qqhcn - Mathbox for Thierry Arnoux

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Theorem qqhcn 34679

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 4137	. . . . . . . 8 ⊢ (NrmRing ∩ DivRing) ⊆ DivRing
2	1	sseli 3909	. . . . . . 7 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) → 𝑅 ∈ DivRing)
3	2	3ad2ant1 1151	. . . . . 6 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑅 ∈ DivRing)
4		simp3 1156	. . . . . 6 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (chr‘𝑅) = 0)
5		eqid 2763	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2763	. . . . . . 7 ⊢ (/_r‘𝑅) = (/_r‘𝑅)
7		eqid 2763	. . . . . . 7 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
8	5, 6, 7	qqhf 34674	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶(Base‘𝑅))
9	3, 4, 8	syl2anc 596	. . . . 5 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶(Base‘𝑅))
10		simpr 490	. . . . . . 7 ⊢ (((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ⁺)
11		qsscn 13100	. . . . . . . . . . . . . 14 ⊢ ℚ ⊆ ℂ
12		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑞 ∈ ℚ)
13	11, 12	sselid 3911	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑞 ∈ ℂ)
14		0cn 11311	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℂ
15		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (abs ∘ − ) = (abs ∘ − )
16	15	cnmetdval 25104	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℂ ∧ 𝑞 ∈ ℂ) → (0(abs ∘ − )𝑞) = (abs‘(0 − 𝑞)))
17	14, 16	mpan 703	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℂ → (0(abs ∘ − )𝑞) = (abs‘(0 − 𝑞)))
18		df-neg 11557	. . . . . . . . . . . . . . . 16 ⊢ -𝑞 = (0 − 𝑞)
19	18	fveq2i 6899	. . . . . . . . . . . . . . 15 ⊢ (abs‘ -𝑞) = (abs‘(0 − 𝑞))
20	19	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℂ → (abs‘ -𝑞) = (abs‘(0 − 𝑞)))
21		absneg 15459	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℂ → (abs‘ -𝑞) = (abs‘𝑞))
22	17, 20, 21	3eqtr2d 2804	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℂ → (0(abs ∘ − )𝑞) = (abs‘𝑞))
23	13, 22	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (0(abs ∘ − )𝑞) = (abs‘𝑞))
24		zssq 13096	. . . . . . . . . . . . . . 15 ⊢ ℤ ⊆ ℚ
25		0z 12717	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℤ
26	24, 25	sselii 3910	. . . . . . . . . . . . . 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 7597	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (0(abs ∘ − )𝑞))
29		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (norm‘𝑅) = (norm‘𝑅)
30		qqhcn.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤMod‘𝑅)
31	29, 30	qqhnm 34678	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑞 ∈ ℚ) → ((norm‘𝑅)‘((ℚHom‘𝑅)‘𝑞)) = (abs‘𝑞))
32	31	adantlr 728	. . . . . . . . . . . 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 2808	. . . . . . . . . . 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 739	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (ℚHom‘𝑅):ℚ⟶(Base‘𝑅))
35	34, 27	ffvelcdmd 7096	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘0) ∈ (Base‘𝑅))
36	34, 12	ffvelcdmd 7096	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑞) ∈ (Base‘𝑅))
37	35, 36	ovresd 7597	. . . . . . . . . . . 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 4136	. . . . . . . . . . . . . . . . 17 ⊢ (NrmRing ∩ DivRing) ⊆ NrmRing
39	38	sseli 3909	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) → 𝑅 ∈ NrmRing)
40	39	3ad2ant1 1151	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑅 ∈ NrmRing)
41	40	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ NrmRing)
42		nrgngp 24996	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
43	41, 42	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ NrmGrp)
44		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
45		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (dist‘𝑅) = (dist‘𝑅)
46	29, 5, 44, 45	ngpdsr 24939	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ DivRing)
49	4	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (chr‘𝑅) = 0)
50	5, 6, 7	qqh0 34672	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘0) = (0_g‘𝑅))
51	48, 49, 50	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘0) = (0_g‘𝑅))
52	51	oveq2d 7446	. . . . . . . . . . . . . 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 24933	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ Grp)
54	43, 53	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ Grp)
55		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
56	5, 55, 44	grpsubid1 19250	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Grp ∧ ((ℚHom‘𝑅)‘𝑞) ∈ (Base‘𝑅)) → (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)(0_g‘𝑅)) = ((ℚHom‘𝑅)‘𝑞))
57	54, 36, 56	syl2anc 596	. . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . 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 6900	. . . . . . . . . . . 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 2802	. . . . . . . . . . 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 2801	. . . . . . . . . 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 5116	. . . . . . . . 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 3158	. . . . . . 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 5110	. . . . . . . 8 ⊢ (𝑑 = 𝑒 → ((0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 ↔ (0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒))
