qqhucn - Mathbox for Thierry Arnoux

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Theorem qqhucn 31235

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 2823	. . . . 5 ⊢ (/_r‘𝑅) = (/_r‘𝑅)
5		eqid 2823	. . . . 5 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅)
6	3, 4, 5	qqhf 31229	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶𝐵)
7	1, 2, 6	syl2anc 586	. . 3 ⊢ (𝜑 → (ℚHom‘𝑅):ℚ⟶𝐵)
8		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ⁺)
9		qqhucn.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ NrmRing)
10		nrgngp 23273	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
11	9, 10	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ NrmGrp)
12	11	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ NrmGrp)
13	7	ffvelrnda 6853	. . . . . . . . . . . . . 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	ffvelrnda 6853	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑞) ∈ 𝐵)
17		eqid 2823	. . . . . . . . . . . . . 14 ⊢ (norm‘𝑅) = (norm‘𝑅)
18		eqid 2823	. . . . . . . . . . . . . 14 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
19		eqid 2823	. . . . . . . . . . . . . 14 ⊢ (dist‘𝑅) = (dist‘𝑅)
20	17, 3, 18, 19	ngpdsr 23216	. . . . . . . . . . . . 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 1367	. . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑝 ∈ ℚ)
24		qsubdrg 20599	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) ∧ (ℂ_fld ↾_s ℚ) ∈ DivRing)
25	24	simpli 486	. . . . . . . . . . . . . . . . . 18 ⊢ ℚ ∈ (SubRing‘ℂ_fld)
26		subrgsubg 19543	. . . . . . . . . . . . . . . . . 18 ⊢ (ℚ ∈ (SubRing‘ℂ_fld) → ℚ ∈ (SubGrp‘ℂ_fld))
27	25, 26	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ℚ ∈ (SubGrp‘ℂ_fld)
28		cnfldsub 20575	. . . . . . . . . . . . . . . . . 18 ⊢ − = (-_g‘ℂ_fld)
29		qqhucn.q	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
30		eqid 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (-_g‘𝑄) = (-_g‘𝑄)
31	28, 29, 30	subgsub 18293	. . . . . . . . . . . . . . . . 17 ⊢ ((ℚ ∈ (SubGrp‘ℂ_fld) ∧ 𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 − 𝑝) = (𝑞(-_g‘𝑄)𝑝))
32	27, 31	mp3an1 1444	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 − 𝑝) = (𝑞(-_g‘𝑄)𝑝))
33	22, 23, 32	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑞 − 𝑝) = (𝑞(-_g‘𝑄)𝑝))
34	33	fveq2d 6676	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘(𝑞 − 𝑝)) = ((ℚHom‘𝑅)‘(𝑞(-_g‘𝑄)𝑝)))
35	3, 4, 5, 29	qqhghm 31231	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
36	1, 2, 35	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
37	36	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
38	29	qrngbas 26197	. . . . . . . . . . . . . . . 16 ⊢ ℚ = (Base‘𝑄)
39	38, 30, 18	ghmsub 18368	. . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((ℚHom‘𝑅)‘(𝑞(-_g‘𝑄)𝑝)) = (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝)))
41	34, 40	eqtr2d 2859	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘𝑞)(-_g‘𝑅)((ℚHom‘𝑅)‘𝑝)) = ((ℚHom‘𝑅)‘(𝑞 − 𝑝)))
42	41	fveq2d 6676	. . . . . . . . . . . 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 4173	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ (NrmRing ∩ DivRing))
44	43	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑅 ∈ (NrmRing ∩ DivRing))
45		qqhucn.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ NrmMod)
46	45	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑍 ∈ NrmMod)
47	2	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (chr‘𝑅) = 0)
48		qsubcl 12370	. . . . . . . . . . . . . 14 ⊢ ((𝑞 ∈ ℚ ∧ 𝑝 ∈ ℚ) → (𝑞 − 𝑝) ∈ ℚ)
49	22, 23, 48	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑞 − 𝑝) ∈ ℚ)
