qqhcn - Mathbox for Thierry Arnoux

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Theorem qqhcn 33937

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 4259	. . . . . . . 8 ⊢ (NrmRing ∩ DivRing) ⊆ DivRing
2	1	sseli 4004	. . . . . . 7 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) → 𝑅 ∈ DivRing)
3	2	3ad2ant1 1133	. . . . . 6 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑅 ∈ DivRing)
4		simp3 1138	. . . . . 6 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (chr‘𝑅) = 0)
5		eqid 2740	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2740	. . . . . . 7 ⊢ (/_r‘𝑅) = (/_r‘𝑅)
7		eqid 2740	. . . . . . 7 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
8	5, 6, 7	qqhf 33932	. . . . . 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 13025	. . . . . . . . . . . . . 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 4006	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → 𝑞 ∈ ℂ)
14		0cn 11282	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℂ
15		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (abs ∘ − ) = (abs ∘ − )
16	15	cnmetdval 24812	. . . . . . . . . . . . . . 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 11523	. . . . . . . . . . . . . . . 16 ⊢ -𝑞 = (0 − 𝑞)
19	18	fveq2i 6923	. . . . . . . . . . . . . . 15 ⊢ (abs‘-𝑞) = (abs‘(0 − 𝑞))
20	19	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℂ → (abs‘-𝑞) = (abs‘(0 − 𝑞)))
21		absneg 15326	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ ℂ → (abs‘-𝑞) = (abs‘𝑞))
22	17, 20, 21	3eqtr2d 2786	. . . . . . . . . . . . 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 13021	. . . . . . . . . . . . . . 15 ⊢ ℤ ⊆ ℚ
25		0z 12650	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℤ
26	24, 25	sselii 4005	. . . . . . . . . . . . . 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 7617	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → (0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (0(abs ∘ − )𝑞))
29		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (norm‘𝑅) = (norm‘𝑅)
30		qqhcn.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤMod‘𝑅)
31	29, 30	qqhnm 33936	. . . . . . . . . . . . 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 2790	. . . . . . . . . . 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	ffvelcdmd 7119	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘0) ∈ (Base‘𝑅))
36	34, 12	ffvelcdmd 7119	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑞) ∈ (Base‘𝑅))
37	35, 36	ovresd 7617	. . . . . . . . . . . 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 4258	. . . . . . . . . . . . . . . . 17 ⊢ (NrmRing ∩ DivRing) ⊆ NrmRing
39	38	sseli 4004	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) → 𝑅 ∈ NrmRing)
40	39	3ad2ant1 1133	. . . . . . . . . . . . . . 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 24704	. . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . 14 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
45		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (dist‘𝑅) = (dist‘𝑅)
46	29, 5, 44, 45	ngpdsr 24639	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 33930	. . . . . . . . . . . . . . . 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 7464	. . . . . . . . . . . . . 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 24633	. . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . 16 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
56	5, 55, 44	grpsubid1 19065	. . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . 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 6924	. . . . . . . . . . . 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 2784	. . . . . . . . . . 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 2783	. . . . . . . . . 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 5176	. . . . . . . . 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 3152	. . . . . . 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 5170	. . . . . . . 8 ⊢ (𝑑 = 𝑒 → ((0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 ↔ (0((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒))
66	65	rspceaimv 3641	. . . . . . 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 3152	. . . . 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 24818	. . . . . . . . 9 ⊢ ℂ_fld ∈ ∞MetSp
71		qex 13026	. . . . . . . . 9 ⊢ ℚ ∈ V
72		ressxms 24559	. . . . . . . . 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 2840	. . . . . . 7 ⊢ 𝑄 ∈ ∞MetSp
