qqhucn - Mathbox for Thierry Arnoux

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Theorem qqhucn 34243

Description: The ℚHom homomorphism is uniformly continuous. (Contributed by Thierry Arnoux, 28-Jan-2018.)

**Hypotheses**
Ref	Expression
qqhucn.b	⊢ 𝐵 = (Base‘𝑅)
qqhucn.q	⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
qqhucn.u	⊢ 𝑈 = (UnifSt‘𝑄)
qqhucn.v	⊢ 𝑉 = (metUnif‘((dist‘𝑅) ↾ (𝐵 × 𝐵)))
qqhucn.z	⊢ 𝑍 = (ℤMod‘𝑅)
qqhucn.1	⊢ (𝜑 → 𝑅 ∈ NrmRing)
qqhucn.2	⊢ (𝜑 → 𝑅 ∈ DivRing)
qqhucn.3	⊢ (𝜑 → 𝑍 ∈ NrmMod)
qqhucn.4	⊢ (𝜑 → (chr‘𝑅) = 0)

**Assertion**
Ref	Expression
qqhucn	⊢ (𝜑 → (ℚHom‘𝑅) ∈ (𝑈 Cn_u𝑉))

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

Step	Hyp	Ref	Expression
1		qqhucn.2	. . . 4 ⊢ (𝜑 → 𝑅 ∈ DivRing)
2		qqhucn.4	. . . 4 ⊢ (𝜑 → (chr‘𝑅) = 0)
3		qqhucn.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
4		eqid 2756	. . . . 5 ⊢ (/_r‘𝑅) = (/_r‘𝑅)
5		eqid 2756	. . . . 5 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
6	3, 4, 5	qqhf 34237	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶𝐵)
7	1, 2, 6	syl2anc 592	. . 3 ⊢ (𝜑 → (ℚHom‘𝑅):ℚ⟶𝐵)
8		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ⁺)
9		qqhucn.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ NrmRing)
10		nrgngp 24695	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
11	9, 10	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ NrmGrp)
12	11	ad2antrr 734	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ NrmGrp)
13	7	ffvelcdmda 7054	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑝) ∈ 𝐵)
14	13	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑝) ∈ 𝐵)
15	7	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ ℚ) → (ℚHom‘𝑅):ℚ⟶𝐵)
16	15	ffvelcdmda 7054	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑞) ∈ 𝐵)
17		eqid 2756	. . . . . . . . . . . . . 14 ⊢ (norm‘𝑅) = (norm‘𝑅)
18		eqid 2756	. . . . . . . . . . . . . 14 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
19		eqid 2756	. . . . . . . . . . . . . 14 ⊢ (dist‘𝑅) = (dist‘𝑅)
20	17, 3, 18, 19	ngpdsr 24638	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ NrmGrp ∧ ((ℚHom‘𝑅)‘𝑝) ∈ 𝐵 ∧ ((ℚHom‘𝑅)‘𝑞) ∈ 𝐵) → (((ℚHom‘𝑅)‘𝑝)(dist‘𝑅)((ℚHom‘𝑅)‘𝑞)) = ((norm‘𝑅)‘(((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝))))
21	12, 14, 16, 20	syl3anc 1386	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘𝑝)(dist‘𝑅)((ℚHom‘𝑅)‘𝑞)) = ((norm‘𝑅)‘(((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝))))
22		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑞 ∈ ℚ)
23		simplr 776	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑝 ∈ ℚ)
24		qsubdrg 21444	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s ℚ) ∈ DivRing)
25	24	simpli 486	. . . . . . . . . . . . . . . . . 18 ⊢ ℚ ∈ (SubRing‘ℂ_fld)
26		subrgsubg 20599	. . . . . . . . . . . . . . . . . 18 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) → ℚ ∈ (SubGrp‘ℂ_fld))
27	25, 26	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ℚ ∈ (SubGrp‘ℂ_fld)
28		cnfldsub 21425	. . . . . . . . . . . . . . . . . 18 ⊢ − = (-_g‘ℂ_fld)
29		qqhucn.q	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
30		eqid 2756	. . . . . . . . . . . . . . . . . 18 ⊢ (-_g‘𝑄) = (-_g‘𝑄)
31	28, 29, 30	subgsub 19156	. . . . . . . . . . . . . . . . 17 ⊢ ((ℚ ∈ (SubGrp‘ℂ_fld) ∧ 𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 − 𝑝) = (𝑞(-_g‘𝑄)𝑝))
