qqhucn - Mathbox for Thierry Arnoux

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Theorem qqhucn 33003

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 2733	. . . . 5 ⊢ (/_r‘𝑅) = (/_r‘𝑅)
5		eqid 2733	. . . . 5 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
6	3, 4, 5	qqhf 32997	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶𝐵)
7	1, 2, 6	syl2anc 585	. . 3 ⊢ (𝜑 → (ℚHom‘𝑅):ℚ⟶𝐵)
8		simpr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ⁺)
9		qqhucn.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ NrmRing)
10		nrgngp 24179	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
11	9, 10	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ NrmGrp)
12	11	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ NrmGrp)
13	7	ffvelcdmda 7087	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑝) ∈ 𝐵)
14	13	adantr 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑝) ∈ 𝐵)
15	7	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ ℚ) → (ℚHom‘𝑅):ℚ⟶𝐵)
16	15	ffvelcdmda 7087	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑞) ∈ 𝐵)
17		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (norm‘𝑅) = (norm‘𝑅)
18		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
19		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (dist‘𝑅) = (dist‘𝑅)
20	17, 3, 18, 19	ngpdsr 24114	. . . . . . . . . . . . 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 1372	. . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑞 ∈ ℚ)
23		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑝 ∈ ℚ)
24		qsubdrg 20997	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s ℚ) ∈ DivRing)
25	24	simpli 485	. . . . . . . . . . . . . . . . . 18 ⊢ ℚ ∈ (SubRing‘ℂ_fld)
26		subrgsubg 20325	. . . . . . . . . . . . . . . . . 18 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) → ℚ ∈ (SubGrp‘ℂ_fld))
27	25, 26	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ℚ ∈ (SubGrp‘ℂ_fld)
28		cnfldsub 20973	. . . . . . . . . . . . . . . . . 18 ⊢ − = (-_g‘ℂ_fld)
29		qqhucn.q	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
30		eqid 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (-_g‘𝑄) = (-_g‘𝑄)
31	28, 29, 30	subgsub 19018	. . . . . . . . . . . . . . . . 17 ⊢ ((ℚ ∈ (SubGrp‘ℂ_fld) ∧ 𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 − 𝑝) = (𝑞(-_g‘𝑄)𝑝))
32	27, 31	mp3an1 1449	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 − 𝑝) = (𝑞(-_g‘𝑄)𝑝))
33	22, 23, 32	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑞 − 𝑝) = (𝑞(-_g‘𝑄)𝑝))
34	33	fveq2d 6896	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘(𝑞 − 𝑝)) = ((ℚHom‘𝑅)‘(𝑞(-_g‘𝑄)𝑝)))
35	3, 4, 5, 29	qqhghm 32999	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
36	1, 2, 35	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
37	36	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
38	29	qrngbas 27122	. . . . . . . . . . . . . . . 16 ⊢ ℚ = (Base‘𝑄)
39	38, 30, 18	ghmsub 19100	. . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘(𝑞(-_g‘𝑄)𝑝)) = (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝)))
41	34, 40	eqtr2d 2774	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝)) = ((ℚHom‘𝑅)‘(𝑞 − 𝑝)))
42	41	fveq2d 6896	. . . . . . . . . . . 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 4195	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ (NrmRing ∩ DivRing))
44	43	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ (NrmRing ∩ DivRing))
45		qqhucn.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ NrmMod)
46	45	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑍 ∈ NrmMod)
47	2	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (chr‘𝑅) = 0)
