅)
46	29, 5, 44, 45	ngpdsr 24434	. . . . . . . . . . . . 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 1370	. . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ DivRing)
49	4	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (chr‘𝑅) = 0)
50	5, 6, 7	qqh0 33428	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘0) = (0_g‘𝑅))
51	48, 49, 50	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘0) = (0_g‘𝑅))
52	51	oveq2d 7428	. . . . . . . . . . . . . 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 24428	. . . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
56	5, 55, 44	grpsubid1 18951	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Grp ∧ ((ℚHom‘𝑅)‘𝑞) ∈ (Base‘𝑅)) → (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)(0_g‘𝑅)) = ((ℚHom‘𝑅)‘𝑞))
57	54, 36, 56	syl2anc 583	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . 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 6895	. . . . . . . . . . . 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 2775	. . . . . . . . . . 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 2774	. . . . . . . . . 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 5158	. . . . . . . . 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 228	. . . . . . . 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 3145	. . . . . . 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 5152	. . . . . . . 8 ⊢ (𝑑 = 𝑒 → ((0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 ↔ (0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒))
66	65	rspceaimv 3617	. . . . . . 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 583	. . . . . 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 3145	. . . . 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 24613	. . . . . . . . 9 ⊢ ℂ_fld ∈ ∞MetSp
71		qex 12952	. . . . . . . . 9 ⊢ ℚ ∈ V
72		ressxms 24354	. . . . . . . . 9 ⊢ ((ℂ_fld ∈ ∞MetSp ∧ ℚ ∈ V) → (ℂ_fld ↾_s ℚ) ∈ ∞MetSp)
73	70, 71, 72	mp2an 689	. . . . . . . 8 ⊢ (ℂ_fld ↾_s ℚ) ∈ ∞MetSp
74	69, 73	eqeltri 2828	. . . . . . 7 ⊢ 𝑄 ∈ ∞MetSp
75	69	qrngbas 27465	. . . . . . . 8 ⊢ ℚ = (Base‘𝑄)
76		cnfldds 21243	. . . . . . . . . 10 ⊢ (abs ∘ − ) = (dist‘ℂ_fld)
77	69, 76	ressds 17362	. . . . . . . . 9 ⊢ (ℚ ∈ V → (abs ∘ − ) = (dist‘𝑄))
78	71, 77	ax-mp 5	. . . . . . . 8 ⊢ (abs ∘ − ) = (dist‘𝑄)
79	75, 78	xmsxmet2 24285	. . . . . . 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 24430	. . . . . . . . 9 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
82	39, 42, 81	3syl 18	. . . . . . . 8 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) → 𝑅 ∈ ∞MetSp)
83	82	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑅 ∈ ∞MetSp)
84	5, 45	xmsxmet2 24285	. . . . . . 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 5977	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (ℚ × ℚ)) = ((dist‘𝑄) ↾ (ℚ × ℚ))
89	87, 75, 88	xmstopn 24277	. . . . . . . 8 ⊢ (𝑄 ∈ ∞MetSp → 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℚ × ℚ))))
90	74, 89	ax-mp 5	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℚ × ℚ)))
91		eqid 2731	. . . . . . 7 ⊢ (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))
92	90, 91	metcnp 24370	. . . . . 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 1370	. . . . 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 710	. . . 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 2731	. . . . . . . 8 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
97	95, 5, 96	xmstopn 24277	. . . . . . 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 7428	. . . . 5 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (𝐽 CnP 𝐾) = (𝐽 CnP (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))))
100	99	fveq1d 6893	. . . 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 2835	. . 3 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ ((𝐽 CnP 𝐾)‘0))
102		cnfldtgp 24707	. . . . . 6 ⊢ ℂ_fld ∈ TopGrp
103		qsubdrg 21286	. . . . . . . 8 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s ℚ) ∈ DivRing)
104	103	simpli 483	. . . . . . 7 ⊢ ℚ ∈ (SubRing‘ℂ_fld)
105		subrgsubg 20475	. . . . . . 7 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) → ℚ ∈ (SubGrp‘ℂ_fld))
106	104, 105	ax-mp 5	. . . . . 6 ⊢ ℚ ∈ (SubGrp‘ℂ_fld)
107	69	subgtgp 23929	. . . . . 6 ⊢ ((ℂ_fld ∈ TopGrp ∧ ℚ ∈ (SubGrp‘ℂ_fld)) → 𝑄 ∈ TopGrp)
108	102, 106, 107	mp2an 689	. . . . 5 ⊢ 𝑄 ∈ TopGrp
109		tgptmd 23903	. . . . 5 ⊢ (𝑄 ∈ TopGrp → 𝑄 ∈ TopMnd)
110	108, 109	mp1i 13	. . . 4 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑄 ∈ TopMnd)
111		nrgtrg 24527	. . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing)
112		trgtmd2 23993	. . . . 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 33432	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
115	3, 4, 114	syl2anc 583	. . . 4 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
116	75, 87, 95	ghmcnp 23939	. . . 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 1370	. . 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 231	. 2 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (0 ∈ ℚ ∧ (ℚHom‘𝑅) ∈ (𝐽 Cn 𝐾)))
119	118	simprd 495	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 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∃wrex 3069 Vcvv 3473 ∩ cin 3947 class class class wbr 5148 × cxp 5674 ↾ cres 5678 ∘ ccom 5680 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 0cc0 11116 < clt 11255 − cmin 11451 -cneg 11452 ℤcz 12565 ℚcq 12939 ℝ⁺crp 12981 abscabs 15188 Basecbs 17151 ↾_s cress 17180 distcds 17213 TopOpenctopn 17374 0_gc0g 17392 Grpcgrp 18861 -_gcsg 18863 SubGrpcsubg 19043 GrpHom cghm 19134 /_rcdvr 20298 SubRingcsubrg 20465 DivRingcdr 20583 ∞Metcxmet 21218 MetOpencmopn 21223 ℂ_fldccnfld 21233 ℤRHomczrh 21359 ℤModczlm 21360 chrcchr 21361 Cn ccn 23048 CnP ccnp 23049 TopMndctmd 23894 TopGrpctgp 23895 TopRingctrg 23980 ∞MetSpcxms 24143 normcnm 24405 NrmGrpcngp 24406 NrmRingcnrg 24408 NrmModcnlm 24409 ℚHomcqqh 33416

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-fi 9412 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ico 13337 df-icc 13338 df-fz 13492 df-fzo 13635 df-fl 13764 df-mod 13842 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-dvds 16205 df-gcd 16443 df-numer 16678 df-denom 16679 df-gz 16870 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-plusf 18570 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-subg 19046 df-ghm 19135 df-cntz 19229 df-od 19444 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-rhm 20370 df-nzr 20411 df-subrng 20442 df-subrg 20467 df-drng 20585 df-abv 20656 df-lmod 20704 df-scaf 20705 df-sra 21019 df-rgmod 21020 df-psmet 21225 df-xmet 21226 df-met 21227 df-bl 21228 df-mopn 21229 df-cnfld 21234 df-zring 21307 df-zrh 21363 df-zlm 21364 df-chr 21365 df-top 22716 df-topon 22733 df-topsp 22755 df-bases 22769 df-cn 23051 df-cnp 23052 df-tx 23386 df-hmeo 23579 df-tmd 23896 df-tgp 23897 df-trg 23984 df-xms 24146 df-ms 24147 df-tms 24148 df-nm 24411 df-ngp 24412 df-nrg 24414 df-nlm 24415 df-qqh 33417

This theorem is referenced by: rrhqima 33458

W3C validator