32	27, 31	mp3an1 1463	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 − 𝑝) = (𝑞(-_g‘𝑄)𝑝))
33	22, 23, 32	syl2anc 592	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑞 − 𝑝) = (𝑞(-_g‘𝑄)𝑝))
34	33	fveq2d 6860	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘(𝑞 − 𝑝)) = ((ℚHom‘𝑅)‘(𝑞(-_g‘𝑄)𝑝)))
35	3, 4, 5, 29	qqhghm 34239	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
36	1, 2, 35	syl2anc 592	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
37	36	ad2antrr 734	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
38	29	qrngbas 27653	. . . . . . . . . . . . . . . 16 ⊢ ℚ = (Base‘𝑄)
39	38, 30, 18	ghmsub 19240	. . . . . . . . . . . . . . 15 ⊢ (((ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅) ∧ 𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → ((ℚHom‘𝑅)‘(𝑞(-_g‘𝑄)𝑝)) = (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝)))
40	37, 22, 23, 39	syl3anc 1386	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘(𝑞(-_g‘𝑄)𝑝)) = (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝)))
41	34, 40	eqtr2d 2792	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝)) = ((ℚHom‘𝑅)‘(𝑞 − 𝑝)))
42	41	fveq2d 6860	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((norm‘𝑅)‘(((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝))) = ((norm‘𝑅)‘((ℚHom‘𝑅)‘(𝑞 − 𝑝))))
43	9, 1	elind 4147	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ (NrmRing ∩ DivRing))
44	43	ad2antrr 734	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ (NrmRing ∩ DivRing))
45		qqhucn.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ NrmMod)
46	45	ad2antrr 734	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑍 ∈ NrmMod)
47	2	ad2antrr 734	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (chr‘𝑅) = 0)
48		qsubcl 12959	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 − 𝑝) ∈ ℚ)
49	22, 23, 48	syl2anc 592	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑞 − 𝑝) ∈ ℚ)
50		qqhucn.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤMod‘𝑅)
51	17, 50	qqhnm 34241	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑞 − 𝑝) ∈ ℚ) → ((norm‘𝑅)‘((ℚHom‘𝑅)‘(𝑞 − 𝑝))) = (abs‘(𝑞 − 𝑝)))
52	44, 46, 47, 49, 51	syl31anc 1388	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((norm‘𝑅)‘((ℚHom‘𝑅)‘(𝑞 − 𝑝))) = (abs‘(𝑞 − 𝑝)))
53	21, 42, 52	3eqtrd 2795	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘𝑝)(dist‘𝑅)((ℚHom‘𝑅)‘𝑞)) = (abs‘(𝑞 − 𝑝)))
54	14, 16	ovresd 7552	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) = (((ℚHom‘𝑅)‘𝑝)(dist‘𝑅)((ℚHom‘𝑅)‘𝑞)))
55		qsscn 12951	. . . . . . . . . . . . . 14 ⊢ ℚ ⊆ ℂ
56	55, 23	sselid 3929	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑝 ∈ ℂ)
57	55, 22	sselid 3929	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑞 ∈ ℂ)
58		eqid 2756	. . . . . . . . . . . . . 14 ⊢ (abs ∘ − ) = (abs ∘ − )
59	58	cnmetdval 24803	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ) → (𝑝(abs ∘ − )𝑞) = (abs‘(𝑝 − 𝑞)))
60	56, 57, 59	syl2anc 592	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝(abs ∘ − )𝑞) = (abs‘(𝑝 − 𝑞)))
61	23, 22	ovresd 7552	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (𝑝(abs ∘ − )𝑞))
62	57, 56	abssubd 15459	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (abs‘(𝑞 − 𝑝)) = (abs‘(𝑝 − 𝑞)))
63	60, 61, 62	3eqtr4d 2801	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (abs‘(𝑞 − 𝑝)))