48		qsubcl 12952	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 − 𝑝) ∈ ℚ)
49	22, 23, 48	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑞 − 𝑝) ∈ ℚ)
50		qqhucn.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤMod‘𝑅)
51	17, 50	qqhnm 33001	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((norm‘𝑅)‘((ℚHom‘𝑅)‘(𝑞 − 𝑝))) = (abs‘(𝑞 − 𝑝)))
53	21, 42, 52	3eqtrd 2777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘𝑝)(dist‘𝑅)((ℚHom‘𝑅)‘𝑞)) = (abs‘(𝑞 − 𝑝)))
54	14, 16	ovresd 7574	. . . . . . . . . . 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 12944	. . . . . . . . . . . . . 14 ⊢ ℚ ⊆ ℂ
56	55, 23	sselid 3981	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑝 ∈ ℂ)
57	55, 22	sselid 3981	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑞 ∈ ℂ)
58		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (abs ∘ − ) = (abs ∘ − )
59	58	cnmetdval 24287	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ) → (𝑝(abs ∘ − )𝑞) = (abs‘(𝑝 − 𝑞)))
60	56, 57, 59	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝(abs ∘ − )𝑞) = (abs‘(𝑝 − 𝑞)))
61	23, 22	ovresd 7574	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (𝑝(abs ∘ − )𝑞))
62	57, 56	abssubd 15400	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (abs‘(𝑞 − 𝑝)) = (abs‘(𝑝 − 𝑞)))
63	60, 61, 62	3eqtr4d 2783	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (abs‘(𝑞 − 𝑝)))
64	53, 54, 63	3eqtr4rd 2784	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)))
65	64	breq1d 5159	. . . . . . . . 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 228	. . . . . . . 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 3147	. . . . . . 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 3147	. . . . . 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 482	. . . . 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 5153	. . . . . . . 8 ⊢ (𝑑 = 𝑒 → ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 ↔ (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒))
71	70	imbi1d 342	. . . . . . 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 3219	. . . . . 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 3613	. . . . 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 585	. . . 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 3147	. . 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 2733	. . . 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 12938	. . . . . 6 ⊢ (0 ∈ ℤ → 0 ∈ ℚ)
80		ne0i 4335	. . . . . 6 ⊢ (0 ∈ ℚ → ℚ ≠ ∅)
81	78, 79, 80	mp2b 10	. . . . 5 ⊢ ℚ ≠ ∅
82	81	a1i 11	. . . 4 ⊢ (𝜑 → ℚ ≠ ∅)
83		drngring 20364	. . . . 5 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
84		eqid 2733	. . . . . 6 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
85	3, 84	ringidcl 20083	. . . . 5 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ 𝐵)
86		ne0i 4335	. . . . 5 ⊢ ((1_r‘𝑅) ∈ 𝐵 → 𝐵 ≠ ∅)
87	1, 83, 85, 86	4syl 19	. . . 4 ⊢ (𝜑 → 𝐵 ≠ ∅)
88		cnfldxms 24293	. . . . . . . 8 ⊢ ℂ_fld ∈ ∞MetSp
89		qex 12945	. . . . . . . 8 ⊢ ℚ ∈ V
90		ressxms 24034	. . . . . . . 8 ⊢ ((ℂ_fld ∈ ∞MetSp ∧ ℚ ∈ V) → (ℂ_fld ↾_s ℚ) ∈ ∞MetSp)
91	88, 89, 90	mp2an 691	. . . . . . 7 ⊢ (ℂ_fld ↾_s ℚ) ∈ ∞MetSp
92	29, 91	eqeltri 2830	. . . . . 6 ⊢ 𝑄 ∈ ∞MetSp
93		cnfldds 20954	. . . . . . . . 9 ⊢ (abs ∘ − ) = (dist‘ℂ_fld)
94	29, 93	ressds 17355	. . . . . . . 8 ⊢ (ℚ ∈ V → (abs ∘ − ) = (dist‘𝑄))
95	89, 94	ax-mp 5	. . . . . . 7 ⊢ (abs ∘ − ) = (dist‘𝑄)
96	38, 95	xmsxmet2 23965	. . . . . 6 ⊢ (𝑄 ∈ ∞MetSp → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (∞Met‘ℚ))
97	92, 96	mp1i 13	. . . . 5 ⊢ (𝜑 → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (∞Met‘ℚ))