66	65	rspceaimv 3586	. . . . . . 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 596	. . . . . 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 3158	. . . . 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 25110	. . . . . . . . 9 ⊢ ℂ_fld ∈ ∞MetSp
71		qex 13101	. . . . . . . . 9 ⊢ ℚ ∈ V
72		ressxms 24859	. . . . . . . . 9 ⊢ ((ℂ_fld ∈ ∞MetSp ∧ ℚ ∈ V) → (ℂ_fld ↾_s ℚ) ∈ ∞MetSp)
73	70, 71, 72	mp2an 705	. . . . . . . 8 ⊢ (ℂ_fld ↾_s ℚ) ∈ ∞MetSp
74	69, 73	eqeltri 2860	. . . . . . 7 ⊢ 𝑄 ∈ ∞MetSp
75	69	qrngbas 27961	. . . . . . . 8 ⊢ ℚ = (Base‘𝑄)
76		cnfldds 21705	. . . . . . . . . 10 ⊢ (abs ∘ − ) = (dist‘ℂ_fld)
77	69, 76	ressds 17596	. . . . . . . . 9 ⊢ (ℚ ∈ V → (abs ∘ − ) = (dist‘𝑄))
78	71, 77	ax-mp 5	. . . . . . . 8 ⊢ (abs ∘ − ) = (dist‘𝑄)
79	75, 78	xmsxmet2 24793	. . . . . . 7 ⊢ (𝑄 ∈ ∞MetSp → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (∞Met‘ℚ))
80	74, 79	mp1i 14	. . . . . 6 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (∞Met‘ℚ))
81		ngpxms 24935	. . . . . . . . 9 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
82	39, 42, 81	3syl 19	. . . . . . . 8 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) → 𝑅 ∈ ∞MetSp)
83	82	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑅 ∈ ∞MetSp)
84	5, 45	xmsxmet2 24793	. . . . . . 7 ⊢ (𝑅 ∈ ∞MetSp → ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) ∈ (∞Met‘(Base‘𝑅)))
85	83, 84	syl 18	. . . . . 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 5971	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (ℚ × ℚ)) = ((dist‘𝑄) ↾ (ℚ × ℚ))
89	87, 75, 88	xmstopn 24785	. . . . . . . 8 ⊢ (𝑄 ∈ ∞MetSp → 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℚ × ℚ))))
90	74, 89	ax-mp 5	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℚ × ℚ)))
91		eqid 2763	. . . . . . 7 ⊢ (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))
92	90, 91	metcnp 24875	. . . . . 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 1398	. . . . 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 726	. . . 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 2763	. . . . . . . 8 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
97	95, 5, 96	xmstopn 24785	. . . . . . 7 ⊢ (𝑅 ∈ ∞MetSp → 𝐾 = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
98	83, 97	syl 18	. . . . . 6 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝐾 = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))
99	98	oveq2d 7446	. . . . 5 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (𝐽 CnP 𝐾) = (𝐽 CnP (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))))
100	99	fveq1d 6898	. . . 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 2867	. . 3 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ ((𝐽 CnP 𝐾)‘0))
102		cnfldtgp 25205	. . . . . 6 ⊢ ℂ_fld ∈ TopGrp
103		qsubdrg 21740	. . . . . . . 8 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s ℚ) ∈ DivRing)
104	103	simpli 489	. . . . . . 7 ⊢ ℚ ∈ (SubRing‘ℂ_fld)
105		subrgsubg 20844	. . . . . . 7 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) → ℚ ∈ (SubGrp‘ℂ_fld))
106	104, 105	ax-mp 5	. . . . . 6 ⊢ ℚ ∈ (SubGrp‘ℂ_fld)
107	69	subgtgp 24439	. . . . . 6 ⊢ ((ℂ_fld ∈ TopGrp ∧ ℚ ∈ (SubGrp‘ℂ_fld)) → 𝑄 ∈ TopGrp)
108	102, 106, 107	mp2an 705	. . . . 5 ⊢ 𝑄 ∈ TopGrp
109		tgptmd 24413	. . . . 5 ⊢ (𝑄 ∈ TopGrp → 𝑄 ∈ TopMnd)
110	108, 109	mp1i 14	. . . 4 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑄 ∈ TopMnd)
111		nrgtrg 25024	. . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing)
112		trgtmd2 24503	. . . . 5 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopMnd)
113	40, 111, 112	3syl 19	. . . 4 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑅 ∈ TopMnd)
114	5, 6, 7, 69	qqhghm 34676	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
115	3, 4, 114	syl2anc 596	. . . 4 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
116	75, 87, 95	ghmcnp 24449	. . . 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 1398	. . 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 501	1 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2146 ∀wral 3080 ∃wrex 3090 Vcvv 3454 ∩ cin 4092 class class class wbr 5106 × cxp 5653 ↾ cres 5657 ∘ ccom 5659 ⟶wf 6543 ‘cfv 6547 (class class class)co 7430 ℂcc 11211 0cc0 11213 < clt 11356 − cmin 11554 -cneg 11555 ℤcz 12706 ℚcq 13088 ℝ⁺crp 13133 abscabs 15416 Basecbs 17402 ↾_s cress 17423 distcds 17452 TopOpenctopn 17607 0_gc0g 17625 Grpcgrp 19159 -_gcsg 19161 SubGrpcsubg 19345 GrpHom cghm 19442 /_rcdvr 20645 SubRingcsubrg 20836 DivRingcdr 20995 ∞Metcxmet 21678 MetOpencmopn 21683 ℂ_fldccnfld 21693 ℤRHomczrh 21820 ℤModczlm 21821 chrcchr 21822 Cn ccn 23557 CnP ccnp 23558 TopMndctmd 24404 TopGrpctgp 24405 TopRingctrg 24490 ∞MetSpcxms 24651 normcnm 24910 NrmGrpcngp 24911 NrmRingcnrg 24913 NrmModcnlm 24914 ℚHomcqqh 34658