64	53, 54, 63	3eqtr4rd 2802	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)))
65	64	breq1d 5104	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 ↔ (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
66	65	biimpd 231	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
67	66	ralrimiva 3148	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ℚ) → ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
68	67	ralrimiva 3148	. . . . . 6 ⊢ (𝜑 → ∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
69	68	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → ∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
70		breq2 5098	. . . . . . . 8 ⊢ (𝑑 = 𝑒 → ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 ↔ (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒))
71	70	imbi1d 343	. . . . . . 7 ⊢ (𝑑 = 𝑒 → (((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒) ↔ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒)))
72	71	2ralbidv 3220	. . . . . 6 ⊢ (𝑑 = 𝑒 → (∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒) ↔ ∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒)))
73	72	rspcev 3576	. . . . 5 ⊢ ((𝑒 ∈ ℝ⁺ ∧ ∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒)) → ∃𝑑 ∈ ℝ⁺ ∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
74	8, 69, 73	syl2anc 592	. . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → ∃𝑑 ∈ ℝ⁺ ∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
75	74	ralrimiva 3148	. . 3 ⊢ (𝜑 → ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))
76		eqid 2756	. . . 4 ⊢ (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))) = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ)))
77		qqhucn.v	. . . 4 ⊢ 𝑉 = (metUnif‘((dist‘𝑅) ↾ (𝐵 × 𝐵)))
78		0z 12569	. . . . . 6 ⊢ 0 ∈ ℤ
79		zq 12945	. . . . . 6 ⊢ (0 ∈ ℤ → 0 ∈ ℚ)
80		ne0i 4288	. . . . . 6 ⊢ (0 ∈ ℚ → ℚ ≠ ∅)
81	78, 79, 80	mp2b 10	. . . . 5 ⊢ ℚ ≠ ∅
82	81	a1i 11	. . . 4 ⊢ (𝜑 → ℚ ≠ ∅)
83		drngring 20758	. . . . 5 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
84		eqid 2756	. . . . . 6 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
85	3, 84	ringidcl 20287	. . . . 5 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ 𝐵)
86		ne0i 4288	. . . . 5 ⊢ ((1_r‘𝑅) ∈ 𝐵 → 𝐵 ≠ ∅)
87	1, 83, 85, 86	4syl 19	. . . 4 ⊢ (𝜑 → 𝐵 ≠ ∅)
88		cnfldxms 24809	. . . . . . . 8 ⊢ ℂ_fld ∈ ∞MetSp
89		qex 12952	. . . . . . . 8 ⊢ ℚ ∈ V
90		ressxms 24558	. . . . . . . 8 ⊢ ((ℂ_fld ∈ ∞MetSp ∧ ℚ ∈ V) → (ℂ_fld ↾_s ℚ) ∈ ∞MetSp)
91	88, 89, 90	mp2an 700	. . . . . . 7 ⊢ (ℂ_fld ↾_s ℚ) ∈ ∞MetSp
92	29, 91	eqeltri 2852	. . . . . 6 ⊢ 𝑄 ∈ ∞MetSp
93		cnfldds 21409	. . . . . . . . 9 ⊢ (abs ∘ − ) = (dist‘ℂ_fld)
94	29, 93	ressds 17415	. . . . . . . 8 ⊢ (ℚ ∈ V → (abs ∘ − ) = (dist‘𝑄))
95	89, 94	ax-mp 5	. . . . . . 7 ⊢ (abs ∘ − ) = (dist‘𝑄)
96	38, 95	xmsxmet2 24492	. . . . . 6 ⊢ (𝑄 ∈ ∞MetSp → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (∞Met‘ℚ))
97	92, 96	mp1i 13	. . . . 5 ⊢ (𝜑 → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (∞Met‘ℚ))
98		xmetpsmet 24381	. . . . 5 ⊢ (((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (∞Met‘ℚ) → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (PsMet‘ℚ))
99	97, 98	syl 17	. . . 4 ⊢ (𝜑 → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (PsMet‘ℚ))
100		ngpxms 24634	. . . . . 6 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
