qqhcn - Mathbox for Thierry Arnoux

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Theorem qqhcn 33992

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 4238	. . . . . . . 8 ⊢ (NrmRing ∩ DivRing) ⊆ DivRing
2	1	sseli 3979	. . . . . . 7 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) → 𝑅 ∈ DivRing)
3	2	3ad2ant1 1134	. . . . . 6 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑅 ∈ DivRing)
4		simp3 1139	. . . . . 6 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (chr‘𝑅) = 0)
5		eqid 2737	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2737	. . . . . . 7 ⊢ (/_r‘𝑅) = (/_r‘𝑅)
7		eqid 2737	. . . . . . 7 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
8	5, 6, 7	qqhf 33987	. . . . . 6 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶(Base‘𝑅))
9	3, 4, 8	syl2anc 584	. . . . 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 13002	. . . . . . . . . . . . . 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 3981	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑞 ∈ ℂ)
14		0cn 11253	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℂ
15		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (abs ∘ − ) = (abs ∘ − )
16	15	cnmetdval 24791	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℂ ∧ 𝑞 ∈ ℂ) → (0(abs ∘ − )𝑞) = (abs‘(0 − 𝑞)))
17	14, 16	mpan 690	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℂ → (0(abs ∘ − )𝑞) = (abs‘(0 − 𝑞)))
18		df-neg 11495	. . . . . . . . . . . . . . . 16 ⊢ -𝑞 = (0 − 𝑞)
19	18	fveq2i 6909	. . . . . . . . . . . . . . 15 ⊢ (abs‘-𝑞) = (abs‘(0 − 𝑞))
20	19	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℂ → (abs‘-𝑞) = (abs‘(0 − 𝑞)))
21		absneg 15316	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℂ → (abs‘-𝑞) = (abs‘𝑞))
22	17, 20, 21	3eqtr2d 2783	. . . . . . . . . . . . 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 12998	. . . . . . . . . . . . . . 15 ⊢ ℤ ⊆ ℚ
25		0z 12624	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℤ
26	24, 25	sselii 3980	. . . . . . . . . . . . . 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 7600	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (0(abs ∘ − )𝑞))
29		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (norm‘𝑅) = (norm‘𝑅)
30		qqhcn.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤMod‘𝑅)
31	29, 30	qqhnm 33991	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑞 ∈ ℚ) → ((norm‘𝑅)‘((ℚHom‘𝑅)‘𝑞)) = (abs‘𝑞))
32	31	adantlr 715	. . . . . . . . . . . 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 2787	. . . . . . . . . . 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 726	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (ℚHom‘𝑅):ℚ⟶(Base‘𝑅))
35	34, 27	ffvelcdmd 7105	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘0) ∈ (Base‘𝑅))
36	34, 12	ffvelcdmd 7105	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑞) ∈ (Base‘𝑅))
37	35, 36	ovresd 7600	. . . . . . . . . . . 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 4237	. . . . . . . . . . . . . . . . 17 ⊢ (NrmRing ∩ DivRing) ⊆ NrmRing
39	38	sseli 3979	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) → 𝑅 ∈ NrmRing)
40	39	3ad2ant1 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑅 ∈ NrmRing)
41	40	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ NrmRing)
42		nrgngp 24683	. . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
45		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (dist‘𝑅) = (dist‘𝑅)
46	29, 5, 44, 45	ngpdsr 24618	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ DivRing)
49	4	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (chr‘𝑅) = 0)
50	5, 6, 7	qqh0 33985	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘0) = (0_g‘𝑅))
51	48, 49, 50	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘0) = (0_g‘𝑅))
52	51	oveq2d 7447	. . . . . . . . . . . . . 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 24612	. . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
56	5, 55, 44	grpsubid1 19043	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Grp ∧ ((ℚHom‘𝑅)‘𝑞) ∈ (Base‘𝑅)) → (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)(0_g‘𝑅)) = ((ℚHom‘𝑅)‘𝑞))
57	54, 36, 56	syl2anc 584	. . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . 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 6910	. . . . . . . . . . . 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 2781	. . . . . . . . . . 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 2780	. . . . . . . . . 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 5153	. . . . . . . . 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 229	. . . . . . . 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 3146	. . . . . . 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 5147	. . . . . . . 8 ⊢ (𝑑 = 𝑒 → ((0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 ↔ (0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒))