50		qqhucn.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤMod‘𝑅)
51	17, 50	qqhnm 31233	. . . . . . . . . . . . 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 1369	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → ((norm‘𝑅)‘((ℚHom‘𝑅)‘(𝑞 − 𝑝))) = (abs‘(𝑞 − 𝑝)))
53	21, 42, 52	3eqtrd 2862	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (((ℚHom‘𝑅)‘𝑝)(dist‘𝑅)((ℚHom‘𝑅)‘𝑞)) = (abs‘(𝑞 − 𝑝)))
54	14, 16	ovresd 7317	. . . . . . . . . . 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 12362	. . . . . . . . . . . . . 14 ⊢ ℚ ⊆ ℂ
56	55, 23	sseldi 3967	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑝 ∈ ℂ)
57	55, 22	sseldi 3967	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → 𝑞 ∈ ℂ)
58		eqid 2823	. . . . . . . . . . . . . 14 ⊢ (abs ∘ − ) = (abs ∘ − )
59	58	cnmetdval 23381	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ) → (𝑝(abs ∘ − )𝑞) = (abs‘(𝑝 − 𝑞)))
60	56, 57, 59	syl2anc 586	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝(abs ∘ − )𝑞) = (abs‘(𝑝 − 𝑞)))
61	23, 22	ovresd 7317	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (𝑝(abs ∘ − )𝑞))
62	57, 56	abssubd 14815	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (abs‘(𝑞 − 𝑝)) = (abs‘(𝑝 − 𝑞)))
63	60, 61, 62	3eqtr4d 2868	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (abs‘(𝑞 − 𝑝)))
64	53, 54, 63	3eqtr4rd 2869	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ ℚ) ∧ 𝑞 ∈ ℚ) → (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) = (((ℚHom‘𝑅)‘𝑝)((dist‘𝑅) ↾ (𝐵 × 𝐵))((ℚHom‘𝑅)‘𝑞)))
65	64	breq1d 5078	. . . . . . . . 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 3184	. . . . . . 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 3184	. . . . . 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 5072	. . . . . . . 8 ⊢ (𝑑 = 𝑒 → ((𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑑 ↔ (𝑝((abs ∘ − ) ↾ (ℚ × ℚ))𝑞) < 𝑒))
71	70	imbi1d 344	. . . . . . 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 3201	. . . . . 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 3625	. . . . 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 586	. . . 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 3184	. . 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 2823	. . . 4 ⊢ (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))) = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ)))
77		qqhucn.v	. . . 4 ⊢ 𝑉 = (metUnif‘((dist‘𝑅) ↾ (𝐵 × 𝐵)))
78		0z 11995	. . . . . 6 ⊢ 0 ∈ ℤ
79		zq 12357	. . . . . 6 ⊢ (0 ∈ ℤ → 0 ∈ ℚ)
80		ne0i 4302	. . . . . 6 ⊢ (0 ∈ ℚ → ℚ ≠ ∅)
81	78, 79, 80	mp2b 10	. . . . 5 ⊢ ℚ ≠ ∅
82	81	a1i 11	. . . 4 ⊢ (𝜑 → ℚ ≠ ∅)
83		drngring 19511	. . . . 5 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
84		eqid 2823	. . . . . 6 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
85	3, 84	ringidcl 19320	. . . . 5 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ 𝐵)
86		ne0i 4302	. . . . 5 ⊢ ((1_r‘𝑅) ∈ 𝐵 → 𝐵 ≠ ∅)
87	1, 83, 85, 86	4syl 19	. . . 4 ⊢ (𝜑 → 𝐵 ≠ ∅)
88		cnfldxms 23387	. . . . . . . 8 ⊢ ℂ_fld ∈ ∞MetSp
89		qex 12363	. . . . . . . 8 ⊢ ℚ ∈ V
90		ressxms 23137	. . . . . . . 8 ⊢ ((ℂ_fld ∈ ∞MetSp ∧ ℚ ∈ V) → (ℂ_fld ↾_s ℚ) ∈ ∞MetSp)
91	88, 89, 90	mp2an 690	. . . . . . 7 ⊢ (ℂ_fld ↾_s ℚ) ∈ ∞MetSp
92	29, 91	eqeltri 2911	. . . . . 6 ⊢ 𝑄 ∈ ∞MetSp
93		cnfldds 20557	. . . . . . . . 9 ⊢ (abs ∘ − ) = (dist‘ℂ_fld)
94	29, 93	ressds 16688	. . . . . . . 8 ⊢ (ℚ ∈ V → (abs ∘ − ) = (dist‘𝑄))
95	89, 94	ax-mp 5	. . . . . . 7 ⊢ (abs ∘ − ) = (dist‘𝑄)
96	38, 95	xmsxmet2 23071	. . . . . 6 ⊢ (𝑄 ∈ ∞MetSp → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (∞Met‘ℚ))
97	92, 96	mp1i 13	. . . . 5 ⊢ (𝜑 → ((abs ∘ − ) ↾ (ℚ × ℚ)) ∈ (∞Met‘ℚ))