75	69	qrngbas 27681	. . . . . . . 8 ⊢ ℚ = (Base‘𝑄)
76		cnfldds 21399	. . . . . . . . . 10 ⊢ (abs ∘ − ) = (dist‘ℂ_fld)
77	69, 76	ressds 17469	. . . . . . . . 9 ⊢ (ℚ ∈ V → (abs ∘ − ) = (dist‘𝑄))
78	71, 77	ax-mp 5	. . . . . . . 8 ⊢ (abs ∘ − ) = (dist‘𝑄)
79	75, 78	xmsxmet2 24490	. . . . . . 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 24635	. . . . . . . . 9 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
82	39, 42, 81	3syl 18	. . . . . . . 8 ⊢ (𝑅 ∈ (NrmRing ∩ DivRing) → 𝑅 ∈ ∞MetSp)
83	82	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑅 ∈ ∞MetSp)
84	5, 45	xmsxmet2 24490	. . . . . . 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 6005	. . . . . . . . 9 ⊢ ((abs ∘ − ) ↾ (ℚ × ℚ)) = ((dist‘𝑄) ↾ (ℚ × ℚ))
89	87, 75, 88	xmstopn 24482	. . . . . . . 8 ⊢ (𝑄 ∈ ∞MetSp → 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℚ × ℚ))))
90	74, 89	ax-mp 5	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (ℚ × ℚ)))
91		eqid 2740	. . . . . . 7 ⊢ (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))) = (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))
92	90, 91	metcnp 24575	. . . . . 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 1371	. . . . 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 2740	. . . . . . . 8 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
97	95, 5, 96	xmstopn 24482	. . . . . . 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 7464	. . . . 5 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (𝐽 CnP 𝐾) = (𝐽 CnP (MetOpen‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))))
100	99	fveq1d 6922	. . . 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 2847	. . 3 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ ((𝐽 CnP 𝐾)‘0))
102		cnfldtgp 24912	. . . . . 6 ⊢ ℂ_fld ∈ TopGrp
103		qsubdrg 21460	. . . . . . . 8 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s ℚ) ∈ DivRing)
104	103	simpli 483	. . . . . . 7 ⊢ ℚ ∈ (SubRing‘ℂ_fld)
105		subrgsubg 20605	. . . . . . 7 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) → ℚ ∈ (SubGrp‘ℂ_fld))
106	104, 105	ax-mp 5	. . . . . 6 ⊢ ℚ ∈ (SubGrp‘ℂ_fld)
107	69	subgtgp 24134	. . . . . 6 ⊢ ((ℂ_fld ∈ TopGrp ∧ ℚ ∈ (SubGrp‘ℂ_fld)) → 𝑄 ∈ TopGrp)
108	102, 106, 107	mp2an 691	. . . . 5 ⊢ 𝑄 ∈ TopGrp
109		tgptmd 24108	. . . . 5 ⊢ (𝑄 ∈ TopGrp → 𝑄 ∈ TopMnd)
110	108, 109	mp1i 13	. . . 4 ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → 𝑄 ∈ TopMnd)
111		nrgtrg 24732	. . . . 5 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing)
112		trgtmd2 24198	. . . . 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 33934	. . . . 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 24144	. . . 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 1371	. . 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 1537 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 Vcvv 3488 ∩ cin 3975 class class class wbr 5166 × cxp 5698 ↾ cres 5702 ∘ ccom 5704 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 0cc0 11184 < clt 11324 − cmin 11520 -cneg 11521 ℤcz 12639 ℚcq 13013 ℝ⁺crp 13057 abscabs 15283 Basecbs 17258 ↾_s cress 17287 distcds 17320 TopOpenctopn 17481 0_gc0g 17499 Grpcgrp 18973 -_gcsg 18975 SubGrpcsubg 19160 GrpHom cghm 19252 /_rcdvr 20426 SubRingcsubrg 20595 DivRingcdr 20751 ∞Metcxmet 21372 MetOpencmopn 21377 ℂ_fldccnfld 21387 ℤRHomczrh 21533 ℤModczlm 21534 chrcchr 21535 Cn ccn 23253 CnP ccnp 23254 TopMndctmd 24099 TopGrpctgp 24100 TopRingctrg 24185 ∞MetSpcxms 24348 normcnm 24610 NrmGrpcngp 24611 NrmRingcnrg 24613 NrmModcnlm 24614 ℚHomcqqh 33918

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-dvds 16303 df-gcd 16541 df-numer 16782 df-denom 16783 df-gz 16977 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-plusf 18677 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-ghm 19253 df-cntz 19357 df-od 19570 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-rhm 20498 df-nzr 20539 df-subrng 20572 df-subrg 20597 df-drng 20753 df-abv 20832 df-lmod 20882 df-scaf 20883 df-sra 21195 df-rgmod 21196 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-cnfld 21388 df-zring 21481 df-zrh 21537 df-zlm 21538 df-chr 21539 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cn 23256 df-cnp 23257 df-tx 23591 df-hmeo 23784 df-tmd 24101 df-tgp 24102 df-trg 24189 df-xms 24351 df-ms 24352 df-tms 24353 df-nm 24616 df-ngp 24617 df-nrg 24619 df-nlm 24620 df-qqh 33919

This theorem is referenced by: rrhqima 33960

W3C validator