101	3, 19	xmsxmet2 24492	. . . . . 6 ⊢ (𝑅 ∈ ∞MetSp → ((dist‘𝑅) ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
102	9, 10, 100, 101	4syl 19	. . . . 5 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵))
103		xmetpsmet 24381	. . . . 5 ⊢ (((dist‘𝑅) ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵) → ((dist‘𝑅) ↾ (𝐵 × 𝐵)) ∈ (PsMet‘𝐵))
104	102, 103	syl 17	. . . 4 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝐵 × 𝐵)) ∈ (PsMet‘𝐵))
105	76, 77, 82, 87, 99, 104	metucn 24604	. . 3 ⊢ (𝜑 → ((ℚHom‘𝑅) ∈ ((metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))) Cn_u𝑉) ↔ ((ℚHom‘𝑅):ℚ⟶𝐵 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑑 ∈ ℝ⁺ ∀𝑝 ∈ ℚ ∀𝑞 ∈ ℚ ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 → (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)) < 𝑒))))
106	7, 75, 105	mpbir2and 721	. 2 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ ((metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))) Cn_u𝑉))
107		qqhucn.u	. . . . . 6 ⊢ 𝑈 = (UnifSt‘𝑄)
108	29	fveq2i 6859	. . . . . 6 ⊢ (UnifSt‘𝑄) = (UnifSt‘(ℂ_fld ↾_s ℚ))
109		ressuss 24295	. . . . . . 7 ⊢ (ℚ ∈ V → (UnifSt‘(ℂ_fld ↾_s ℚ)) = ((UnifSt‘ℂ_fld) ↾_t (ℚ × ℚ)))
110	89, 109	ax-mp 5	. . . . . 6 ⊢ (UnifSt‘(ℂ_fld ↾_s ℚ)) = ((UnifSt‘ℂ_fld) ↾_t (ℚ × ℚ))
111	107, 108, 110	3eqtri 2783	. . . . 5 ⊢ 𝑈 = ((UnifSt‘ℂ_fld) ↾_t (ℚ × ℚ))
112		eqid 2756	. . . . . . 7 ⊢ (UnifSt‘ℂ_fld) = (UnifSt‘ℂ_fld)
113	112	cnflduss 25391	. . . . . 6 ⊢ (UnifSt‘ℂ_fld) = (metUnif‘(abs ∘ − ))
114	113	oveq1i 7395	. . . . 5 ⊢ ((UnifSt‘ℂ_fld) ↾_t (ℚ × ℚ)) = ((metUnif‘(abs ∘ − )) ↾_t (ℚ × ℚ))
115		cnxmet 24805	. . . . . . 7 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
116		xmetpsmet 24381	. . . . . . 7 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (abs ∘ − ) ∈ (PsMet‘ℂ))
117	115, 116	ax-mp 5	. . . . . 6 ⊢ (abs ∘ − ) ∈ (PsMet‘ℂ)
118		restmetu 24603	. . . . . 6 ⊢ ((ℚ ≠ ∅ ∧ (abs ∘ − ) ∈ (PsMet‘ℂ) ∧ ℚ ⊆ ℂ) → ((metUnif‘(abs ∘ − )) ↾_t (ℚ × ℚ)) = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))))
119	81, 117, 55, 118	mp3an 1476	. . . . 5 ⊢ ((metUnif‘(abs ∘ − )) ↾_t (ℚ × ℚ)) = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ)))
120	111, 114, 119	3eqtri 2783	. . . 4 ⊢ 𝑈 = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ)))
121	120	a1i 11	. . 3 ⊢ (𝜑 → 𝑈 = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))))
122	121	oveq1d 7400	. 2 ⊢ (𝜑 → (𝑈 Cn_u𝑉) = ((metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))) Cn_u𝑉))
123	106, 122	eleqtrrd 2859	1 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ (𝑈 Cn_u𝑉))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1554 ∈ wcel 2136 ≠ wne 2951 ∀wral 3070 ∃wrex 3080 Vcvv 3448 ∩ cin 3898 ⊆ wss 3899 ∅c0 4280 class class class wbr 5094 × cxp 5638 ↾ cres 5642 ∘ ccom 5644 ⟶wf 6506 ‘cfv 6510 (class class class)co 7385 ℂcc 11061 0cc0 11063 < clt 11206 − cmin 11404 ℤcz 12558 ℚcq 12939 ℝ⁺crp 12983 abscabs 15237 Basecbs 17221 ↾_s cress 17242 distcds 17271 ↾_t crest 17425 -_gcsg 18953 SubGrpcsubg 19138 GrpHom cghm 19229 1_rcur 20203 Ringcrg 20255 /_rcdvr 20421 SubRingcsubrg 20591 DivRingcdr 20751 PsMetcpsmet 21381 ∞Metcxmet 21382 metUnifcmetu 21388 ℂ_fldccnfld 21397 ℤRHomczrh 21524 ℤModczlm 21525 chrcchr 21526 UnifStcuss 24286 Cn_ucucn 24307 ∞MetSpcxms 24350 normcnm 24609 NrmGrpcngp 24610 NrmRingcnrg 24612 NrmModcnlm 24613 ℚHomcqqh 34221