66	65	rspceaimv 3628	. . . . . . 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 584	. . . . . 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 3146	. . . . 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 24797	. . . . . . . . 9 ⊢ ℂ_fld ∈ ∞MetSp
71		qex 13003	. . . . . . . . 9 ⊢ ℚ ∈ V
72		ressxms 24538	. . . . . . . . 9 ⊢ ((ℂ_fld ∈ ∞MetSp ∧ ℚ ∈ V) → (ℂ_fld ↾_s ℚ) ∈ ∞MetSp)
73	70, 71, 72	mp2an 692	. . . . . . . 8 ⊢ (ℂ_fld ↾_s ℚ) ∈ ∞MetSp
74	69, 73	eqeltri 2837	. . . . . . 7 ⊢ 𝑄 ∈ ∞MetSp
75	69	qrngbas 27663	. . . . . . . 8 ⊢ ℚ = (Base‘𝑄)
76		cnfldds 21376	. . . . . . . . . 10 ⊢ (abs ∘ − ) = (dist‘ℂ_fld)
77	69, 76	ressds 17454	. . . . . . . . 9 ⊢ (ℚ ∈ V → (abs ∘ − ) = (dist‘𝑄))
78	71, 77	ax-mp 5	. . . . . . . 8 ⊢ (abs ∘ − ) = (dist‘𝑄)
79	75, 78	xmsxmet2 24469	. . . . . . 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 24614	. . . . . . . . 9 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
82	39, 42, 81	3syl 18	. . . . . . . 8 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) → 𝑅 ∈ ∞MetSp)
83	82	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑅 ∈ ∞MetSp)
84	5, 45	xmsxmet2 24469	. . . . . . 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 5993	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (ℚ × ℚ)) = ((dist‘𝑄) ↾ (ℚ × ℚ))
89	87, 75, 88	xmstopn 24461	. . . . . . . 8 ⊢ (𝑄 ∈ ∞MetSp → 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℚ × ℚ))))
90	74, 89	ax-mp 5	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℚ × ℚ)))
91		eqid 2737	. . . . . . 7 ⊢ (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))
92	90, 91	metcnp 24554	. . . . . 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 1373	. . . . 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 713	. . . 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 2737	. . . . . . . 8 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
97	95, 5, 96	xmstopn 24461	. . . . . . 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 7447	. . . . 5 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (𝐽 CnP 𝐾) = (𝐽 CnP (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))))
100	99	fveq1d 6908	. . . 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 2844	. . 3 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ ((𝐽 CnP 𝐾)‘0))
102		cnfldtgp 24893	. . . . . 6 ⊢ ℂ_fld ∈ TopGrp
103		qsubdrg 21437	. . . . . . . 8 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s ℚ) ∈ DivRing)
104	103	simpli 483	. . . . . . 7 ⊢ ℚ ∈ (SubRing‘ℂ_fld)
105		subrgsubg 20577	. . . . . . 7 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) → ℚ ∈ (SubGrp‘ℂ_fld))
106	104, 105	ax-mp 5	. . . . . 6 ⊢ ℚ ∈ (SubGrp‘ℂ_fld)
107	69	subgtgp 24113	. . . . . 6 ⊢ ((ℂ_fld ∈ TopGrp ∧ ℚ ∈ (SubGrp‘ℂ_fld)) → 𝑄 ∈ TopGrp)
108	102, 106, 107	mp2an 692	. . . . 5 ⊢ 𝑄 ∈ TopGrp
109		tgptmd 24087	. . . . 5 ⊢ (𝑄 ∈ TopGrp → 𝑄 ∈ TopMnd)
110	108, 109	mp1i 13	. . . 4 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑄 ∈ TopMnd)
111		nrgtrg 24711	. . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing)
112		trgtmd2 24177	. . . . 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 33989	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
115	3, 4, 114	syl2anc 584	. . . 4 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
116	75, 87, 95	ghmcnp 24123	. . . 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 1373	. . 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 232	. 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ∩ cin 3950 class class class wbr 5143 × cxp 5683 ↾ cres 5687 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 < clt 11295 − cmin 11492 -cneg 11493 ℤcz 12613 ℚcq 12990 ℝ⁺crp 13034 abscabs 15273 Basecbs 17247 ↾_s cress 17274 distcds 17306 TopOpenctopn 17466 0_gc0g 17484 Grpcgrp 18951 -_gcsg 18953 SubGrpcsubg 19138 GrpHom cghm 19230 /_rcdvr 20400 SubRingcsubrg 20569 DivRingcdr 20729 ∞Metcxmet 21349 MetOpencmopn 21354 ℂ_fldccnfld 21364 ℤRHomczrh 21510 ℤModczlm 21511 chrcchr 21512 Cn ccn 23232 CnP ccnp 23233 TopMndctmd 24078 TopGrpctgp 24079 TopRingctrg 24164 ∞MetSpcxms 24327 normcnm 24589 NrmGrpcngp 24590 NrmRingcnrg 24592 NrmModcnlm 24593 ℚHomcqqh 33971

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-numer 16772 df-denom 16773 df-gz 16968 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-plusf 18652 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-od 19546 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-nzr 20513 df-subrng 20546 df-subrg 20570 df-drng 20731 df-abv 20810 df-lmod 20860 df-scaf 20861 df-sra 21172 df-rgmod 21173 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-zring 21458 df-zrh 21514 df-zlm 21515 df-chr 21516 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cn 23235 df-cnp 23236 df-tx 23570 df-hmeo 23763 df-tmd 24080 df-tgp 24081 df-trg 24168 df-xms 24330 df-ms 24331 df-tms 24332 df-nm 24595 df-ngp 24596 df-nrg 24598 df-nlm 24599 df-qqh 33972

This theorem is referenced by: rrhqima 34015

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