98		xmetpsmet 22960	. . . . 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 23212	. . . . . 6 ⊢ (𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp)
101	3, 19	xmsxmet2 23071	. . . . . 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 22960	. . . . 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 23183	. . 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 711	. 2 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ ((metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))) Cn_u𝑉))
107		qqhucn.u	. . . . . 6 ⊢ 𝑈 = (UnifSt‘𝑄)
108	29	fveq2i 6675	. . . . . 6 ⊢ (UnifSt‘𝑄) = (UnifSt‘(ℂ_fld ↾_s ℚ))
109		ressuss 22874	. . . . . . 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 2850	. . . . 5 ⊢ 𝑈 = ((UnifSt‘ℂ_fld) ↾_t (ℚ × ℚ))
112		eqid 2823	. . . . . . 7 ⊢ (UnifSt‘ℂ_fld) = (UnifSt‘ℂ_fld)
113	112	cnflduss 23961	. . . . . 6 ⊢ (UnifSt‘ℂ_fld) = (metUnif‘(abs ∘ − ))
114	113	oveq1i 7168	. . . . 5 ⊢ ((UnifSt‘ℂ_fld) ↾_t (ℚ × ℚ)) = ((metUnif‘(abs ∘ − )) ↾_t (ℚ × ℚ))
115		cnxmet 23383	. . . . . . 7 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
116		xmetpsmet 22960	. . . . . . 7 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → (abs ∘ − ) ∈ (PsMet‘ℂ))
117	115, 116	ax-mp 5	. . . . . 6 ⊢ (abs ∘ − ) ∈ (PsMet‘ℂ)
118		restmetu 23182	. . . . . 6 ⊢ ((ℚ ≠ ∅ ∧ (abs ∘ − ) ∈ (PsMet‘ℂ) ∧ ℚ ⊆ ℂ) → ((metUnif‘(abs ∘ − )) ↾_t (ℚ × ℚ)) = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))))
119	81, 117, 55, 118	mp3an 1457	. . . . 5 ⊢ ((metUnif‘(abs ∘ − )) ↾_t (ℚ × ℚ)) = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ)))
120	111, 114, 119	3eqtri 2850	. . . 4 ⊢ 𝑈 = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ)))
121	120	a1i 11	. . 3 ⊢ (𝜑 → 𝑈 = (metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))))
122	121	oveq1d 7173	. 2 ⊢ (𝜑 → (𝑈 Cn_u𝑉) = ((metUnif‘((abs ∘ − ) ↾ (ℚ × ℚ))) Cn_u𝑉))
123	106, 122	eleqtrrd 2918	1 ⊢ (𝜑 → (ℚHom‘𝑅) ∈ (𝑈 Cn_u𝑉))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 Vcvv 3496 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 class class class wbr 5068 × cxp 5555 ↾ cres 5559 ∘ ccom 5561 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 < clt 10677 − cmin 10872 ℤcz 11984 ℚcq 12351 ℝ⁺crp 12392 abscabs 14595 Basecbs 16485 ↾_s cress 16486 distcds 16576 ↾_t crest 16696 -_gcsg 18107 SubGrpcsubg 18275 GrpHom cghm 18357 1_rcur 19253 Ringcrg 19299 /_rcdvr 19434 DivRingcdr 19504 SubRingcsubrg 19533 PsMetcpsmet 20531 ∞Metcxmet 20532 metUnifcmetu 20538 ℂ_fldccnfld 20547 ℤRHomczrh 20649 ℤModczlm 20650 chrcchr 20651 UnifStcuss 22864 Cn_ucucn 22886 ∞MetSpcxms 22929 normcnm 23188 NrmGrpcngp 23189 NrmRingcnrg 23191 NrmModcnlm 23192 ℚHomcqqh 31215

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ico 12747 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-dvds 15610 df-gcd 15846 df-numer 16077 df-denom 16078 df-gz 16268 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-rest 16698 df-topn 16699 df-0g 16717 df-topgen 16719 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-ghm 18358 df-od 18658 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-rnghom 19469 df-drng 19506 df-subrg 19535 df-abv 19590 df-lmod 19638 df-nzr 20033 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-metu 20546 df-cnfld 20548 df-zring 20620 df-zrh 20653 df-zlm 20654 df-chr 20655 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-fil 22456 df-ust 22811 df-uss 22867 df-ucn 22887 df-xms 22932 df-ms 22933 df-nm 23194 df-ngp 23195 df-nrg 23197 df-nlm 23198 df-qqh 31216

This theorem is referenced by: rrhcn 31240

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