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-rep 5221 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707 ax-cnex 11119 ax-resscn 11120 ax-1cn 11121 ax-icn 11122 ax-addcl 11123 ax-addrcl 11124 ax-mulcl 11125 ax-mulrcl 11126 ax-mulcom 11127 ax-addass 11128 ax-mulass 11129 ax-distr 11130 ax-i2m1 11131 ax-1ne0 11132 ax-1rid 11133 ax-rnegex 11134 ax-rrecex 11135 ax-cnre 11136 ax-pre-lttri 11137 ax-pre-lttrn 11138 ax-pre-ltadd 11139 ax-pre-mulgt0 11140 ax-pre-sup 11141 ax-addf 11142 ax-mulf 11143

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-nel 3056 df-ral 3071 df-rex 3081 df-rmo 3361 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-iun 4945 df-br 5095 df-opab 5157 df-mpt 5176 df-tr 5202 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-riota 7342 df-ov 7388 df-oprab 7389 df-mpo 7390 df-om 7836 df-1st 7959 df-2nd 7960 df-tpos 8194 df-frecs 8250 df-wrecs 8281 df-recs 8330 df-rdg 8369 df-1o 8425 df-er 8666 df-map 8798 df-en 8917 df-dom 8918 df-sdom 8919 df-fin 8920 df-sup 9378 df-inf 9379 df-pnf 11208 df-mnf 11209 df-xr 11210 df-ltxr 11211 df-le 11212 df-sub 11406 df-neg 11407 df-div 11835 df-nn 12201 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12472 df-z 12559 df-dec 12679 df-uz 12830 df-q 12940 df-rp 12984 df-xneg 13104 df-xadd 13105 df-xmul 13106 df-ico 13345 df-fz 13503 df-fzo 13650 df-fl 13792 df-mod 13870 df-seq 14005 df-exp 14065 df-cj 15102 df-re 15103 df-im 15104 df-sqrt 15238 df-abs 15239 df-dvds 16263 df-gcd 16505 df-numer 16746 df-denom 16747 df-gz 16942 df-struct 17159 df-sets 17176 df-slot 17194 df-ndx 17206 df-base 17222 df-ress 17243 df-plusg 17275 df-mulr 17276 df-starv 17277 df-sca 17278 df-vsca 17279 df-ip 17280 df-tset 17281 df-ple 17282 df-ds 17284 df-unif 17285 df-rest 17427 df-topn 17428 df-0g 17446 df-topgen 17448 df-mgm 18650 df-sgrp 18729 df-mnd 18745 df-mhm 18793 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19230 df-od 19544 df-cmn 19798 df-abl 19799 df-mgp 20163 df-rng 20175 df-ur 20204 df-ring 20257 df-cring 20258 df-oppr 20358 df-dvdsr 20378 df-unit 20379 df-invr 20409 df-dvr 20422 df-rhm 20493 df-nzr 20535 df-subrng 20568 df-subrg 20592 df-drng 20753 df-abv 20831 df-lmod 20902 df-psmet 21389 df-xmet 21390 df-met 21391 df-bl 21392 df-mopn 21393 df-fbas 21394 df-fg 21395 df-metu 21396 df-cnfld 21398 df-zring 21472 df-zrh 21528 df-zlm 21529 df-chr 21530 df-top 22927 df-topon 22944 df-topsp 22966 df-bases 22979 df-fil 23879 df-ust 24234 df-uss 24289 df-ucn 24308 df-xms 24353 df-ms 24354 df-nm 24615 df-ngp 24616 df-nrg 24618 df-nlm 24619 df-qqh 34222

This theorem is referenced by: rrhcn 34248

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