98		xmetpsmet 23854	. . . . 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 24110	. . . . . 6 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
101	3, 19	xmsxmet2 23965	. . . . . 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 23854	. . . . 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 24080	. . 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 712	. 2 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ ((metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))) Cn_u𝑉))
107		qqhucn.u	. . . . . 6 ⊢ 𝑈 = (UnifSt‘𝑄)
108	29	fveq2i 6895	. . . . . 6 ⊢ (UnifSt‘𝑄) = (UnifSt‘(ℂ_fld ↾_s ℚ))
109		ressuss 23767	. . . . . . 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 2765	. . . . 5 ⊢ 𝑈 = ((UnifSt‘ℂ_fld) ↾_t (ℚ × ℚ))
112		eqid 2733	. . . . . . 7 ⊢ (UnifSt‘ℂ_fld) = (UnifSt‘ℂ_fld)
113	112	cnflduss 24873	. . . . . 6 ⊢ (UnifSt‘ℂ_fld) = (metUnif‘(abs ∘ − ))
114	113	oveq1i 7419	. . . . 5 ⊢ ((UnifSt‘ℂ_fld) ↾_t (ℚ × ℚ)) = ((metUnif‘(abs ∘ − )) ↾_t (ℚ × ℚ))
115		cnxmet 24289	. . . . . . 7 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
116		xmetpsmet 23854	. . . . . . 7 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (abs ∘ − ) ∈ (PsMet‘ℂ))
117	115, 116	ax-mp 5	. . . . . 6 ⊢ (abs ∘ − ) ∈ (PsMet‘ℂ)
118		restmetu 24079	. . . . . 6 ⊢ ((ℚ ≠ ∅ ∧ (abs ∘ − ) ∈ (PsMet‘ℂ) ∧ ℚ ⊆ ℂ) → ((metUnif‘(abs ∘ − )) ↾_t (ℚ × ℚ)) = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))))
119	81, 117, 55, 118	mp3an 1462	. . . . 5 ⊢ ((metUnif‘(abs ∘ − )) ↾_t (ℚ × ℚ)) = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ)))
120	111, 114, 119	3eqtri 2765	. . . 4 ⊢ 𝑈 = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ)))
121	120	a1i 11	. . 3 ⊢ (𝜑 → 𝑈 = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))))
122	121	oveq1d 7424	. 2 ⊢ (𝜑 → (𝑈 Cn_u𝑉) = ((metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))) Cn_u𝑉))
123	106, 122	eleqtrrd 2837	1 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ (𝑈 Cn_u𝑉))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 class class class wbr 5149 × cxp 5675 ↾ cres 5679 ∘ ccom 5681 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 0cc0 11110 < clt 11248 − cmin 11444 ℤcz 12558 ℚcq 12932 ℝ⁺crp 12974 abscabs 15181 Basecbs 17144 ↾_s cress 17173 distcds 17206 ↾_t crest 17366 -_gcsg 18821 SubGrpcsubg 19000 GrpHom cghm 19089 1_rcur 20004 Ringcrg 20056 /_rcdvr 20214 SubRingcsubrg 20315 DivRingcdr 20357 PsMetcpsmet 20928 ∞Metcxmet 20929 metUnifcmetu 20935 ℂ_fldccnfld 20944 ℤRHomczrh 21049 ℤModczlm 21050 chrcchr 21051 UnifStcuss 23758 Cn_ucucn 23780 ∞MetSpcxms 23823 normcnm 24085 NrmGrpcngp 24086 NrmRingcnrg 24088 NrmModcnlm 24089 ℚHomcqqh 32983

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ico 13330 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-dvds 16198 df-gcd 16436 df-numer 16671 df-denom 16672 df-gz 16863 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-rest 17368 df-topn 17369 df-0g 17387 df-topgen 17389 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mulg 18951 df-subg 19003 df-ghm 19090 df-od 19396 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-oppr 20150 df-dvdsr 20171 df-unit 20172 df-invr 20202 df-dvr 20215 df-rnghom 20251 df-nzr 20292 df-subrg 20317 df-drng 20359 df-abv 20425 df-lmod 20473 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-metu 20943 df-cnfld 20945 df-zring 21018 df-zrh 21053 df-zlm 21054 df-chr 21055 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-fil 23350 df-ust 23705 df-uss 23761 df-ucn 23781 df-xms 23826 df-ms 23827 df-nm 24091 df-ngp 24092 df-nrg 24094 df-nlm 24095 df-qqh 32984

This theorem is referenced by: rrhcn 33008

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