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 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290 ax-pre-sup 11291 ax-addf 11292 ax-mulf 11293

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 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7703 df-om 7888 df-1st 8011 df-2nd 8012 df-supp 8191 df-tpos 8257 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-1o 8490 df-2o 8491 df-er 8731 df-map 8863 df-ixp 8940 df-en 8988 df-dom 8989 df-sdom 8990 df-fin 8991 df-fsupp 9368 df-fi 9416 df-sup 9447 df-inf 9448 df-oi 9517 df-card 10033 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-div 11987 df-nn 12349 df-2 12418 df-3 12419 df-4 12420 df-5 12421 df-6 12422 df-7 12423 df-8 12424 df-9 12425 df-n0 12620 df-z 12707 df-dec 12828 df-uz 12979 df-q 13089 df-rp 13134 df-xneg 13254 df-xadd 13255 df-xmul 13256 df-ico 13495 df-icc 13496 df-fz 13653 df-fzo 13802 df-fl 13945 df-mod 14023 df-seq 14158 df-exp 14218 df-hash 14488 df-cj 15281 df-re 15282 df-im 15283 df-sqrt 15417 df-abs 15418 df-dvds 16438 df-gcd 16680 df-numer 16926 df-denom 16927 df-gz 17123 df-struct 17340 df-sets 17357 df-slot 17375 df-ndx 17387 df-base 17403 df-ress 17424 df-plusg 17456 df-mulr 17457 df-starv 17458 df-sca 17459 df-vsca 17460 df-ip 17461 df-tset 17462 df-ple 17463 df-ds 17465 df-unif 17466 df-hom 17467 df-cco 17468 df-rest 17608 df-topn 17609 df-0g 17627 df-gsum 17628 df-topgen 17629 df-pt 17630 df-prds 17633 df-xrs 17689 df-qtop 17694 df-imas 17695 df-xps 17697 df-mre 17771 df-mrc 17772 df-acs 17774 df-plusf 18830 df-mgm 18831 df-sgrp 18923 df-mnd 18939 df-mhm 18993 df-submnd 18994 df-grp 19162 df-minusg 19163 df-sbg 19164 df-mulg 19293 df-subg 19348 df-ghm 19443 df-cntz 19546 df-od 19757 df-cmn 20011 df-abl 20012 df-mgp 20376 df-rng 20390 df-ur 20423 df-ring 20476 df-cring 20477 df-oppr 20582 df-dvdsr 20602 df-unit 20603 df-invr 20633 df-dvr 20646 df-rhm 20717 df-nzr 20778 df-subrng 20813 df-subrg 20837 df-drng 20997 df-abv 21081 df-lmod 21152 df-scaf 21153 df-sra 21463 df-rgmod 21464 df-psmet 21685 df-xmet 21686 df-met 21687 df-bl 21688 df-mopn 21689 df-cnfld 21694 df-zring 21768 df-zrh 21824 df-zlm 21825 df-chr 21826 df-top 23227 df-topon 23244 df-topsp 23266 df-bases 23279 df-cn 23560 df-cnp 23561 df-tx 23896 df-hmeo 24089 df-tmd 24406 df-tgp 24407 df-trg 24494 df-xms 24654 df-ms 24655 df-tms 24656 df-nm 24916 df-ngp 24917 df-nrg 24919 df-nlm 24920 df-qqh 34659

This theorem is used by: rrhqima 34702

W3C validator